A THOULESS-LIKE EFFECT IN THE DYSON HIERARCHICAL MODEL WITH CONTINUOUS

SYMMETRY

PAVEL BLEHER AND PÉTER MAJOR

Abstract. In this paper we study Dyson’s classical r-component hierarchical model with a Hamiltonian function which has a continuous O(r)-symmetry, r ≥ 2. This is a one-dimensional ferromagnetic model with a long range interaction potential U(i, j) = −l(d(i, j))d^{−2}(i, j), where d(i, j) denotes the hierarchical distance. We are interested in the case when l_{n} = l(2^{n}), n = 1, 2, …, is an increasing sequence, with a subexponential growth as n → ∞. For a class of free measures, we prove a conjecture of Dyson. This conjecture states that the convergence of the series l_{1}^{−1} + l_{2}^{−1} + … is a necessary and sufficient condition of the existence of phase transition in the model under consideration, and the spontaneous magnetization vanishes at the critical point, i.e., there is no Thouless’ effect. We find, however, that the distribution of the normalized mean spin at the critical temperature T_{c} tends to the uniform distribution on the unit sphere in ℝ^{r} as the volume tends to infinity, a phenomenon which resembles the Thouless effect. We prove that the limit distribution of the normalized mean spin is Gaussian for T > T_{c}, and it is non-Gaussian for T ≤ T_{c}. We also show that the density of the limit distribution of the normalized mean spin for T ≤ T_{c} is a nice analytic function which can be found from the unique solution of a nonlinear fixed point integral equation. Finally, we determine some critical asymptotics and show that the divergence of the correlation length and magnetic susceptibility is super-polynomial as T → T_{c}.

Contents

1. Introduction. Formulation of the Main Results.
2. Analytic Reformulation of the Problem. Strategy of the Proof.
3. Formulation of Auxiliary Theorems.
4. Basic Estimates in the Low Temperature Region.
5. Estimates in the Intermediate Region. The proof of Theorem 3.1.
6. Estimates in the High Temperature Region. The proof of Theorem 3.3.
7. Estimates in the Low Temperature Region. The proof of Theorem 3.2.
8. Estimates Near the Critical Point. The proof of Theorems 3.4, 1.3 and 1.5.

Appendices A and B References

Key words and phrases. Dyson’s hierarchical model, continuous symmetry, Thouless’ effect, renormalization transformation, limit distribution of the average spin, superpolynomial critical asymptotics.

1. Introduction. Formulation of the Main Results.

In this paper we investigate Dyson’s hierarchical vector-valued model with continuous symmetry. The model consists of spin variables σ(j) ∈ ℝ^{r}, j ∈ ℕ = {1, 2, … }, where r ≥ 2. We define the hierarchical distance d(·, ·) on ℕ as

d(j, k) = 2^{n(j,k)−1} for j ≠ k

with

n(j, k) = {min n: there is an integer l such that (l − 1)2^{n} < j, k ≤ l2^{n}}
if j ≠ k,

and d(j, j) = 0. The Hamiltonian of the ferromagnetic Dyson’s hierarchical r-component model in the volume V_{n} = {1, 2, …, 2^{n}} is

H_{n}(σ) = − ∑_{1≤j<k≤2^{n}} \frac{l(d(j, k))}{d^{2}(j, k)} σ(j)σ(k),  (1.1)

where σ(j)σ(k) denotes a scalar product in ℝ^{r}, and l(t) is a positive function. In this paper we will be interested in the case when l(t) is a positive increasing function such that

lim_{t→∞} l(t) = ∞;  lim_{t→∞} \frac{l(t)}{t^{ε}} = 0, for all ε > 0.

Since the hierarchical distance d(j, k) for j ≠ k takes the values 2^{n}, n = 0, 1, 2, …, only, we consider the function l(t) for t = 2^{n} only and define

l_{n} = l(2^{n}).

Let ν(dx) be a probability measure on ℝ^{r}. Then the Gibbs measure in V_{n} at a temperature T > 0 with free boundary conditions and the free measure ν(dx) is defined as

μ_{n}(dx; T) = Z_{n}^{−1}(T) exp {−βH_{n}(x)} ∏_{j=1}^{2^{n}} ν(dx_{j}), β = T^{−1}.

We will assume that the free measure ν(dx) is invariant with respect to the group O(r) of orthogonal transformations, i.e., ν(UA) = ν(A) for all U ∈ O(r) and all Borel sets A ∈ B(ℝ^{r}). Then the Gibbs measure μ_{n}(dx; T) is O(r)-invariant as well,

μ_{n}(UA_{1}, …, UA_{2^{n}}; T) = μ_{n}(A_{1}, …, A_{2^{n}}; T), for all U ∈ O(r),
A_{j} ∈ B(ℝ^{r}), j = 1, …, 2^{n}.

In [Dys2], Dyson proved the following theorem (see also [Dys3]). Assume that r = 3, and ν(dx) is a uniform measure on the unit sphere in ℝ^{3}. This is the classical Heisenberg hierarchical model.

Theorem 1.1. (see [Dys2]). The classical Heisenberg hierarchical model has a phase transition if

B = ∑_{n=1}^{∞} l_{n}^{−1} < ∞.  (1.2)

It has a long-range order so long as β > B.

Dyson also formulated the following conjecture (see [Dys2]): “It also seems likely that for sequences l_{n} which are positive and increasing with n the condition (1.2) is necessary for a phase transition in Heisenberg hierarchical models.” The goal of this paper is to prove Dyson’s conjecture for a class of hierarchical models and to study the limit distribution of the normalized mean spin both below and above the critical temperature if condition (1.2) holds. Dyson’s proof is a clever application of correlation inequalities. Our approach is based on an analytical study of the renormalization group transformation for the hierarchical models.

The renormalization group (RG) approach to the Dyson hierarchical models was initiated in the works of Bleher and Sinai [BS1]–[BS3] (see also the monograph [Sin] and the review [Ble], and references therein). The Dyson hierarchical models are of a great interest because for this model the RG transformation reduces to a nonlinear integral equation, and this allows a study of critical phenomena unavailable in other models. The works of Bleher and Sinai were concerned with the critical phenomena and phase transitions in the scalar Dyson hierarchical models. They were extended to the study of critical phenomena and phase transitions in the vector Dyson hierarchical models with continuous symmetry in the works of Bleher and Major [BM1]–[BM5]. The present paper is a continuation of the works [BM1]–[BM5].

We apply a perturbation technique which works if the free measure ν(dx) is a small perturbation of the Gaussian measure. Hence, we cannot treat the case when ν(dx) is a uniform measure on the unit sphere. On the other hand, we will consider arbitrary spin dimension r ≥ 2. We will focus on free measures ν(dx), which have a density function p(x) on ℝ^{r} such that p(x) is close, in an appropriate sense, to the density function

p_{0}(x) = C(κ) exp \{−\frac{|x|^{2}}{2} − κ \frac{|x|^{4}}{4} \}  (1.3)

with a sufficiently small parameter κ > 0. Precise conditions on p(x) are given below. We also will assume some regularity conditions about the sequence l_{n} = l(2^{n}) (see below).

We are investigating the following question. Let p_{n}(x, T) denote the den-

sity function of the mean spin 2^{−n} ∑_{j=1}^{2^{n}} σ(j), where (σ(1), …, σ(2^{n})) is a

μ_{n}(T)-distributed random vector. Because of the rotational invariance of the model, the function p_{n}(x, T) is a function of |x|. We are interested in the limit behaviour of the function p_{n}(x, T) as n → ∞, with an appropriate normalization. In our papers [BM1]–[BM5] this problem was considered for the Hamiltonian

H_{n}(σ) = − ∑_{1≤j<k≤2^{n}} \frac{1}{d^{α}(j, k)} σ(j)σ(k),

where 1 < α < 2. Observe that if α ≤ 1 then the thermodynamic limit of the model does not exist, and if α ≥ 2 then there is no phase transition, hence the range 1 < α < 2 is natural. We distinguished in [BM1]–[BM5] the three cases for α:

(i) 1 < α < 3/2, (ii) α = 3/2, and (iii) 3/2 < α < 2.

The difference between these cases appears in the asymptotic behavior of p_{n}(x, T) at small T. When T is small the spontaneous magnetization M(T) is positive, and the function p_{n}(x, T) is concentrated in a narrow spherical shell near the sphere |x| = M(T). The question is what the width of this shell is and what the limiting shape of p_{n}(x, T) is like along the radius after an appropriate rescaling. In case (i), the width is of the order of 2^{−n/2} and the limit shape of p_{n}(x, T) is Gaussian (see [BM1]). In case (ii), there is a logarithmic correction in the asymptotics of the width, but the limit shape is still Gaussian (see [BM4]). In case (iii), the width of the shell has a nonstandard asymptotics of the order of 2^{−n(2−α)}, and the limit shape of p_{n}(x, T) along the radius (after a rescaling) is a non-Gaussian function which is a solution of a nonlinear integral equation (see [BM3] and the review [BM2]). In the present paper we are interested in the marginal potential l(d(j, k))/d^{2}(j, k), with an extra factor l(t) of a sub-polynomial growth.

Before formulating the main results we would like to discuss the importance of Dyson’s condition (1.2). In the case of the Ising hierarchical model (r = 1), Dyson proved in [Dys2] that there exists a “weakest” interaction function l(t) for which the hierarchical model (1.1) has a phase transition. This function is l(t) = log log t, which corresponds to l_{n} = log n. Dyson has proved that if

lim_{n→∞} \frac{l_{n}}{log n} = 0,

then the spontaneous magnetization is equal to zero for all temperatures T > 0. On the other hand, if

\frac{l_{n}}{log n} > ε for all n > 0 with some ε > 0,

then the spontaneous magnetization is positive at sufficiently low temperatures T > 0. In the borderline model, when

l_{n} = J log n, J > 0,

Dyson proved that the spontaneous magnetization M(T) has a jump at the critical temperature T_{c}. The existence of the jump for the 1D Ising model with long-range interaction was first predicted by Thouless (see [Tho], and also the works [YA] of Anderson, Yuval and [Ham] of Hamann and references therein) for the translationally invariant Ising model with the interaction

H(σ) = − ∑_{j,k} \frac{σ(j)σ(k)}{(j − k)^{2}}.  (1.4)

This phenomenon (the jump of M(T) at T = T_{c}) is called the Thouless effect. The existence of a phase transition in the ferromagnetic one-dimensional Ising model with 1/(j − k)^{2} interaction energy was proved by Fröhlich and Spencer in [FS]. A rigorous proof of the existence of the Thouless effect in the Ising model with the inverse square interaction (1.4) was given by Aizenman, J. Chayes, L. Chayes, and Newman [ACCN]. Simon proved in [Sim] the absence of continuous symmetry breaking in the one-dimensional r-component Heisenberg model with the interaction (1.4), in the case when r ≥ 2.

Dyson formulated a general heuristic principle in [Dys2] which tells us when one should expect the Thouless effect in a 1D long-range ferromagnetic model: It should occur for the “weakest” interaction (if it exists) for which a phase transition appears. Dyson wrote that in the hierarchical model “in the Ising case, there exists a borderline model l_{n} = log n which is the ‘weakest’ ferromagnet for which a transition occurs, and this borderline model shows a Thouless effect. In the Heisenberg case there exists no borderline model, since there is no ‘most slowly converging’ series (1.2). Thus we do not expect to find a Thouless effect in any one-dimensional Heisenberg hierarchical ferromagnet.” This conjecture of Dyson, about the absence of a Thouless effect in the Heisenberg case, plays a very essential role in our investigation. We show that in the class of the r-component hierarchical models under consideration, the spontaneous magnetization M(T) approaches zero as T approaches the critical temperature, i.e., there is no Thouless effect. On the other hand, we observe a phenomenon which resembles the Thouless effect: at T = T_{c} the rescaled distribution

M̄_{n}^{r}(T_{c})p_{n}(M_{n}(T_{c})x, T_{c}) dx,  M̄_{n}(T) = (∫_{ℝ^{r}} |x|^{2}p_{n}(x, T) dx)^{1/2} ,

approaches, as n → ∞, a uniform measure on the unit sphere in ℝ^{r}, r ≥ 2. Thus, although the spontaneous magnetization M(T_{c}) = lim_{n→∞} M̄_{n}(T_{c}) is equal to zero at the critical point, the distribution of the normalized mean spin converges to a uniform measure on the unit sphere. This is a “remnant” of the spontaneous magnetization at the critical temperature T_{c}.

To formulate our results we will need some conditions on the sequence l_{n} = l(2^{n}). We need different conditions on l_{n} in different theorems. We formulate the conditions we shall later apply.

Conditions on the sequence l_{n}, n = 0, 1, 2, . . .. Let us introduce the notation

c_{n} = \frac{l_{n}}{l_{n−1}} , n = 0, 1, …, with l_{−1} = 1.

Condition 1.

l_{0} = 1; 1 ≤ c_{n} ≤ 1.01 for all n;  lim_{n→∞} c_{n} = 1.  (1.5)

Remark. The requirement l_{0} = 1 is not a real condition, it can be reached by a rescaling of the temperature. We use it just for a normalization.

Condition 2.

lim_{n→∞} l_{n} ∑_{j=n}^{∞} l_{j}^{−1} = ∞.

Moreover, the above condition is uniform in the following sense: For all ε > 0 there are some numbers K(ε) > 0 and L(ε) > 0 such that

l_{n} ∑_{j=n}^{n+K(ε)} l_{j}^{−1} ≥ ε^{−1}

for all n > L(ε).

Condition 3.

sup_{1<n<∞} ∑_{k=1}^{n} (l_{k} ∑_{j=k}^{n} l_{j}^{−1})^{−2} < ∞.

Condition 4.

∑_{n=1}^{∞} l_{j}^{−1} > 400 κ^{−1}.

Condition 5.

\frac{l_{n}}{l_{n+k}} > η̄ for all n = 0, 1, 2, …, and all k = 1, …, L.

The numbers κ, η̄ > 0, and L ∈ ℕ in these conditions will be chosen later. An example of sequences l_{n} satisfying Conditions 1–5 is given in the following proposition.

Proposition 1.2. The sequence

l_{n} = (1 + an)^{λ},  a > 0, λ > 1,  (1.6)

satisfies Conditions 2 and 3 for all a > 0 and λ > 1. There exists a number a_{0} = a_{0}(λ) > 0 such that this sequence satisfies Condition 1 for all 0 < a < a_{0}, a number a_{1} = a_{1}(κ, λ) > 0 such that this sequence satisfies Condition 4 for all 0 < a < a_{1}, and finally there exists a number a_{2} = a_{2}(η̄, L) > 0 such that this sequence satisfies Condition 5 for all 0 < a < a_{2}.

Thus, for all λ > 1 there exists a number

a_{3} = a_{3}(λ, κ, η̄, L) = min{a_{0}(λ), a_{1}(κ, λ), a_{2}(η̄, L)} > 0

such that for all 0 < a < a_{3}, the sequence (1.6) satisfy Conditions 1—5. We prove Proposition 1.2 in Appendix B below. Now we describe the class of initial densities we shall consider.

Class of initial densities. We say that a probability density p(x) on ℝ^{r} belongs to the class P_{κ} if

p(x) = C(1 + ε(|x|^{2})) exp (−\frac{|x|^{2}}{2} − κ \frac{|x|^{4}}{4} ) ,  (1.7)

where C > 0 is a norming factor, and

∥ε(t)∥_{C^{4}(ℝ^{1})} < 0.01.  (1.8)

Now we formulate our main results. We denote by p_{n}(x, T) the distribution of the mean spin 2^{−n}[σ(1) + ⋯ + σ(2^{n})] with respect to the Gibbs measure μ_{n}(dx; T) and put

M̄_{n}(T) = (∫_{ℝ^{r}} |x|^{2}p_{n}(x, T) dx)^{1/2} .  (1.9)

By p̃_{n}(x, T) we denote the rescaled density function

p̃_{n}(x, T) = M̄_{n}^{r}(T)p_{n}(M̄_{n}(T)x, T)  (1.10)

and by ν̃_{n,T} (dx) the corresponding probability distribution

ν̃_{n,T} (dx) = p̃_{n}(x, T) dx.  (1.11)

Formulation of the main results. We fix a sufficiently small positive number η which will be the same through the whole paper. For instance, η = 10^{−100} is a good choice. Define the following number N = N(η):

N = min{n: l_{n} > η^{−1}}.  (1.12)

Assume that an arbitrary number η̄ in the interval 0 < η̄ ≤ η is fixed. (The number η̄ appears in Condition 5).

Theorem 1.3. (Necessity of Dyson’s condition). Let us consider the case when

∑_{n=1}^{∞} l_{n}^{−1} = ∞.

Then there exists a number κ_{0} = κ_{0}(N) such that for all 0 < κ < κ_{0} the following statements hold.

Assume that the density p(x) = \frac{ν(dx)}{dx} belongs to the class P_{κ} and the sequence {l_{n}, n ≥ 0} satisfies Conditions 1–3. Then there exists a constant L = L(η̄, κ) such that if the sequence {l_{n}, n ≥ 0} satisfies Condition 5, then for all T > 0, there exists the limit

lim_{n→∞} 2^{n}M̄_{n}^{2}(T) = χ(T) > 0.  (1.13)

In particular, the spontaneous magnetization satisfies the relation

M(T) = lim_{n→∞} M̄_{n}(T) = 0.

In addition, the distribution ν̃_{n,T} (dx) tends weakly to the r-dimensional standard normal distribution as n → ∞.

To formulate our results for the case when the Dyson condition (1.2) holds, we define a function p̄_{n}(t, T) by the formula

p_{n}(x, T) = C_{n}(T)^{−1}p̄_{n}(|x|, T),  (1.14)

for t = |x| > 0 and p̄_{n}(t, T) = 0 for t < 0. The norming constant C_{n}(T) is chosen in such a way that p̄_{n}(t, T) is a probability density function, i.e.

∫_{0}^{∞} p̄_{n}(t, T) dt = 1.

We will call p̄_{n}(t, T) the probability density of the mean spin distribution along the radius.

In Parts 2 and 3 we will describe the limit behaviour of an appropriate rescaling of the probability density p̄_{n}(t, T) for T = T_{c} and T < T_{c}. Then we will formulate a Corollary which gives a good asymptotics for the norming constants C_{n}(T) in (1.14). In such a way we get a good asymptotics for the probability density functions p_{n}(x, T) for T ≤ T_{c}. To do this we introduce the notations

\begin{matrix}M̂_{n}(T) = ∫_{−∞}^{∞} tp̄_{n}(t, T) dt, \\ V_{n}(T) = (∫_{−∞}^{∞}(t − M̂_{n}(T))^{2}p̄_{n}(t, T) dt)^{1/2} , \end{matrix}  (1.15)

and the rescaled probability density

π_{n}(t, T) = V_{n}(T)p̄_{n} (M̂_{n}(T) + V_{n}(T) t, T )  (1.16)

which can be rewritte in an equivalent form as

p̄_{n}(t, T) = \frac{1}{V_{n}(T)}π_{n} (\frac{t − M̂_{n}(T)}{V_{n}(T)}, T ) .  (1.17)

Observe that, in general, M̂_{n}(T) and M̄_{n}(T), which is defined in (1.9), are different, but as we will see later,

lim_{n→∞}[M̂_{n}(T) − M̄_{n}(T)] = 0.

Our aim is to prove that in the case when the Dyson condition (1.2) holds, there exists a critical temperature T_{c} such that the spontaneous magnetization M(T) = lim_{n→∞} M̂_{n}(T) is positive for T < T_{c} and it is zero for T ≥ T_{c}. For T < T_{c} the density function p̄_{n}(t, T) is concentrated near the point t = M̂_{n}(T), and the function π_{n}(t, T) represents a rescaled distribution of p̄_{n}(t, T) near this point. We want to prove that π_{n}(t, T) tends to a limit π(t) as n → ∞. It turns out that this limit does exist, and the limit function π(t) is a nice analytic function, although it is non-Gaussian. The function π(t) is expressed in terms of a solution of a nonlinear fixed point equation, and the next proposition concerns the existence of such a solution. Introduce the space of probability densities p(t) on the line

A = {p(t): ∫_{−∞}^{∞} e^{ε|t|}p(t) dt < ∞ for some ε = ε(p(t)) > 0} .

Consider also the subspace A_{0} ⊂ A,

A_{0} = {p(t): p(t) ∈ A, ∫_{−∞}^{∞} tp(t) dt = 0} .

Proposition 1.4. There exists a unique probability density function g ∈ A_{0} which satisfies the following fixed point equation:

\begin{aligned}g(t) = \frac{2}{π^{\frac{r−1}{2}}} & ∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} e^{−|v|^{2}}g (t − \frac{r − 1}{4} − u + \frac{|v|^{2}}{2} ) \\ & × g (t − \frac{r − 1}{4} + u + \frac{|v|^{2}}{2} ) du dv. \end{aligned}  (1.18)

The density g(t) can be extended to an entire function on the complex plane, and for real t it satisfies the estimate

0 < g(t) < C_{ε} exp{−(2 − ε)|t|}, for all ε > 0.  (1.19)

For a proof of Proposition 1.4 see the proof of Lemmas 12 and 13 in [BM3]. It is worth noticing that the Fourier transform of g,

g̃(ξ) = ∫_{−∞}^{∞} e^{iξt}g(t) dt,

solves the equation

g̃(ξ) = \frac{e^{\frac{iξ(r−1)}{4}} g̃^{2}(\frac{ξ}{2})}{(1 + \frac{iξ}{2} )^{\frac{r−1}{2}}}.  (1.20)

Using the probability density g(t) of Proposition 1.4, we introduce a probability density π(t) on the line of the form

π(t) = ce^{−2bt/3}g(bt − a),  (1.21)

where the numbers b > 0, c > 0, and a are chosen in such a way that

∫_{−∞}^{∞} π(t) dt = 1, ∫_{−∞}^{∞} t π(t) dt = 0, ∫_{−∞}^{∞} t^{2}π(t) dt = 1.  (1.22)

Observe that such a, b, c exist and are unique. Indeed, after the change of variable u = bt − a, the second equation in (1.22) gives a as

a = −\frac{∫_{−∞}^{∞} ue^{−2u/3}g(u) du}{∫_{−∞}^{∞} e^{−2u/3}g(u) du}.  (1.23)

Then the first and third equations determine b and c uniquely from the system,

\{ \begin{matrix}\frac{c}{b} ∫_{−∞}^{∞} e^{−2(u+a)/3}g(u) du = 1, \\ \frac{c}{b^{3}} ∫_{−∞}^{∞}(u + a)^{2}e^{−2(u+a)/3}g(u) du = 1. \end{matrix}  (1.24)

Estimate (1.19) secures the convergence of the integrals in (1.23) and (1.24). Now we formulate

Theorem 1.5. Assume that

∑_{n=1}^{∞} l_{n}^{−1} < ∞.  (1.25)

Then there exists a number κ_{0} = κ_{0}(N), where N is defined in (1.12), such that for all 0 < κ < κ_{0} the following statements hold.

Assume that the density p(x) = \frac{ν(dx)}{dx} belongs to the class P_{κ}, and the sequence {l_{n}, n ≥ 0} satisfies Conditions 1–4. Then there exists a constant L = L(η̄, κ) such that if the sequence {l_{n}, n ≥ 0} satisfies Condition 5, then there exists a critical temperature T_{c} > 0 with the following properties:

(1) If T > T_{c} then

lim_{n→∞} 2^{n}M̄_{n}^{2}(T) = χ(T) > 0,  (1.26)

and the distribution ν̃_{n,T} (dx) tends weakly, as n → ∞, to the rdimensional standard normal distribution. The function χ(T) in (1.26) satisfies the following estimates near the critical point: There exists a temperature T_{0} > T_{c} and numbers C_{2} > C_{1} > 0 such that for all T_{0} > T > T_{c} there exists a number n̄(T) such that

\begin{matrix}C_{1} ∑_{k=n̄(T)}^{∞} l_{k}^{−1} < T − T_{c} ≤ C_{2} ∑_{k=n̄(T)}^{∞} l_{k}^{−1}, \\ C_{1} \frac{2^{n̄(T)}}{l_{n̄(T)}} < χ(T) < C_{2} \frac{2^{n̄(T)}}{l_{n̄(T)}} . \end{matrix}  (1.27)

(The number ξ(T) = 2^{n̄(T)} is the correlation length.)

(2) At T = T_{c}, lim_{n→∞} M_{n}(T_{c}) = 0 (there is no Thouless’ effect), and

moreover

lim_{n→∞} L_{n}^{−1}M_{n}(T_{c}) = 1,  (1.28)

where M_{n}(T) is defined in (2.12), and

L_{n} = (\frac{r − 1}{6} ∑_{j=n}^{∞} l_{j}^{−1})^{1/2} .  (1.29)

(Condition (1.25) implies that lim_{n→∞} L_{n} = 0.)

Let us define the rescaled version ρ_{n}(t) of the probability density function p̄_{n}(t, T_{c}) as

ρ_{n}(t) = \frac{M̂_{n}(T_{c})}{d_{n}} p̄_{n} (M̂_{n}(T_{c}) (1 + \frac{t}{d_{n}} ) , T_{c}) ,  (1.30)

where M̂_{n}(T) is defined in (1.15), and

d_{n} = \frac{(r − 1)l_{n}}{2b} ∑_{k=n}^{∞} l_{k}^{−1}.  (1.31)

(Observe that lim_{n→∞} d_{n} = ∞ by Condition 2 on {l_{n}}.) The function ρ_{n}(t) is defined on the half-line [−d_{n}, ∞). Then

lim_{n→∞} ∥ρ_{n}(t) − π(t)∥ = 0,  (1.32)

where the probability density π(t) is defined in equations (1.21), (1.22) and

∥f(t)∥ = ∑_{j=0}^{2} sup_{t≥−d_{n}} \{e^{|t|/3} |\frac{d^{j}f(t)}{dt^{j}} | \}  (1.33)

(3) If T < T_{c}, then the numbers M̂_{n}(T) and V_{n}(T) defined in formula (1.15) satisfy the following relations: The limit

lim_{n→∞} M̂_{n}(T) = M(T) > 0  (1.34)

exists, and

C_{1}|T − T_{c}|^{1/2} < M(T) < C_{2}|T − T_{c}|^{1/2}.  (1.35)

In addition,

lim_{n→∞} l_{n}V_{n}(T) = γ(T) = \frac{bT}{3M(T)} > 0  (1.36)

with the number b appeared in formula (1.21), and

lim_{n→∞} ∥π_{n}(t, T) − π(t)∥ = 0  (1.37)

where the probability densities π_{n}(t, T) and π(t) are defined in equations (1.16) and (1.21), (1.22), respectively, and ∥f(t)∥ is defined in (1.33), with d_{n} = \frac{M̂_{n}(T)}{V_{n}(T)}.

Theorems 1.3 and 1.5 are the central results of the present paper. Let us make some remarks about Theorem 1.5. Relations (1.26) and (1.28) imply that

M(T) = lim_{n→∞} M̂_{n}(T) = 0, for all T ≥ T_{c},

i.e. the spontaneous magnetization M(T) vanishes at T ≥ T_{c}. Relation (1.35) implies that

lim_{T →T_{c}^{−}} M(T) = 0,

with the classical critical exponent 1/2 for the magnetization.

The number n̄(T) in (1.27) is very important for our investigation in the subsequent sections. It shows how many iterations of the recursive equation (renormalization group transformation) is needed to reach the “high temperature region” (see Section 3 below for precise definitions). The quantity ξ(T) = 2^{n̄(T)} is the correlation length. Usually the correlation length has a power-like asymptotics ξ(T) ≍ |T − T_{c}|^{−ν} as T → T_{c} where ν is the critical exponent of the correlation length (see, e.g., [Fish] or [WK]). It follows from (1.27) that in the case under consideration, ξ(T) grows super-polynomially as T → T_{c}^{+}. For instance, if l_{n} is a sequence determined by equation (1.6) then ξ(T) grows like exp [C_{0}(T − T_{c})^{1/(λ−1)}]. Similarly, (1.27) implies that the magnetic susceptibility χ(T) diverges super-polynomially as T → T_{c}^{+}.

Relation (1.36) shows that the mean square deviation of the mean spin along the radius behaves, when n → ∞, as

V_{n}(T) ∼ \frac{bT}{3M(T)l_{n}} , T < T_{c},

so that it goes to zero very slowly as n → ∞ (comparing with the standard behavior of C2^{−n/2}). In fact, it goes to zero sub-polynomially with respect to the number of spins 2^{n}. And according to (1.31), at T = T_{c} the scaled mean square deviation of the mean spin along the radius, d_{n}^{−1}, goes to zero even slower, than at T < T_{c}, namely,

d_{n}^{−1} ∼ \frac{2b}{r − 1} (l_{n} ∑_{k=n}^{∞} l_{k}^{−1})^{−1} , T = T_{c}.

On the other hand, observe that by (1.32) and (1.37) the limit distribution density π(t) of the normalized mean spin along the radius is the same for all T < T_{c} and for T = T_{c} as well.

Let us say some words about our methods. The questions we investigate in this paper lead to a problem of the following type: We have a starting probability density function p_{0}(x, T) which depends on a parameter T, the temperature, and we apply the powers of an appropriately defined nonlinear operator Q to it. This operator Q is the renormalization group operator. We want to describe the behavior of the sequence of functions p_{n}(x, T) = Q^{n}p_{0}(x, T), n = 1, 2, . . .. In particular, we want to understand how the behavior of this sequence of functions p_{n}(x, T), n = 1, 2, …, depends on the parameter T. Our investigation shows that if the function p_{n}(x, T) is essentially concentrated around the origin, then a negligible error is committed when p_{n+1}(x, T) = Qp_{n}(x, T) is replaced by the convolution of the function p_{n}(x, T) with itself, and this is the case for all n if the parameter T is large. The replacement of the operator Q by the convolution is called the high temperature approximation.

On the other hand, if the function p_{n}(x, T) is essentially concentrated in a narrow shell far from the origin, and this is the case for all n if the parameter T is small, then another good approximation of the function p_{n+1}(x, T) = Q_{n}p_{n}(x, T) is possible. This is called the low temperature approximation. The high temperature approximation actually means the application of the standard methods of classical probability theory. The low temperature approximation applied in this paper is a natural modification of the methods in our paper [BM3] where a similar problem was investigated. But in the present paper we have to make a more careful and detailed analysis. The reason for it is that while in [BM3] it was enough to investigate only very low temperatures T, now we have to follow carefully when the high and when the low temperature approximation is applicable. Moreover, — and this is a most important part of this paper, — to describe the behavior of the functions p_{n}(·, T) for all temperatures T we have to follow the behavior of these functions also in the case when neither the high nor the low temperature approximation is applicable. This is the so called intermediate region. (See Section 3 for precise definitions.)

We study the intermediate region in Section 5. There we show that if the function p_{n}(x, T) “is not very far from the origin”, namely, the low temperature approximation is not applicable for it, then the functions p_{n+k}(x, T) are getting closer and closer to the origin as the index n + k is increasing. Moreover, after finitely many steps k the high temperature approximation is already applicable, and the number of steps k we need to get into this situation can be bounded by a constant independent of the parameter T. The proof given in Section 5 contains arguments essentially different from the rest of the paper. Here we heavily exploit that the numbers c_{n} = \frac{l_{n}}{l_{n−1}} are very close to one. Informally speaking, the sequence of numbers c_{n} − 1 behaves like a small parameter, and this “small parameter” enables us to handle our model near the critical temperature.

The setup of the rest of the paper is the following. In Section 2 we give an analytic reformulation of the problem and connect Dyson’s condition (1.2) with an approximate recursive formula for some quantities M_{n}(T) related to the spontaneous magnetization (see (2.20) below). In Section 3 we introduce a notion of low and high temperature regions together with an intermediate region. Then we formulate the basic auxiliary theorems about the characterization of these regions. In Sections 4, 5, and 6 we prove the main estimates concerning the low temperature region, the intermediate region, and the high temperature region, respectively. In Section 7 we prove the convergence of the recursive iterations to the fixed point for all T < T_{c}. Finally, in Section 8 we prove Theorem 3.4 concerning some asymptotics near the critical point T_{c} and derive Theorems 1.3 and 1.5 from the auxiliary theorems.

2. Analytic Reformulation of the Problem. Strategy of the Proof.

The hierarchical structure of the Hamiltonian (1.1) leads to the following recursive equation for the density functions p_{n}(x, T) (see, e.g., Appendix A to the paper [BM3]):

p_{n+1}(x, T) = C_{n}(T) ∫_{ℝ^{r}} exp (\frac{l_{n}}{T} (x^{2} − u^{2})) p_{n}(x − u, T)p_{n}(x + u, T) du,

(2.1)

for n ≥ 0, where p_{0}(x, T) = p_{0}(x) is defined in (1.7),

l_{n} = l(2^{n}),

and C_{n}(T) is an appropriate norming constant which turns p_{n+1}(x, T) into a density function. We are interested in the asymptotic behaviour of the functions p_{n}(x, T) as n → ∞. For the sake of simplicity we will assume that ε(t) = 0 in (1.7), so that p_{0}(x) coincides with (1.3). All the proofs below are easily extended to the case of nonzero ε(t) satisfying estimate (1.8).

Define

c_{n} = \frac{l_{n}}{l_{n−1}} , n = 0, 1, … with l_{−1} = 1,  (2.2)

A_{n} = 1 + ∑_{j=1}^{∞} \frac{c_{n+1}}{2} ⋯ \frac{c_{n+j}}{2} = 1 + l_{n}^{−1} ∑_{j=1}^{∞} 2^{−j}l_{n+j},  n = 0, 1, …. (2.3)

Then

l_{n} = ∏_{j=0}^{n} c_{j}, n ≥ 0,  (2.4)

and

l_{n}A_{n} = l_{n} + \frac{l_{n+1}A_{n+1}}{2} .  (2.5)

Indeed, by (2.3),

l_{n}A_{n} = l_{n} + ∑_{j=1}^{∞} 2^{−j}l_{n+j} = ∑_{j=0}^{∞} 2^{−j}l_{n+j},

hence

l_{n}A_{n} − l_{n} = ∑_{j=1}^{∞} 2^{−j}l_{n+j} = \frac{1}{2} ∑_{j=0}^{∞} 2^{−j}l_{n+1+j} = \frac{l_{n+1}A_{n+1}}{2} ,  (2.6)

and (2.5) follows.

Define

q_{n}(x, T) = Λ_{n}(T)^{−1} exp (\frac{A_{n}l_{n}x^{2}}{2} ) p_{n}(\sqrt{T} x, T),  (2.7)

where Λ_{n}(T) > 0 is a norming constant such that

∫_{ℝ^{r}} q_{n}(x, T) dx = 1.

Let

c^{(n)} = (1 + A_{n}) l_{n}, n = 0, 1, 2, … (2.8)

Then it follows from equations (2.1) and (2.5) that

q_{n+1}(x, T) = \frac{1}{Z_{n}(T)} ∫_{ℝ^{r}} e^{−c^{(n)}u^{2}}q_{n}(x − u, T)q_{n}(x + u, T) du.  (2.9)

Also, by (1.3),

q_{0}(x, T) = \frac{1}{Z_{0}(T)} exp \{(c_{0}A_{0} − T)\frac{|x|^{2}}{2} − κT^{2} \frac{|x|^{4}}{4} \} .  (2.10)

The norming constants Z_{n}(T) in the previous formulas are determined by the condition that

∫_{ℝ^{r}} q_{n}(x, T) dx = 1.

Thus, the functions q_{n}(x, T) are defined recursively by formulas (2.9) and (2.10). Our goal is to derive an asymptotics of the functions q_{n}(x, T) as n → ∞. Then the asymptotics of the functions p_{n}(x, T) can be found by means of formula (2.7). The advantage of the functions q_{n}(x, T) is that their recursive equation (2.9) does not depend on T.

The method of paper [BM3] can be adapted in the study of the low temperature approximation. We shall follow this approach. Due to the rotational symmetry of the Hamiltonian (1.1), the function q_{n}(x, T) depends only on |x|. Define the function q̄_{n}(t, T), t ∈ ℝ^{1}, n = 0, 1, 2, …, such that

q_{n}(x, T) = C_{n}(T) q̄_{n}(|x|, T),  (2.11)

with a norming constant C_{n}(T) such that

∫_{0}^{∞} q̄_{n}(t, T) dt = 1.

We will define

q̄_{n}(t, T) = 0 for t < 0.

Put also

M_{n}(T) = ∫_{0}^{∞} t q̄_{n}(t, T) dt, n = 0, 1, …,  (2.12)

and define the rescaled probability density functions

f_{n}(t, T) = \frac{1}{c^{(n)}} q̄_{n} (M_{n}(T) + \frac{t}{c^{(n)}} , T ) , t ∈ ℝ^{1}, n = 0, 1, …. (2.13)

Then

q̄_{n}(t, T) = c^{(n)}f_{n} (c^{(n)}(t − M_{n}(T)), T ) ,  (2.14)

and

∫_{−∞}^{∞} f_{n}(t, T) dt = 1, ∫_{−∞}^{∞} tf_{n}(t, T) dt = 0.

The order parameter M_{n}(T) in (2.12) is very convenient for the asymptotic recursive analysis. Later we will relate it to the parameters M̄_{n}(T) and M̂_{n}(T) introduced in formulae (1.9) and (1.15), respectively.

A low temperature approximation can be applied in the case when M_{n}(T) is relatively large, comparing with the size of the neighborhood of M_{n}(T) in which the function f_{n}(t, T) is essentially concentrated. In this case we follow the behaviour of the pair (f_{n}(t, T), M_{n}(T)). To describe this procedure introduce the notation c = {c^{(n)}, n = 0, 1, … }. The rotational invariance of the function q_{n}(·, T) suggests the definition of the operator

Q̄_{n,M}^{c}f(t) = ∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} exp \{−\frac{u^{2}}{c^{(n)}} − v^{2}\}

× f (c^{(n)} (\sqrt{(M + \frac{t}{c^{(n+1)}} + \frac{u}{c^{(n)}} )^{2} + \frac{v^{2}}{c^{(n)}}} − M))

× f (c^{(n)} (\sqrt{(M + \frac{t}{c^{(n+1)}} − \frac{u}{c^{(n)}} )^{2} + \frac{v^{2}}{c^{(n)}}} − M)) du dv.

Formula (2.9) together with the definition of the function f_{n}(t, T) yields that

q̄_{n+1} (M_{n}(T) + \frac{t}{c^{(n+1)}} , T ) = \frac{c^{(n+1)}}{Z_{n}(T)}Q̄_{n,M_{n}(T)}^{c}f_{n}(t, T)

with

Z_{n}(T) = ∫_{−c^{(n+1)}M_{n}(T)}^{∞} Q̄_{n,M_{n}(T)}^{c}f_{n}(t, T) dt.

The norming constant Z_{n}(T) is determined by the condition

∫_{0}^{∞} q̄_{n+1}(t, T) dt = 1.

Define also

m_{n}(T) = m_{n}(f_{n}(t, T)) = \frac{1}{Z_{n}(T)} ∫_{−c^{(n+1)}M_{n}(T)}^{∞} t Q̄_{n,M_{n}(T)}^{c}f_{n}(t, T) dt (2.15)

and

Q_{n,M_{n}(T)}^{c}f_{n}(t, T) = \frac{1}{Z_{n}(T)}Q̄_{n,M_{n}(T)}^{c}f_{n}(t + m_{n}(T), T).

Then

f_{n+1}(t, T) = Q_{n,M_{n}(T)}^{c}f_{n}(t, T) and M_{n+1}(T) = M_{n}(T) + \frac{m_{n}(T)}{c^{(n+1)}}. (2.16)

To formulate a good approximation of the operator Q_{n,M_{n}(T)}^{c}, let us introduce the numbers

c̄_{n} = \frac{c^{(n)}}{c^{(n−1)}} = \frac{(1 + A_{n})l_{n}}{(1 + A_{n−1})l_{n−1}} , n = 1, 2, ….  (2.17)

The arguments of the function f in the definition of the operator Q̄_{n,M}^{c},

ℓ_{n,M}^{c,±}(t, u, v) = c^{(n)} (\sqrt{(M + \frac{t}{c^{(n+1)}} ± \frac{u}{c^{(n)}} )^{2} + \frac{v^{2}}{c^{(n)}}} − M) , (2.18)

can be well approximated by a simpler expression because of the estimate

|ℓ_{n,M}^{c,±}(t, u, v) − (\frac{t}{c̄_{n+1}} ± u + \frac{v^{2}}{2M} )| ≤ 100 (\frac{|v|^{4}}{c^{(n)}M^{3}} + \frac{t^{2} + u^{2}}{c^{(n)}M} )

which holds for |t| < \frac{1}{4}c^{(n+1)}M, |u| < \frac{1}{4}c^{(n)}M and v^{2} < c^{(n)}M^{2}. This estimate suggests that for low temperatures T, when M_{n}(T) is not small, the operator Q̄_{n,M_{n}(T)}^{c} can be well approximated by the operator T̄_{n,M_{n}(T)}^{c} defined as

\begin{aligned}T̄_{n,M_{n}(T)}^{c}f(t, T) & = ∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} e^{−v^{2}}f (\frac{t}{c̄_{n+1}} + u + \frac{v^{2}}{2M_{n}(T)}, T ) \\ & × f (\frac{t}{c̄_{n+1}} − u + \frac{v^{2}}{2M_{n}(T)}, T ) du dv. \end{aligned} (2.19)

The elaboration of the above indicated method will be called the low temperature approximation. It works well when M_{n}(T) is much larger than the range where the function f_{n}(t, T) is essentially concentrated. For n = 0 the starting value M_{0}(T) at very low temperatures T > 0 is very large. In this case the low temperature expansion can be applied. As we shall see later, the approximation of Q̄_{n,M_{n}(T)}^{c} by Tc̄ ; n,M_{n}(T) yields that

M_{n+1}(T) ∼ M_{n}(T) − \frac{r − 1}{4c^{(n)}M_{n}(T)},  (2.20)

which, in turn, implies that

M_{n+1}^{2}(T) ∼ M_{n}^{2}(T) − \frac{r − 1}{2c^{(n)}}.  (2.21)

It follows from (2.3) and (1.5) that

2 ≤ A_{n} ≤ 2.03,  lim_{n→∞} A_{n} = 2,  (2.22)

hence if Condition 1 is satisfied, then not only lim_{n→∞} c_{n} = 1, but also lim_{n→∞} c̄_{n} = 1, and by (2.8),

3 ≤ \frac{c^{(n)}}{l_{n}} ≤ 3.03,  lim_{n→∞} \frac{c^{(n)}}{l_{n}} = 3.  (2.23)

This allows us to rewrite (2.21) as

M_{n+1}^{2}(T) ∼ M_{n}^{2}(T) − \frac{r − 1}{6l_{n}}  (2.24)

This formula underlines the importance of the Dyson condition (1.2). Namely, if the series

B = ∑_{n=1}^{∞} l_{n}^{−1}  (2.25)

converges then M_{n}(T) remains large for all n if T > 0 is small. Indeed, assume that T < c_{0}A_{0}/2. Then it follows from (2.10) that M_{0}^{2}(T) > C(κT^{2})^{−1}, hence by (2.24), neglecting the error term,

M_{n}^{2}(T) ≥ M_{0}^{2}(T) − \frac{r − 1}{6} ∑_{n=0}^{∞} l_{n}^{−1} ≥ C(κT^{2})^{−1} − C_{1} ≫ 1

for all n if T > 0 is small, which was stated. On the other hand, if the series (2.25) diverges, then for some n, M_{n}(T) becomes small, and the approximation (2.20) becomes inapplicable.

The low temperature approximation can be applied when M_{n}(T) is not small. When M_{n}(T) is small a different approximation is natural. If the function q_{n}(x, T) is essentially concentrated in a ball whose radius is much less than (c^{(n)})^{−1/2}, then a small error is committed if the kernel function e^{−c^{(n)}u^{2}} in formula (2.9) is omitted. This means that the formula expressing q_{n+1}(x) by q_{n}(x) can be well approximated through the convolution q_{n+1}(x) = q_{n} ∗ q_{n}(x). This approximation will be called the high temperature approximation. If the high temperature approximation can be applied for q_{n}(x, T), then the function q_{n+1}(x, T) is even more strongly concentrated around zero. Hence, as a detailed analysis will show, if at a temperature T it can be applied for a certain n_{0}, then it can be applied for all n ≥ n_{0}.

Finally, there are such pairs (n, T) for which the function q_{n}(x, T) can be studied neither by the low nor by the high temperature approximation. We call the set of such pairs an intermediate region. We shall prove that if the sequence c^{(n)} sufficiently slowly tends to infinity and the function q_{n}(x, T) is out of the region where the low temperature approximation is applicable, then the density function q_{n+1}(x, T) will be more strongly concentrated around zero than the function q_{n}(x, T). Moreover, in finitely many steps the function q_{n+k}(x, T) will be so strongly concentrated around zero that after this step the high temperature approximation is applicable. It is important that the number of steps k needed to get into the high temperature region can be bounded independently of the parameter T.

The main part of the paper consists of an elaboration of the above heuristic argument.

3. Formulation of Auxiliary Theorems.

To describe the region where the low temperature approximation will be applied we define some sequences β_{n}(T) which depend on the temperature T. Define recursively,

\begin{aligned}β_{N}(T) & = \frac{(c^{(N)})^{2}}{2^{N}} , \\ β_{n+1}(T) & = (\frac{c̄_{n+1}^{2}}{2} + \sqrt{\frac{β_{n}(T)}{c^{(n)}}}) β_{n}(T) + \frac{10}{M_{n}^{2}(T)} for n ≥ N, \end{aligned} (3.1)

where the number N is defined in (1.12), c̄_{n} in (2.17) and M_{n}(T) in (2.12). As it will be seen later, these numbers measure how strongly the functions f_{n}(x, T) are concentrated around zero. We define the low temperature region, where low temperature approximation will be applied.

Definition of the low temperature region. A pair (n, T) is in the low temperature region if the following properties (1) and (2) hold.

(1) 0 < T ≤ c_{0}A_{0}/2, where A_{0} was defined in (2.3).

(2) Either 0 ≤ n ≤ N with the number N introduced in (1.12) or n > N and \frac{β_{n−1}(T)}{c^{(n−1)}} ≤ η with the number η appearing also in (1.12).

The temperature T is in the low temperature region if the pair (n, T) is in the low temperature region for all numbers n. Let us remark that by (2.4)

and (1.5)

1 ≤ l_{n} = ∏_{j=1}^{n} c_{j} ≤ 1.01^{n},

hence by (2.23),

3 ≤ c^{(n)} ≤ 3.03 · 1.01^{n}.  (3.2)

Therefore, by (3.1),

\frac{β_{N}(T)}{c^{(N)}} = \frac{c^{(N)}}{2^{N}} ≤ \frac{1}{c^{(N)}} ≤ η  (3.3)

hence the pair (N + 1, T) is in the low temperature region if T ≤ c_{0}A_{0}/2. Since β_{n+1}(T) ≥ \frac{10}{M_{n}^{2}(T)} the pair (n, T) can get out of the low temperature region only if M_{n}(T) becomes very small.

To define the high temperature region introduce the notations

\begin{aligned}h_{n}(x, T) & = (c^{(n)})^{−r/2} q_{n} (\frac{x}{\sqrt{c^{(n)}}} , T ) , \\ D_{n}^{2}(T) & = ∫_{ℝ^{r}} x^{2}h_{n}(x, T) dx. \end{aligned}  (3.4)

where the function q_{n}(x, T) is defined in (2.7). Let us also introduce the probability measure H_{n,T},

H_{n,T} (A) = ∫_{A} h_{n}(x, T) dx, A ⊂ ℝ^{r},  (3.5)

on ℝ^{r}.

Definition of the high temperature region. A pair (n, T) is in the high temperature region if D_{n}^{2}(T) < e^{−1/η^{2}} with the number η in formula (1.12), where D_{n}^{2}(T) is defined in (3.4). The temperature T is in the high temperature region if there exists a threshold index n_{0}(T) such that (n, T) is in the high temperature region for all n ≥ n_{0}(T).

It may happen that a pair (n, T) belongs neither to the low nor to the high temperature region. Then we say that (n, T) belongs to the intermediate region. Let us remark that we introduced two numbers N and η in formula (1.12), and in the formulation of the subsequent results N and η will denote these numbers. The following result is very important for us.

Theorem 3.1. There exists a number κ_{0} = κ_{0}(N) such that for all 0 < κ < κ_{0} (where κ appears in formula (1.3)) and 0 < η̄ < η there is a number L = L(η̄, κ) for which the following is true. Assume that Conditions 1 and 5 (with η̄ and this number L = L(η̄, κ)) hold. We consider such temperatures T for which there are numbers n such that the pair (n, T) does not belong to the low temperature region. Let n̄(T) ≥ 0 be the smallest number n with this property.

If the pair (n̄(T), T) does not belong to the high temperature region (which means that (n̄(T), T) is in the intermediate region), then there exist some numbers K = K(η̄, κ) > 0, η̃ = η̃(η̄, κ) > 0, and k = k(η̄, κ) ∈ ℕ such that

D_{n̄(T)}^{2}(T) < K,  η̃ < D_{n̄(T)+k}^{2}(T) < e^{−1/η^{2}}.

This implies in particular that the pair (n̄(T)+k, T) with this index k belongs to the high temperature region.

We shall also prove the following corollary of Theorem 3.1. (See the Remark after the proof of Lemma 6.1.)

Corollary. Under the conditions of Theorem 3.1 all temperatures T > 0 belong either to the low or to the high temperature region. If the Dyson condition (1.2) holds, then all sufficiently low temperatures belong to the low and all sufficiently high temperatures to the high temperature region. If the Dyson condition (1.2) is violated, then all temperatures T > 0 belong to the high temperature region.

The next theorem concerns the low temperature region.

Theorem 3.2. There exists a number κ_{0} = κ_{0}(N) such that for all 0 < κ < κ_{0} the following is true. Assume that the Dyson condition (1.2) and Conditions 1 and 2 hold. Assume that the temperature T is in the low temperature region. Then the numbers M_{n}(T) defined in (2.12) have a limit,

lim_{n→∞} M_{n}(T) = M_{∞}(T),  (3.6)

and

lim_{n→∞} \frac{M_{n}^{2}(T) − M_{∞}^{2}(T)}{\frac{r−1}{2} ∑_{k=n}^{∞} \frac{1}{c^{(k)}}} = 1.  (3.7)

In addition,

lim_{n→∞} ∥\frac{1}{M_{n}(T)}f_{n} (\frac{t}{M_{n}(T)}, T ) − g (t)∥ = 0,  (3.8)

where

∥f(t)∥ = ∑_{j=0}^{2} sup_{t≥−c^{(n)}M_{n}(T)} e^{|t|} |\frac{d^{j}f(t)}{d t^{j}} | ,  (3.9)

f_{n}(t, T) is introduced in (2.13), and the probability density g(t) is defined as a solution of the fixed point equation (1.18).

Part (3) of Theorem 1.5, with the exception of estimate (1.35), follows from Theorem 3.2 and the additional relation M_{∞}(T) > 0 if T < T_{c} which follows from the results in Theorem 3.4 formulated at the end of this section. Indeed, we can express the function p_{n}(x, T) in terms of f_{n}(t, T). Namely, by (2.7), (2.11), and (2.14)

\begin{aligned}p_{n}(x, T) & = L_{n}^{−1}(T) exp (−\frac{A_{n}l_{n}|x|^{2}}{2T} ) \\ & × f_{n} (\frac{c^{(n)}}{\sqrt{T}} (|x| − \sqrt{T} M_{n}(T)) , T ) \end{aligned}  (3.10)

with an approriate norming constant L_{n}(T). Let us write that |x|^{2} =

(\sqrt{T} M_{n}(T) + |x| − \sqrt{T} M_{n}(T))^{2}, hence

exp (−\frac{A_{n}l_{n}|x|^{2}}{2T} ) = exp\{−\frac{A_{n}l_{n}}{2T} [T M_{n}^{2}(T)

+2\sqrt{T} M_{n}(T)(|x| − \sqrt{T} M_{n}(T)) + (|x| − \sqrt{T} M_{n}(T))^{2}]\},

and substitute it into (3.10). This leads to the equation

p_{n}(x, T) = L̃_{n}^{−1}(T)f̃_{n} (\frac{|x| − M̃_{n}(T)}{Ṽ_{n}(T)}, T )  (3.11)

with an appropriate norming constant L̃_{n}(T), where

M̃_{n}(T) = \sqrt{T} M_{n}(T),  Ṽ_{n}(T) = \frac{\sqrt{T}}{c^{(n)}M_{n}(T)},
f̃_{n}(t, T) = f_{n} (\frac{t}{M_{n}(T)}, T ) exp (−\frac{A_{n}l_{n}t}{c^{(n)}} − ε_{n}(t, T)) ,
ε_{n}(t, T) = \frac{A_{n}l_{n}t^{2}}{2(c^{(n)})^{2}M_{n}^{2}(T)}.

Observe that by (2.22) and (2.23)

lim_{n→∞} \frac{A_{n}l_{n}}{c^{(n)}} = \frac{2}{3},  lim_{n→∞} \frac{A_{n}l_{n}}{2(c^{(n)})^{2}M_{n}^{2}(T)} = 0,

hence (3.8) implies that there is some C_{0} > 0 such that

lim_{n→∞} ∥\frac{1}{M_{n}(T)}f̃_{n}(t, T) − C_{0}g (t) e^{−2t/3}∥′ = 0,

where

∥f(t)∥′ = ∑_{j=0}^{2} sup_{t≥−c^{(n)}M_{n}^{2}(T)} e^{|t|/3} |\frac{d^{j}f(t)}{d t^{j}} | .

This also implies that there exist some real number a and C′ > 0 such that

lim_{n→∞} ∥\frac{1}{M_{n}(T)}f̃_{n}(t − a, T) − C′g (t − a) e^{−2t/3}∥′ = 0,  (3.12)

with such numbers a and C′ > 0 for which the relations

∫_{∞}^{∞} C′g(t − a)e^{−2t/3} dt = 1 and ∫_{∞}^{∞} C′tg(t − a)e^{−2t/3} dt = 0

hold.

Let us define for all b > 0 the function π(t|b) = C′bg(bt − a)e^{−2bt/3} dt. These functions satisy the relations

∫_{∞}^{∞} C′bπ(t|b) dt = 1 and ∫_{∞}^{∞} C′btπ(t|b) dt = 0.

Moreover, the number b > 0 can be chosen in such a way that the identity

∫_{∞}^{∞} C′bt^{2}π(t|b) dt = b^{−2} ∫_{∞}^{∞} C′b(bt)^{2}π(t|b) dt
= b^{−2} ∫_{∞}^{∞} C′t^{2}g(t − a)e^{2t/3} dt = 1

also holds. Let us define the function π(t) = π(t|b) with this parameter b. In such a way we constructed a function π(t) that satisfies relations (1.21) and (1.22). Moreover, if we define the functions

π̃_{n}(t, T) = C_{n}(T)f̃_{n}(bt − a, T)  (3.13)

with these numbers a and b and with such a norming constant C_{n}(T) for which

∫_{−∞}^{∞} π̃_{n}(t, T) dt = 1,

then these functions satisfy the relation

lim_{n→∞} ∥π̃_{n}(t) − π(t)∥′ = 0  (3.14)

because of relations (3.12). Because of (3.14) we also have ∫_{−∞}^{∞} π̃_{n}(t, T) dt =

1, lim_{n→∞} ∫_{−∞}^{∞} tπ̃_{n}(t, T) dt = 0, lim_{n→∞} ∫_{−∞}^{∞} t^{2}π̃_{n}(t, T) dt = 1, and because of

(3.11) and (3.13)

π̃_{n}(t, T) = C′_{n}(T)^{−1}p̄_{n}((bt − a)Ṽ_{N}(T) + M̃_{n}(T), T)
= bṼ_{n}(T)p̄_{n}(bt − a)Ṽ_{N}(T) + M̃_{n}(T), T).

(The normalization constant in the second identity of the last formula is determined by the fact that both p̄_{n}(t, T) and π̃_{n}t, T) are probability density functions).

Hence

∫_{−∞}^{∞} tπ̃_{n}(t, T) dt = bṼ_{n}(T) ∫_{−∞}^{∞} tp̄_{n}(bt − a)Ṽ_{N}(T) + M̃_{n}(T), T) dt

= ∫_{−∞}^{∞} \frac{t − M̃_{n}(T) + aṼ_{n}(T)}{bṼ_{n}(T)}p̄_{n}(t, T) dt  (3.15)

= \frac{M̄_{n}(T) − M̃_{n}(T) + aṼ_{n}(T)}{bṼ_{n}(T)} → 0 as n → ∞,

and

∫_{−∞}^{∞} t^{2}π̃_{n}(t, T) dt = bṼ_{n}(T) ∫_{−∞}^{∞} t^{2}p̄_{n}(bt − a)Ṽ_{N}(T) + M̃_{n}(T), T) dt

= ∫_{−∞}^{∞} (\frac{(t − M̄_{n}(T)) + (M̄_{n}(T) − M̃_{n}(T) + aṼ_{n}(T))}{bṼ_{n}(T)})^{2} p̄_{n}(t, T) dt

= \frac{V_{n}^{2}(T) + ((M̄_{n}(T) − M̃_{n}(T)) + aṼ_{n}(T))^{2}}{b^{2}Ṽ_{n}^{2}(T)} → 1 as n → ∞. (3.16)

Relations (3.15) and (3.16) together with Theorem 3.2, the inequqailty M_{∞}(T) > 0 and the definition of the quantities M̃_{n}(T) and Ṽ_{n}(T) imply relations (1.34) and (1.36). Indeed, by Theorem 3.2 lim_{n→∞} M̃_{n}(T) =

M(T) with M(T) = \sqrt{T} M_{∞}(T), and since Ṽ_{n}(T) → 0 as n → ∞ relation (3.15) implies that lim_{n→∞}(M̄_{n}(T) − M̃_{n}(T)) = 0. Formula (1.34) follows from these relations with the above defined number M(T). Relations (3.15) and (3.16) together imply that lim_{n→∞} \frac{V_{n}(T)}{Ṽ_{n}(T)} = b. On the other hand,

lim_{n→∞} l_{n}Ṽ_{n}(T) lim_{n→∞} \frac{c^{(n)}}{3} \frac{T}{c^{(n)}\sqrt{T} M_{n}(T)} = \frac{T}{3M(T)}. These relations imply (1.36).

Finally to prove relation (1.37) let us observe how the functions π_{n}(t, T) and π̃_{n}(t, T) can be expressed with the help of the function p̄_{n}(t, T). Besides, both are probability density functions, and the integrals ∫_{−∞}^{∞} tπ̃_{n}(t, T) dt and ∫_{−∞}^{∞} t^{2}π̃_{n}(t, T) dt tend to zero and 1 as n → ∞, while the correponding integrals for π_{n}(t, T) are equal exactly to these limit values for all parameters n. This implies that the identity π_{n}(t, T) = (1 + ε_{n})π̃_{n}((1 + ε_{n})t + δ_{n}, T) with such numbers ε_{n} = ε_{n}(T) and δ_{n} = δ_{n}(T) for which lim_{n→∞} ε_{n} = 0, and lim_{n→∞} δ_{n} = 0. It can be proved with the help of this observation that relation (3.14) remains valid if we replace the functions π̃_{n}(t, T) by π_{n}(t, T) in it, and this means that formula (1.37) is valid.

Now we formulate a theorem about the high temperature region. Put

h̃_{n}(x, T) = 2^{−rn/2}q_{n} (2^{−n/2}x, T ) = (\frac{c^{(n)}}{2^{n}} )^{r/2} h_{n} (\sqrt{\frac{c^{(n)}}{2^{n}}} x, T ) , (3.17)

and define the probability measures

H̃_{n,T} (A) = ∫_{A} h̃_{n}(x, T) dx, A ⊂ ℝ^{r}  (3.18)

on ℝ^{r}.

Theorem 3.3. There exists a number κ_{0} = κ_{0}(N) such that for all 0 < κ < κ_{0} and 0 < η̄ < η there exists a number L = L(η̄, κ) such that the following is true. Assume that Conditions 1 and 5 (with η̄ and this L = L(η̄.κ)) hold, and T is in the high temperature region. Then the measures H̃_{n,T} defined in (3.18) converge weakly to the normal distribution on ℝ^{r} with expectation zero and covariance matrix σ^{2}(T)I with some σ^{2}(T) > 0, where I denotes the identity matrix.

If T belongs to the high temperature region, but the pair n = (0, T) does not belong to it, (i.e. the temperature T is not too high), then the inequality

C_{1} \frac{2^{n̄(T)}}{c^{(n̄(T))}} ≤ σ^{2}(T) ≤ C_{2} \frac{2^{n̄(T)}}{c^{(n̄(T))}}  (3.19)

also holds with some C_{2} > C_{1} > 0, where n̄(T) is defined in Theorem 3.1.

Remark. Not only the convergence of the measures H̃_{n,T} but also the convergence of their density functions h̃_{n}(x, T) could be proved. But the proof of the convergence of the distribution is simpler, and it is also sufficient for our purposes.

Corollary. Let H̄_{n,T} denote the probability measure on ℝ^{r} with the density function

2^{−rn/2}T^{r}p_{n}(2^{−n/2}\sqrt{T} x, T).

Under the conditions of Theorem 3.3 the measures H̄_{n,T} have the same Gaussian limit as the measures H̃_{n,T} defined in Theorem 3.3 as n → ∞.

Our last theorem concerns the critical point. We want to show that there is a critical temperature T_{c} such that above it all temperatures belong to the high and below it all temperatures belong to the low temperature region. We also want to describe the situation in the neighborhood of the critical temperature in more detail. In Theorem 3.4 we state such a result.

Theorem 3.4. There exists a number κ_{0} = κ_{0}(N) such that for all 0 < κ < κ_{0} there exists a number L = L(η̄, κ) such that the following is true.

Assume that Conditions 1–4 are satisfied. Then for a fixed n the set of temperatures T for which (n, T) belongs to the low temperature region forms an interval (0, T_{n}], and the sequence T_{n}, n = 1, 2, …, is monotone decreasing in n. Define the critical temperature T_{c} as the limit, T_{c} = lim_{n→∞} T_{n}. Then c_{0}A_{0}/4 > T_{c} > 0. The function M_{∞}(T) = lim_{n→∞} M_{n}(T) exists in the interval (0, T_{c}], and for a fixed n the function M_{n}(T) is strictly decreasing on the interval (0, T_{n}]. The relation M_{∞}(T_{c}) = 0 holds. If T_{c} + ε > T > T_{c} with some ε > 0, then the inequality

C_{1} ∑_{k=n̄(T)}^{∞} \frac{1}{c^{(k)}} < T − T_{c} < C_{2} ∑_{k=n̄(T)}^{∞} \frac{1}{c^{(k)}}  (3.20)

holds with some appropriate numbers C_{2} > C_{1} > 0, where n̄(T) is defined in Theorem 3.1. If T_{c} − ε < T < T_{c} with a sufficiently small ε > 0, then

C_{1}(T_{c} − T)^{1/2} < M_{∞}(T) < C_{2}(T_{c} − T)^{1/2}.  (3.21)

4. Basic Estimates in the Low Temperature Region.

In this section we give some basic estimates on the function f_{n}(x, T) and its derivatives (with respect to the variable x) if the pair (n, T) is in the low temperature region. These estimates state in particular, that in the definition of the functions f_{n}(x, T) the right scaling was chosen. With the scaling in formula (2.13) the function f_{n}(x, T) is essentially concentrated in a finite interval whose size depends only on M_{n}(T). Both the results and proofs are closely related to those of Sections 3—6 in paper [BM3].

First we consider the case of small indices 0 ≤ n ≤ N, where the number N defined in (1.12) (cf. Section 4 in [BM3]), and we begin with n = 0. Assume that T < c_{0}A_{0}/2 and κ > 0 is small (exact conditions on the smallness of κ will be given later). In this case the function q̄_{0}(x, T) has its maximum in the points M̄_{0}(T) (see (2.10)), where

M̄_{0}(T) = (\frac{A_{0}c_{0} − T}{κT^{2}} )^{1/2}  (4.1)

is a large number. From (2.10) we obtain that

\frac{1}{c^{(0)}} q̄_{0} (M̄_{0}(T) + \frac{x}{c^{(0)}} , T )  (4.2)

= \frac{1}{Z_{0}(T)} exp \{− (A_{0}c_{0} − T) (\frac{x}{c^{(0)}} )^{2} (1 + \frac{x}{2c^{(0)}M̄_{0}(T)})^{2}\} ,

where

Z_{0}(T) = ∫_{−M̄_{0}(T)}^{∞} exp \{−(A_{0}c_{0} − T) (\frac{x}{c^{(0)}} )^{2} (1 + \frac{x}{2c^{(0)}M̂_{0}(T)})^{2}\} dx.

(4.3)

It can be proved by means of the identity

c^{(0)}(M_{0}(T) − M̄_{0}(T))  (4.4)

= \frac{∫_{−c^{(0)}M̄_{0}(T)}^{∞} x exp \{−(c_{0}A_{0} − T) (\frac{x}{c^{(0)}} )^{2} (1 + \frac{x}{2c^{(0)}M̄_{0}(T)})^{2}\} dx}{∫_{−c^{(0)}M̄_{0}(T)}^{∞} exp \{−(A_{0}c_{0} − T) (\frac{x}{c^{(0)}} )^{2} (1 + \frac{x}{2c^{(0)}M̄_{0}(T)})^{2}\} dx}

that

|M_{0}(T) − M̄_{0}(T)| ≤ \frac{const.}{M_{0}(T)} ≤ const. \sqrt{κ} T,  (4.5)

where M_{0}(T) is defined (2.12). This shows that M̄_{0}(T) is a very good approximation to M_{0}(T). Straightforward calculation yields with the help of formulas (4.1) and (4.3) that

|Z_{0}(T) − \frac{c^{(0)}\sqrt{π}}{\sqrt{(A_{0}c_{0} − T)}}| ≤ const. \sqrt{κ} T,  (4.6)

and from (4.1)–(4.6) we obtain that

|\frac{∂^{j}}{∂x^{j}} (f_{0}(x, T) − \frac{\sqrt{A_{0}c_{0} − T}}{c^{(0)}\sqrt{π}} exp \{−(A_{0}c_{0} − T) (\frac{x}{c^{0}} )^{2}\})| (4.7)
≤ const. κ^{1/4}e^{−2|x|/c^{(0)}} if |x| < log κ^{−1}, j = 0, 1, 2,

and

|\frac{∂^{j}}{∂x^{j}} f_{0}(x, T)| ≤ C exp \{−\frac{(A_{0}c_{0} − T)}{4c^{(0)}} |2x + \frac{x^{2}}{c^{(0)}M_{0}^{2}(T)}|\}

for x ≥ −c^{(0)}M_{0}(T), j = 0, 1, 2.  (4.8)

A relatively small error is committed if M_{n} is very large and the arguments ℓ_{n,M_{n}}^{±}(x, u, v) (defined in formula (2.18)) of the function f_{n} in the operator Q̄_{n,M}^{c}f_{n} are replaced by x ± u. Exploiting this fact one can prove, using a natural adaptation of the proof of Proposition 1 of paper [BM3], the following

Proposition 4.1. There exists a number κ_{0} = κ_{0}(N) such that if

(i) 0 < κ < κ_{0}, and (ii) 0 < T ≤ c_{0}A_{0}/2, then the relations

|\frac{∂^{j}}{∂x^{j}} (f_{n}(x, T) − \begin{matrix}√ & \overline{A_{0}c_{0} − T} \\ √ & \overline{π} \end{matrix} \frac{2^{n}}{c^{(n)}} exp \{−2^{n}(A_{0}c_{0} − T) (\frac{x}{c^{(n)}} )^{2}\})|
≤ B(n)κ^{1/4}e^{−2^{n+1}|x|/c^{(n)}}, if |x| < 2^{−n} log κ^{−1}, j = 0, 1, 2,

|\frac{∂^{j}}{∂x^{j}} f_{n}(x, T)| ≤ B(n) exp \{−\frac{(A_{0}c_{0} − T)}{4} \frac{2^{n}}{c^{(n)}} |2x + \frac{x^{2}}{c^{(n)}M_{n}^{2}(T)}|\}

for x ≥ −c^{(n)}M_{n}(T), j = 0, 1, 2,

and

|M_{n}(T) − M̄_{0}(T)| ≤ B(n)\sqrt{κ} T  (4.9)

hold for all 0 ≤ n ≤ N with the function M̄_{0}(T) defined in (4.1) and a function B(n) which depends neither on T nor on κ.

We formulate and prove, similarly to paper [BM3], certain inductive hypotheses about the behaviour of the functions f_{n}(x, T) for n ≥ N if the pair (n, T) is in the low temperature region. In the formulation of these hypotheses we apply the sequence β_{n}(T) defined in (3.1) and the sequence α_{n}(T) defined as

\begin{aligned}α_{N}(T) & = \frac{1}{200} \frac{(c^{(N)})^{2}}{2^{N}} , \\ α_{n+1}(T) & = (\frac{c̄_{n+1}^{2}}{2} − \sqrt{\frac{β_{n}(T)}{c^{(n)}}}) α_{n}(T) + \frac{10^{−12}}{M_{n}^{2}(T)} for n ≥ N. \end{aligned}  (4.10)

To formulate the inductive hypotheses we also introduce a regularization of the functions f_{n}(x, T).

Definition of the regularization of the functions f_{n}(x, T). Let us fix a C^{∞}-function φ(x), −∞ < x < ∞, such that φ(x) = 1 for |x| ≤ 1, 0 ≤ φ(x) ≤ 1 if 1 ≤ x ≤ 2 and φ(x) = 0 for |x| ≥ 2. Then the regularization of the function f_{n}(x, T) is

φ_{n}(f_{n}(x, T)) = A_{n}φ (\frac{x + B_{n}}{100\sqrt{c^{(n)}}} ) f_{n}(x + B_{n}, T),

with norming constants A_{n} and B_{n} such that

∫_{−∞}^{∞} φ_{n}(f_{n}(x, T)) dx = 1, ∫_{−∞}^{∞} xφ_{n}(f_{n}(x, T)) dx = 0.

Now we formulate the inductive hypotheses.

Hypothesis I(n).

|\frac{∂^{j}f_{n}(x, T)}{∂x^{j}} | ≤ \frac{C}{β_{n}(T)^{(j+1)/2}} exp \{−\frac{1}{\sqrt{β_{n}(T)}} |2x + \frac{x^{2}}{c^{(n)}M_{n}(T)}|\}
for j = 0, 1, 2, x ≥ −c^{(n)}M_{n}(T),

with a universal constant C > 0. One could choose, e.g., C = 10^{20}.

Hypothesis J(n).

|φ̃_{n}f_{n}(t + is, T)| ≤ \frac{e^{β_{n}(T)s^{2}}}{1 + α_{n}(T)t^{2}} if |s| ≤ \frac{2}{\sqrt{β_{n+1}(T)}},

where

φ̃_{n}f_{n}(t + is, T) = ∫ e^{i(t+is)x}φ(f_{n}(x, T)) dx.

Corollary of Proposition 4.1. Under the conditions of Proposition 4.1, the inductive hypotheses I(n) and J(n) hold for n = N with a universal constant C > 0 in hypothesis I(n). (For instance, one can choose C = 10^{5}.)

Before formulating the main result of this Section, we introduce the operators T_{n}. They are appropriate scaling of the operators T̄_{n,M_{n}(T)}^{c} defined in formula (2.19), but these operators will be applied only for the regularization of the functions f_{n}(x, T) and not for the functions f_{n}(x, T) themselves. Put

\begin{matrix}T_{n}φ_{n}(f_{n}(x, T)) = \frac{2}{c̄_{n+1}π^{\frac{r−1}{2}}} ∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} e^{−v^{2}} \\ × φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} − \frac{r − 1}{4M_{n}(T)} + u + \frac{v^{2}}{2M_{n}(T)}, T )) \\ × φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} − \frac{r − 1}{4M_{n}(T)} − u + \frac{v^{2}}{2M_{n}(T)}, T )) du dv, \end{matrix} (4.11)

with the constants c̄_{n} defined in (2.17) and the number V (S^{r−2}) introduced in Proposition 1.4. The main result of this section is the following

Proposition 4.2. There exists κ_{0} = κ_{0}(N) > 0 such that if (i) the inductive hypotheses I(n) and J(n) hold for the function f_{n}(x, T), (ii) 0 < κ < κ_{0}, (κ appears in formula (1.3)), and (iii) the pairs (m, T) belong to the low temperature region for all

0 ≤ m ≤ n, then the inductive hypotheses I(n + 1) and J(n + 1) hold for the function f_{n+1}(x, T). Also there exist universal constants C_{1}, K_{1}, K_{2} and K_{3} such that the following estimates hold:

(a)

\begin{matrix}M_{n+1}(T) = M_{n}(T) − \frac{r − 1}{4c^{(n)}M_{n}(T)} + \frac{γ_{n}(T)}{c^{(n)}}, \\ where |γ_{n}(T)| ≤ C_{1} \frac{β_{n+1}(T)}{c^{(n+1)}} \sqrt{β_{n+1}(T)} \end{matrix}  (4.12)

(b)

1 ≤ \frac{β_{n+1}(T)}{α_{n+1}(T)} ≤ K_{1},  (4.13)

(c) For x > −c^{(n+1)}M_{n+1}(T) and j = 0, 1, 2,

\begin{matrix}|\frac{∂^{j}}{∂x^{j}} [f_{n+1}(x, T) − T_{n}φ_{n}(f_{n}(x, T))]| ≤ \frac{K_{2}C^{4}}{β_{n+1}^{(j+1)/2}(T)} \frac{β_{n}(T)}{c^{(n)}} \\ × [ exp \{−\frac{1}{\sqrt{β_{n+1}(T)}} |2x + \frac{x^{2}}{c^{(n+1)}M_{n+1}(T)}|\} \\ + exp \{−\frac{2|x|}{\sqrt{β_{n+1}(T)}}\} ]. \end{matrix}  (4.14)

(d) For x ∈ ℝ^{1} and j = 0, 1, 2, 3, 4,

|\frac{∂^{j}}{∂x^{j}} T_{n}φ_{n}(f_{n}(x, T))| ≤ \frac{K_{3}C^{2}}{β_{n+1}^{(j+1)/2}(T)} exp \{−\frac{2|x|}{\sqrt{β_{n+1}(T)}}\} .  (4.15)

The proof of Proposition 4.2 is based on the observation that the operator T_{n} approximates the operator Q_{n,M_{n}(T)}^{c} very well, and it has a relatively simple structure. Namely, it can be written by writing the vectors v ∈ ℝ^{r−1} in formula (4.11) in spherical coordinates in the form

T_{n}φ_{n}(f_{n}(x, T)) = \frac{4}{c̄_{n+1}Γ(\frac{r−1}{2})} ∫_{0}^{∞} w^{r−2}e^{−w^{2}}

φ_{n}(f_{n}) ∗ φ_{n}(f_{n}) (\frac{2x}{c̄_{n+1}} + \frac{w^{2}}{M_{n}(T)} − \frac{r − 1}{2M_{n}(T)}, T ) dw,

where w = |v|. Then we get with the substitution \frac{w^{2}}{M_{n}(T)} = u that

T_{n}φ_{n}(f_{n}(x, T)) = \frac{2}{c̄_{n+1}Γ(\frac{r−1}{2})} ∫_{0}^{∞} M_{n}(T)(M_{n}(T)u)^{(r−3)/2}e^{−M_{n}(T)u}

φ_{n}(f_{n}) ∗ φ_{n}(f_{n}) (\frac{2x}{c̄_{n+1}} + u − \frac{r − 1}{2M_{n}(T)}, T ) du

= \frac{2}{c̄_{n+1}} φ_{n}(f_{n}) ∗ φ_{n}(f_{n}) ∗ k_{M_{n}(T)}^{−} (\frac{2x}{c̄_{n+1}} − \frac{r − 1}{2M_{n}(T)}) ,  (4.16)

where ∗ denotes convolution, k_{M_{n}(T)}^{−}(x) = k_{M_{n}(T)}(−x), and k_{M_{n}(T)}(x) =

M_{n}(T)k (M_{n}(T)x) with k(x) = \frac{1}{\sqrt{πx}}e^{−x} k(x) = \frac{x^{(r−3)/2}e^{−x}}{Γ( \frac{r−1}{2} )} for x > 0, and

k(x) = 0 for x ≤ 0.

The operator T_{n} has a certain contraction property which can be expressed in the Fourier space. the Fourier transform of T̃_{n}φ̃_{n}(f_{n}(ξ, T)) can be expressed with the help of formula (4.16). One gets that

T̃_{n}φ̃_{n}(f_{n}(ξ, T))

= exp \{i\frac{(r − 1)c̄_{n+1}}{4M_{n}(T)} ξ\} k̃ (−\frac{c̄_{n+1}ξ}{2M_{n}(T)}) [φ̃_{n} (f_{n} (\frac{c̄_{n+1}}{2} ξ, T ))]^{2} ,

= \begin{matrix}exp & \{i\frac{(r−1)c̄_{n+1}}{4M_{n}(T)} ξ\} \\ (1 + i\frac{c̄_{n+1}ξ}{2M_{n}(T)}) & (r−1)/2 \end{matrix} [φ̃_{n} (f_{n} (\frac{c̄_{n+1}}{2} ξ, T ))]^{2} .  (4.17)

In this calculation we have exploited that k(x) is the density function of the gamma distribution with parameter \frac{r−1}{2}, whose characteristic function equals (1 − iξ)^{−(r−1)/2}. It follows from formula (4.17) that T_{n}φ_{n}(f_{n}(x, T)) is the density function of a random variable with expectation zero.

The proof of Proposition 4.2 is a natural adaptation of the proof of the corresponding result (of Proposition 3) in paper [BM3]. Hence we only explain the main points and the necessary modifications.

Because of the inductive property I(n) f_{n}(x, T) is essentially concentrated in a neighbourhood of the origin of size \sqrt{β_{n}(T)}, and if (n, T) is in the low temperature domain and η > 0 is chosen sufficiently small, then \frac{|x|}{100\sqrt{c^{(n)}}} ≤

\frac{η}{10} for |x| ≤ \sqrt{β_{n}(T)}, and the function f_{n}(x, T) (disregarding the scaling with the numbers A_{n} and B_{n}) is not changing in the typical region by the regularization of the function f_{n}(x, T). This is the reason why such a regularization works well.

The proof of Proposition 4.2 contains several estimates. First we list those results whose proof apply the bound on f_{n}(x, T) formulated in the Inductive hypothesis I(n). One can bound the differences

\frac{∂^{j}}{∂x^{j}} (Q̄_{n,M_{n}(T)}^{c}f_{n}(x, T) − Q̄_{n,M_{n}(T)}^{c}φ_{n}(f_{n}(x, T))) (Lemma 4 in [BM3]),

\frac{∂^{j}}{∂x^{j}} (Q̄_{n,M_{n}(T)}^{c}φ_{n}(f_{n}(x, T)) − T̄_{n,M_{n}(T)}^{c}φ_{n}(f_{n}(x, T))) (Lemma 5 in [BM3]),

with the help of Property I(n) similarly to paper [BM3]. The absolute value of these expressions can be bounded for all ε > 0 by

\frac{β_{n}}{c^{(n)}} \frac{C_{1}(ε)C^{2}}{β_{n}^{(j+1)/2}(T)} exp \{−\frac{2(1 − ε)}{c̄_{n+1}\sqrt{β_{n}(T)}} |2x + \frac{x^{2}}{c^{(n+1)}M_{n}(T)}|\}

with some appropriate constant C_{1}(ε) > 0 if f_{n}(x, T) satisfies Condition I(n).

The main difference between these estimates and the analogous results in paper [BM3] is that the upper bounds given for the above expressions contain a small multiplying factor \frac{β_{n}(T)}{c^{(n)}}. In paper [BM3] the multiplying factors 2^{−n} and 1/c^{(n)} appear instead of this term. In the proof of this paper we had to make some modifications, because while in paper [BM3] only very low temperatures were considered when M_{n}(T) is strongly separated from zero, now we want to give an upper bound under the weaker condition formulated in the definition of the low temperature region. The proofs are very similar. The only essential difference is that in the present case the typical region, where a good asymptotic approximation must be given is chosen as the interval |x| < 10\sqrt{c^{(n)}}, i.e. it does not depend on the value of M_{n}(T).

Also the expression Q̄_{n,M_{n}(T)}^{c}f_{n}(x, T) can be bounded together with their first two derivatives with the help of Property I(n) in the same way as in Lemma 3 of paper [BM3]. But this estimate is useful only for large x. It can be proved, similarly to the proof of the corresponding result in paper [BM3] (lemma 7) that the scaling constants which appear in the formulas expressing Q_{n,M_{n}(T)}^{c} through Q̄_{n,M_{n}(T)}^{c} and T_{n} through T̄_{n,M_{n}(T)}^{c} are

very close to each other. Here again the multiplying factor \frac{β_{n}(T)}{c^{(n)}} appears in the error term instead of the multiplying factor 1/c^{(n)} in paper [BM3]. This Lemma 7 in [BM3] is a technical result which expresses the difference of the functions T̄_{n,M_{n}(T)}^{c}F_{1}(x) and T̄_{n,M_{n}(T)}^{c}F_{2}(x) together with its derivatives if we have a control on the difference of the original functions F_{1}(x) and F_{2}(x). We gain such kind of information from the inductive hypothesis I(n). They give a good control on the difference f_{n+1}(x, T) − T_{n}φ_{n}(f_{n}(x, t)). The consequences of these results are formulated in Proposition 2 in paper [BM3]. These results also imply an estimate on the Fourier transforms φ̃_{n+1}(f_{n+1}(ξ, T)) − T̃_{n}φ̃_{n}(f_{n}(ξ, T))) and T̃_{n}φ̃_{n}(f_{n}(ξ, T)) and also on their analytic continuation. This is done in lemma 8 in paper [[BM3]. Now again the analogous result holds under the conditions of the present paper with the difference that the term c^{−n} must be replaced \frac{β_{n}(T)}{c^{(n)}}. The estimate obtained for T̃_{n}φ̃_{n}(f_{n}(ξ, T)) in such a way is relatively weak, it is useful only for large ξ.

The above results are not sufficient to prove Proposition 4.2. In particular, they do not explain why the right scaling was chosen in the definition of the function f_{n}(x, T). Their role is to bound the error which is committed when Q_{n,M_{n}(T)}^{c}f_{n}(x, T) is replaced by T_{n}φ(f_{n}(x, T)). The function T_{n}φ_{n}(f_{n}(x, T)) together with its derivatives and Fourier transform can be well bounded by means of formula (4.17) and the inverse Fourier transform. In the estimations leading to such bounds the inductive hypothesis J(n) plays a crucial role. The proof of Lemma 9 in paper [BM3] can be adapted to the present case without any essential difficulty. But, the parameters α_{n}, β_{n} and c must be replaced by α_{n}(T), β_{n}(T) and c̄_{n+1} in the present case.

Proposition 4.2 can be proved similarly to its analog, Proposition 3 in paper [BM3]. The notation must be adapted to the notation of the present paper. Besides, the small coefficient c^{−n/2} appearing in the proof of Propo-

sition 3 in [BM3] must be replaced by \sqrt{\frac{β_{n}(T)}{c^{(n)}}}. There is one point where a really new argument is needed in the proof. This argument requires a more detailed discussion. It is the proof of relation (4.13), i.e. of the fact that α_{n}(T) and β_{n}(T) have the same order of magnitude. Their ratio must be bounded by a number independent of η. The proof of the analogous result in paper [BM3] exploited the fact that in the model of that paper the sequence c^{(n)} tended to infinity exponentially fast. In the present case this property does not hold any longer, hence a different argument is needed. The validity of relation (4.13) has a different cause for relatively small and large indices n.

For large n it can be shown that both β_{n}(T) and α_{n}(T) have the same order of magnitude as M_{n}^{−2}(T), and for large n these relations imply (4.13). If n is relatively small and M_{0}(T) is large, then M_{n}^{−2}(T) is much less than α_{n}(T) and β_{n}(T). In this case the above indicated argument does not work, but it can be proved that for such indices n the numbers β_{n}(T) are decreasing exponentially fast, and the proof of relation (4.13) for such n is based on this fact.

To distinguish between small and large indices n define the number

N_{1}(T) = \{min n: n ≥ N, and β_{n+1}(T) ≤ \frac{100}{M_{n}^{2}(T)}\} ,
(N_{1}(T) = ∞ if there is no such n). (4.18)

where the number N was defined in formula (1.12). We shall later see that N_{1}(T) < ∞ for all 0 < T ≤ c_{0}A_{0}/2.

First we prove relation (4.13) under the additional condition n ≤ N_{1}(T).

In this case β_{m+1}(T) ≤ \frac{c_{m+1}^{2}}{2} β_{m}(T) + \frac{β_{m}(T)}{10} for m ≤ n, and because of Condition 1

β_{m+1}(T) ≤ \frac{2}{3}β_{m}(T) if m ≤ N_{1}(T)  (4.19)

for all N ≤ m ≤ n.

Hence \sqrt{\frac{β_{m+1}(T)}{c^{(m+1)}}} ≤ \frac{5}{6} \sqrt{\frac{β_{m}(T)}{c^{(m)}}}, \sqrt{\frac{β_{m}(T)}{c^{(m)}}} ≤ (\frac{5}{6})^{m−N}\sqrt{\frac{β_{N} (T)}{c^{(N)}}},

1 ≤ \frac{β_{m+1}(T)}{α_{m+1}(T)} ≤ max ( \frac{\frac{c{2 \atop m+1}}{2} +\sqrt{\frac{βm(T)}{c^{(m)}}}}{\frac{c{2 \atop m+1}}{2} −\sqrt{\frac{βm(T)}{c^{(m)}}}} · \frac{β_{m}(T)}{α_{m}(T)}, 10^{13})

≤ max (exp \{5\sqrt{\frac{β_{m}(T)}{c^{(m)}}}\} · \frac{β_{m}(T)}{α_{m}(T)}, 10^{13})

for N ≤ m ≤ n, and

\frac{β_{n+1}(T)}{α_{n+1}(T)} ≤ max (\frac{β_{N}(T)}{α_{N}(T)}, 10^{13}) exp \{5 ∑_{m=N}^{n} \sqrt{\frac{β_{m}(T)}{c^{m}(T)}}\} ≤ K.

The above argument together with the observation that β_{N}(T) ≫ M_{N}^{−2}(T) if the parameter t > 0 in (1.3) is sufficiently small and T ≤ c_{0}A_{0}/2 imply that N < N_{1}(T), and the pair (n, T) is in the low temperature region for all n ≤ N_{1}(T). The latter property follows from the fact that by formula (4.19) the sequence \frac{β_{n}(T)}{c^{(n)}} is monotone decreasing for N ≤ n ≤ N_{1}(T).

In the case n > N_{1}(T) we can prove by induction with respect to n together with the inductive proof of Proposition 4.2 that

β_{n+1}(T) ≤ \frac{100}{M_{n}^{2}(T)} if n ≥ N_{1}(T).
and (n, T) is in the low temperature region. (4.20)

By applying formula (4.20) for n − 1 and the fact that (n, T) is in the low temperature region we get that the term γ_{n−1}(T) in formula (4.12) can be bounded as

|γ_{n−1}(T)| ≤ \frac{β_{n}(T)}{c^{(n)}} \sqrt{β_{n}(T)} ≤ η \frac{10}{M_{n−1}(T)} ≤ \frac{1}{8C_{1}M_{n−1}(T)}  (4.21)

with the same number C_{1} which appears in (4.12) if the number η > 0 was chosen sufficiently small. Then formula (4.12) implies that M_{n}(T) ≤ M_{n+1}(T). Hence we get by applying again formula (4.20) with n − 1 that M_{n}(T) < M_{n−1}(T), and

β_{n+1}(T) ≤ \frac{2}{3}β_{n}(T) + \frac{10}{M_{n}^{2}(T)} ≤ \frac{200}{3M_{n−1}^{2}(T)} + \frac{10}{M_{n}^{2}(T)} ≤ \frac{100}{M_{n}^{2}(T)}.

This means that formula (4.20) also holds for n. Relation (4.20) together with the definition of the sequence α_{n}(T) imply that for n ≥ N_{1}(T)

α_{n+1}(T) ≥ \frac{10^{−12}}{M_{n}^{2}(T)} ≥ 10^{−14}β_{n}(T),

i.e. formula (4.13) is also valid for n > N_{1}(T) if (n, T) is in the low temperature domain. With the help of this argument Proposition 4.2 can be proved by an adaptation of the proof of the corresponding result in [BM3].

We formulate and prove a lemma which describes some properties of the numbers β_{n}(T) in the cases when n ≤ N_{1}(T) or n ≥ N_{1}(T). Several parts of it were already proved in the previous arguments.

Lemma 4.3. Let 0 < T ≤ c_{0}A_{0}/2. If the parameter κ > 0 in formula (1.3) is sufficiently small, then the following statements are valid:

(1) The number N_{1}(T) defined in (4.18) is finite, and N_{1}(T) > N.

(2) The pair (N_{1}(T), T) is in the low temperature region.

(3) The relations (4.19), (4.20) hold.

(4) If n ≥ N_{1}(T) and (n, T) is in the low temperature region then

M_{n}(T) − \frac{3}{8c^{(n)}M_{n}(T)} ≤ M_{n+1}(T) ≤ M_{n}(T) − \frac{1}{8c^{(n)}M_{n}(T)}.  (4.22)

(5) If N ≤ n ≤ N_{1}(T) then

\begin{matrix}M_{n}(T) − \frac{1}{4c^{(n)}M_{n}(T)} − η (\frac{2}{3})^{(n−N)/2} ≤ M_{n+1}(T) \\ ≤ M_{n}(T) − \frac{1}{4c^{(n)}M_{n}(T)} + η (\frac{2}{3})^{(n−N)/2} . \end{matrix}  (4.23)

(6) We have that

N_{1}(T) − N ≤ 10 log(1/κT^{2}).  (4.24)

(7) If M_{n}(T) < 10 then n ≥ N_{1}(T).

Proof of Lemma 4.3. Formulas (4.19) and (4.20) were already proved in the previous argument, and since (N, T) is in the low temperature region, i.e. β_{N}(T) ≥ ηc^{N}, relation (4.19) implies that (n, T) is in the low temperature region for all N ≤ n ≤ N_{1}(T). Formula (4.22) follows from formula (4.21) with the replacement of n − 1 by n and formula (4.12). By relation (4.19) β_{n}(T) ≤ (\frac{2}{3})^{n−N} if N ≤ n ≤ N_{1}(T). Hence it follows from (4.12) that

M_{n+1}(T) ≤ M_{n}(T)+\frac{β_{n+1}(T)}{c^{(n)}} \frac{\sqrt{β_{n+1}(T)}}{c^{(n)}} ≤ M_{n}(T)+η (\frac{2}{3})^{(n−N)/2} , (4.25)

and even relation (4.23) holds in this case.

Relation (4.25) and the estimate obtained for β_{n}(T) imply that M_{n}^{2}(T) ≤ (M_{N}(T) + 1)^{2} ≤ 2M_{N}^{2}(T) and β_{n+1}(T)M_{n}^{2}(T) ≤ 2M_{N}^{2}(T) (\frac{2}{3})^{n−N} if n ≤ N_{1}(T). This relation together with the definition of the index N_{1}(T) defined in (4.18) imply that 2M_{N}^{2}(T) (\frac{2}{3})^{n−N} ≥ 100 if n < N_{1}(T). Applying the last formula for n = N_{1}(T) − 1 we get that (N_{1}(T) − 1 − N)) log \frac{3}{2} ≤

log \frac{M_{N}^{2} (T)}{50}. Since M_{N}^{2}(T) ∼ const. \frac{1}{κT^{2}} this relation implies that N_{1}(T) is finite, and moreover it satisfies (4.24). Finally, if the inequalities M_{n}(T) ≤ 10 and n < N_{1}(T) held simultaneously, then the inequality M_{n}^{2}(T)β_{n+1}(T) ≤ 100 (\frac{2}{3})^{n−N} ≤ 100 would also hold. This relation contradicts to the assumption n < N_{1}(T). Hence also the last statement of Lemma 4.3 holds. □

The previous results enable us to describe the different behaviour of the model in the cases when the Dyson condition (1.2) is satisfied and when it is not. This will be done in Lemma 4.4. It shows that if (1.2) is not satisfied then for all T there is a pair (n, T) which does not belong to the low temperature region, while if (1.2) is satisfied, then all sufficiently low temperatures T belong to the low temperature region. In the latter case the asymptotic behaviour of the spontaneous magnetization M_{n}(T) can be described for large n. The description of the behaviour of the function q_{n}(x, T) in the case when T does not belong to the low temperature region needs further investigation, and this will be done in Sections 5 and 6. A more detailed investigation of the case when T belongs to the low temperature region will be done in Section 7. We finish this section with the proof of a result about the behaviour of the magnetization M_{n}(T) at low temperatures T > 0 which will be useful in the subsequent part of the paper.

Lemma 4.4. Let 0 < T ≤ c_{0}A_{0}/2, and let the parameter κ > 0 be sufficiently small. If the Dyson condition (1.2) is not satisfied, then for all T > 0 there is some n = n(T) for which (n, T) does not belong to the low temperature region. If, on the other hand, condition (1.2) is satisfied, then T belongs to the low temperature region for sufficiently small T > 0. In this case relation (3.6) and, under the additional Condition 2, also relation (3.7) hold.

Proof of Lemma 4.4. It follows from formulas (4.22) and (4.23) that

−\frac{1}{c^{(n)}} ≤ M_{n+1}^{2}(T) − M_{n}^{2}(T) ≤ −\frac{1}{8c^{(n)}}  (4.26)

if n ≥ N_{1}(T) and the pair (n, T) is in the low temperature region, and

\begin{matrix}− \frac{1}{2c^{(n)}} − 10 (\frac{2}{3})^{n−N} (M_{N}(T) + 1) ≤ M_{n+1}^{2}(T) − M_{n}^{2}(T) \\ ≤ −\frac{1}{2c^{(n)}} + 10 (\frac{2}{3})^{n−N} (M_{N}(T) + 1) \end{matrix}  (4.27)

if N ≤ n ≤ N_{1}(T). Formula (4.26) can be obtained by taking square in formula (4.22) and observing that c^{(n)}M_{n}(T)^{2} > 10η^{−1}. Formula (4.27) can be deduced similarly from (4.23) by observing first that the right-hand side of (4.23) implies that M_{n}(T) ≤ M_{N}(T) + 1 for N ≤ n ≤ N_{1}(T).

Formulas (4.26) and (4.27) imply that if a temperature T > 0 is in the low temperature region, then

∑_{k=N}^{n} \frac{1}{c^{(n)}} ≤ 8(M_{N}^{2}(T)−M_{n}^{2}(T))+30(M_{N}(T)+1) ≤ 8M_{N}^{2}(T)+30(M_{N}(T)+1)

for all n ≥ N, where the number N is defined in (1.12). Since the right-hand side of the last formula does not depend on n, this implies that (1.2) holds.

In the other direction, if (1.2) holds, then since by Proposition 4.1

lim_{T →∞} M_{0}(T) = lim_{T →∞} M_{N}(T) = ∞,

there is some number T̄ ≤ c_{0}A_{0}/2 such that for all temperatures 0 < T ≤ T̄

M_{N}^{2}(T) > 8 ∑_{n=N}^{∞} \frac{1}{c^{(n)}} + 30M_{n}(T) + 31. If T > 0 satisfies the above inequality,

then the left-hand side of the inequalities (4.26) and (4.27) imply that if the pair (n, T) is in the low temperature domain and n ≥ N_{1}(T), then

M_{n}^{2}(T) > M_{N}^{2}(T) − 8 ∑_{n=N}^{n} \frac{1}{c^{(n)}} 30(M_{n}(T) + 1)) ≥ 1.

Hence M_{n}^{2}(T) > 1 for all n, and T is in the low temperature region.

Let T > 0 be in the low temperature region. If n > m > N_{1}(T), then by (4.26)

|M_{n}^{2}(T) − M_{m}^{2}(T)| ≤ ∑_{k=m}^{n} \frac{1}{c^{(k)}} .

Since in this case Condition 1 holds, the last relation implies that M_{n}^{2}(T), n = 1, 2, …, is a Cauchy sequence, and relation (3.6) holds. We claim that if Condition 2 also holds, then for any ε > 0

−\frac{r − 1 + ε}{2c^{(n)}} ≤ M_{n+1}^{2}(T) − M_{n}^{2}(T) ≤ −\frac{r − 1 − ε}{2c^{(n)}}  (4.28)

if n ≥ n(ε). Relation (3.7) is a consequence of (4.28). Relation (4.28) can be deduced from (4.12) and (4.20) if we show that for any temperature T > 0 in the low temperature region

lim_{n→∞} \frac{β_{n}(T)}{c^{(n)}} = 0.  (4.29)

Relation (4.29) holds under Condition 2, since by (4.26) in this case for all n > N_{1}(T),

M_{n}^{2}(T) ≥ lim_{k→∞} (M_{n}^{2}(T) − M_{k}^{2}(T)) ≥ \frac{1}{8} ∑_{k=n}^{∞} \frac{1}{c^{(k)}} ,

and

\frac{β_{n}(T)}{c^{(n)}} ≤ \frac{100}{M_{n−1}^{2}(T)c^{(n)}} ≤ 800 (c^{(n)} ∑_{k=n−1}^{∞} \frac{1}{c^{(k)}} )^{−1} .

Under Condition 2 the last expression tends to zero as n → ∞. This implies formula (4.28). Lemma 4.4 is proved. □

5. Estimates in the Intermediate Region. The proof of Theorem 3.1.

In this section we give some estimates on q_{n}(x, T) when the pair (n, T) belongs neither to the low nor to the high temperature region and prove Theorem 3.1 with their help.

Let us consider the number n̄ = n̄(T) introduced in the formulation of Theorem 3.1, namely

n̄(T) = min{n: D_{n}^{2}(T) < e^{−1/η^{2}}}.

In Lemmas 5.1 and 5.2 we shall prove some estimates about a scaled version of the function q_{n̄(T)}(x, T), where q_{n}(x, T) was defined in (2.7). In Lemma 5.1 the case T ≤ c_{0}A_{0}, and in Lemma 5.2 the case T ≥ c_{0}A_{0} will be considered. Lemmas 5.1 and 5.2 yield some estimates on the tail-behaviour of a scaled version of the function q_{n̄(T)}(x, T). This will be needed to start an inductive procedure for all n ≥ n̄(T) which state that the functions q_{n}(x, T) become more and more strongly concentrated around zero as the index n is increasing. This procedure is based on Lemmas 5.3 and 5.4. The role of Lemma 5.3 is to give an appropriate lower bound for the norming constant Z_{n}(T) in the definition of the function q_{n}(x, T). Then in Lemma 5.4 we prove some contraction property of the operator which maps an appropriate scaled version of the distribution function with density function const. q̄_{n−1}(|x|, T) to an appropriate scaled version of the distribution function with density const. q̄_{n}(|x|, T), x ∈ ℝ^{r}. The proof of Lemma 5.4 will exploit the rotation symmetry of the model. Theorem 3.1 will be proved by means of these lemmas.

To formulate these results we introduce some notations. Let us introduce the functions

ĥ_{n}(x, T) = (c^{(n̄(T))})^{−r/2} q_{n} (\frac{x}{\sqrt{c^{n̄(T)}}} , T ) , x ∈ ℝ^{r},  (5.1)

and measures

Ĥ_{n,T} (A) = ∫_{A} ĥ_{n}(x, T) dx, A ⊂ ℝ^{r},  (5.2)

in the space ℝ^{r}. Define also the function

Ĥ_{n,T} (R) = Ĥ_{n,T} ({x: |x| ≥ R}) for R ≥ 0.  (5.3)

The functions ĥ_{n,T} and measures Ĥ_{n,T} are similar to the functions h_{n,T} and measures H_{n,T} defined in (3.4) and (3.5). The only difference is that the scaling of q_{n}(x, T) in (5.2) and (5.3) is made by means of c^{(n̄(T))} instead of c^{(n)}. If Condition 5 is satisfied with a sufficiently small η̄ and sufficiently large L(η̄, T), and n − n̄(T) is not too large, then the approximation of c^{(n)} by c^{(n̄(T))} is sufficiently good for our purposes. Hence it will be enough to have a good control on the measure Ĥ_{n,T}. In Lemma 5.3 we give a bound on it for large |x| and in Lemma 5.4 we prove an estimate which enables to bound Ĥ_{n,T} (R) for small R too.

With the help of these results we can prove that starting from n̄ = n̄(T) after finitely many steps k the pair (n̄ + k, T) is in the high temperature region. Moreover, this number k can be bounded from above independently of the temperature T. First we formulate Lemma 5.1.

Lemma 5.1. Under the conditions of Proposition 4.2, the function h_{n̄(T)}(x, T) defined in (3.4) satisfies the inequality

h_{n̄(T)}(x, T) ≤ exp \{\frac{K}{η} − \frac{|x|^{2}}{10} \} if T ≤ c_{0}A_{0}/2  (5.4)

with an appropriate K > 0. For T ≤ c_{0}A_{0}/2 the pair (n̄(T), T) does not belong to the high temperature region, and there exists some η̃ = η̃(η) such that the function Ĥ_{n,T} (·) defined in (5.3) satisfies the inequality

Ĥ_{n̄(T),T} (η̃^{−1}) ≤ 1/2, if T ≤ c_{0}A_{0}/2,  (5.5)

i.e. for T ≤ c_{0}A_{0}/2 there is a ball with its center in the origin whose radius depends only on η, and whose Ĥ_{n̄(T),T} measure is greater than 1/2.

Proof of Lemma 5.1. Let us introduce the function

h̄_{n}(x, T) = \frac{1}{\sqrt{c^{(n)}}} q̄_{n} (\frac{x}{\sqrt{c^{(n)}}} , T ) , x ≥ 0

with the function q̄_{n} introduced in (2.11). Observe that

∫_{0}^{∞} h̄_{n}(x, T) dx = 1.

Let us apply Proposition 4.2 with the choice n = n̄(T) − 1. Since hypothesis I(n) holds for n = n̄, we obtain that

f_{n̄(T)}(x, T) ≤ \frac{K}{β_{n̄(T)−1}^{1/2}(T)} exp \{−\frac{1}{\sqrt{β_{n̄(T)}(T)}} |2x + \frac{x^{2}}{c^{(n̄(T))}M_{n̄(T)}(T)}|\}

if x > −c^{(n̄(T))}M_{n̄(T)}(T)

with some universal constant K > 0. It follows from this relation that the function h̄_{n̄(T)}(x, T) = \sqrt{c^{(n̄(T))}}f_{n̄(T)} (\sqrt{c^{(n̄(T))}}x − c^{(n̄(T))}M_{n}(T)), T ) satisfies the inequality

h̄_{n̄(T)}(x, T) ≤ K (\frac{c^{(n̄(T))}}{β_{n̄(T)−1}(T)})^{1/2}

exp \{\frac{1}{\sqrt{β_{n̄(T)}(T)}} (c^{(n̄(T))}M_{n̄(T)}(T) − \frac{x^{2}}{M_{n̄(T)}(T)})\} .

The inequalities β_{n̄(T)}(T) > ηc^{(n̄(T))} and β_{n̄(T)−1}(T) ≤ ηc^{(n̄(T)−1)} hold.

Lemma 4.3 implies that the fractions \frac{β_{n̄(T)}(T)}{β_{n̄(T)−1}(T)}, \frac{M_{n̄(T)}(T)}{M_{n̄(T)−1}(T)} and

β_{n̄(T)}(T)M_{n̄(T)}(T)^{2} are separated both from zero and infinity, hence

\frac{c^{(n̄(T))}}{β_{n̄(T)−1}(T)} ≤ \frac{const.}{η} ,

\frac{c^{(n̄(T))}M_{n̄(T)}(T)}{\sqrt{β_{n̄(T)}(T)}} ≤ \frac{const.}{η}

and

\frac{1}{M_{n̄(T)}(T)\sqrt{β_{n̄(T)}(T)}} ≥ \frac{1}{20}.

These inequalities together with the last relation imply that

h̄_{n̄(T)}(x, T) ≤ e^{K̄/η}e^{−x^{2}/20} x ≥ 0,  (5.6)

with an appropriate K̄ > 0. Since

h_{n̄(T)}(x, T) = C(T)h̄_{n̄(T)}(|x|, T), x ∈ ℝ^{r},  (5.7)

with an appropriate number C(T) > 0, estimate (5.4) can be deduced from (5.6) if we give a good upper bound for the constant C(T) in (5.7). Observe that because of (5.7)

C(T)^{−1} = ∫_{ℝ^{r}} h̄_{n̄(T)}(|x|, T) dx = Vol(S^{r−1}) ∫_{0}^{∞} x^{r−1}h̄_{n̄(T)}(x, T) dx, (5.8)

hence

C(T)^{−1} = Vol(S^{r−1}) ∫_{0}^{∞} x^{r−1}h̄_{n̄(T)}(x, T) dx
≥ Vol(S^{r−1})R^{r−1} (1 − ∫_{0}^{R} h̄_{n̄(T)}(x, T) dx)

for any R > 0. On the other hand, by formula (5.6)

∫_{0}^{R} h̄_{n̄(T)}(x, T) dx ≤ \frac{1}{2}

if 0 < T ≤ c_{0}A_{0}/2 and R ≤ e^{−K/η} with a sufficiently large K > 0. Hence C(T)^{−1} ≥ \frac{1}{2}Vol(S^{r−1})e^{−K(r−1)}. This means that C(T) ≤ e^{Kr/η} in (5.7), and inequality (5.4) follows from (5.6).

We shall prove that n̄(T) does not belong to the high temperature region with the help of the following estimate on D_{n̄(T)}^{2}(T). In its proof we shall apply formula (5.8).

D_{n̄(T)}^{2}(T) = ∫_{ℝ^{r}} |x|^{2}h_{n̄(T)}(x, T) dx = Vol(S^{r−1}) ∫_{0}^{∞} x^{r+1}h_{n̄(T)}(x, T) dx
= C(T)Vol(S^{r−1}) ∫_{0}^{∞} x^{r+1}h̄_{n̄(T)}(x, T) dx

≥ C(T)Vol(S^{r−1}) (∫_{0}^{∞} x^{r−1}h̄_{n̄(T)}(x, T) dx)^{(r+1/(r−1)}

= (∫_{0}^{∞} x^{r−1}h̄_{n̄(T)}(x, T) dx)^{2/(r−1)}

≥ (∫_{0}^{∞} xh̄_{n̄(T)}(x, T) dx)^{2} = (M_{n̄(T)}\sqrt{c^{(n̄(T))}})^{2} .

Since M_{n}^{2} ≥ \frac{10}{β_{n+1}} if n ≥ N + 1, N + 1 is in the low temperature region if T ≤ c_{0}A_{0}/2, (see (3.3) and the subsequent sentence in our discussion), and \frac{M_{n̄(T)}}{M_{n̄(T)−1}} ≤ const., hence

D_{n̄(T)}^{2}(T) ≥ M_{n̄(T)}^{2}c^{(n̄(T))} ≥ const.\frac{c^{(n̄(T))}}{β_{n̄(T)}} ≥ \frac{const.}{η}.

This implies that n̄(t) is not in the high temperature region. □

In the next Lemma 5.2 we shall formulate some properties of the function h_{n}(x, T) defined in (3.4) in the case n = 0. For the sake of a better discussion we define the function h̄_{n}(x, T), x ≥ 0, by the formula h̄_{n}(x, T) = h_{n}(|x|, T), and from now on h̄_{n}(x, T) means this function. (It differs slightly from the function h̄_{n}(x, T) applied in the proof of Lemma 5.1 where a different norming constant was applied.)

If T ≥ c_{0}A_{0}/2, then n̄(T) = 0, and

h̄_{n̄(T)}(|x|, T) = const. q_{0} ((c^{(0})^{−1/2}x, T ) ,

where q_{0}(x, T) is defined in (2.10). Hence

h̄_{n̄(T)}(x, T) = \frac{1}{Z_{0}(T)} exp \{(\frac{A_{0}c_{0} − T}{c^{(0)}} ) \frac{x^{2}}{2} − κT^{2} \frac{x^{4}}{4(c^{(0)})^{2}} \} if T ≥ c_{0}A_{0}/2

(5.9)

with the norming constant (for the function h_{n̄(T)}(x, T))

Z_{0}(T) = Vol(S^{r−1}) ∫_{0}^{∞} x^{r−1} exp \{(\frac{A_{0}c_{0} − T}{c^{(0)}} ) \frac{x^{2}}{2} − κT^{2} \frac{x^{4}}{4(c^{(0)})^{2}} \} dx.

(5.10)

Using formulas (5.9) and (5.10), we will prove the following

Lemma 5.2. There is a constant κ_{0} = κ_{0}(N) > 0 such that if 0 < κ < κ_{0}, Condition 1 is satisfied, and T ≥ c_{0}A_{0}/2, then n̄(T) = 0, and

h̄_{n̄(T)}(x, T) ≤ const. T^{r/2} exp \{\frac{1}{κ} − \frac{T}{2c^{(0)}} x^{2}\} if T ≥ \frac{c_{0}A_{0}}{2} . (5.11)
h̄_{n̄(T)}(x, T) ≤ const. T^{r/2} exp \{\frac{T x^{2}}{2c^{(0)}} \} if T ≥ \frac{c_{0}A_{0}}{2} .  (5.12)
h̄_{n̄(T)}(x, T) ≤ const. e^{−T x^{2}/4} if T ≥ 10A_{0}, and x ≥ T^{−1/3}. (5.13)

The const. in formulas (5.11)—(5.13) depend only on the dimension r of the model.

The pair (n̄(T), T) belongs to the high temperature region if T is very large, e.g. if T ≥ e^{−1/η^{9}}, and it does not belong to it if T > 0 is relatively small, e.g. if T ≤ η^{−100}. If (n̄(T), T) does not belong to the high temperature region, then the function h_{n̄(T)}(x, T) defined in formula (3.4) satisfies the inequality

h_{n̄(T)}(x, T) ≤ exp{K(η, κ) − α|x|^{2}}  (5.14)

with a constant α = α(η) > 0 and an appropriate number K(η, κ) depending only on κ and η. In this case there is a constant B = B(η, κ) > 0 in such a way that the quantity Ĥ_{n̄(T),T} (·) defined in (5.3) satisfies the inequality

Ĥ_{n̄(T),T} (B) ≤ \frac{1}{2}.  (5.15)

This means that if the pair (n̄(T), T) is not in the high temperature region (and T ≥ c_{0}A_{0}/2), then there is a radius B = B(η, κ) such that the Ĥ_{n̄(T),T} measure of the ball {x: |x| ≤ B(η, κ)} is bigger than 1/2.

If (n̄(T), T) = (0, T) is in the high temperature region, then

Ĥ_{n̄(T),T} (x) ≤ K_{1}e^{−K_{2}η^{2}x^{2}} for all x > 0  (5.16)

with some universal constants K_{1} > 0 and K_{2} > 0.

Proof of Lemma 5.2. First we estimate the norming factor Z_{0}(T) from below. Let us observe that

\frac{A_{0}c_{0} − T}{c^{(0)}} \frac{x^{2}}{2} − κT^{2} \frac{x^{4}}{4(c^{(0)})^{2}} ≥ −T x^{2} (\frac{1}{2c^{(0)}} + \frac{1}{4(c^{(0)})^{2}} ) ≥ −10T x^{2},

if κT x^{2} ≤ 1 and Condition 1 holds. Hence

Z_{0}(T) ≥ Vol (S^{r−1}) ∫_{0}^{1/\sqrt{κT}} x^{r−1}e^{−10T x^{2}} dx  (5.17)
= Vol (S^{r−1}) ∫_{0}^{1/\sqrt{κ}} \frac{x^{r−1}e^{−10x^{2}}}{T^{r/2}} dx ≥ const.T^{−r/2}.

Now, if T ≥ c_{0}A_{0}/2, then

\frac{A_{0}c_{0} − T}{c^{(0)}} \frac{x^{2}}{2} − κT^{2} \frac{x^{4}}{4(c^{(0)})^{2}} ≤ −\frac{1}{2} \frac{T x^{2}}{c^{(0)}} + max_{x: x≥0} (\frac{T x^{2}}{c^{(0)}} − \frac{κ}{4} (\frac{T x^{2}}{c^{(0)}} )^{2})

= −\frac{T}{2c^{(0)}} x^{2} + \frac{1}{κ},

and combining this with (5.17), we obtain (5.11). The estimate \frac{A_{0}c_{0}−T}{c^{(0)}} \frac{x^{2}}{2} − κT^{2} \frac{x^{4}}{4(c^{(0)})^{2}} ≤ \frac{T x^{2}}{2c^{(0)}} yields (5.12).

If T ≥ 10A_{0}, then

\frac{A_{0}c_{0} − T}{c^{(0)}} \frac{x^{2}}{2} − κT^{2} \frac{x^{4}}{4(c^{(0)})^{2}} ≤ −\frac{T}{3} x^{2} ≤ −\frac{T}{4} x^{2} − \frac{T^{1/3}}{12} , for |x| ≥ T^{−1/3},

which together with (5.17) imply inequality (5.13).

Furthermore, (5.13) implies that if T > e^{−1/η^{9}}, then the pair (0, T) belongs to the high temperature region. Indeed, if we estimate the integral expressing D_{0}^{2}(T), then by this relation the contribution of the domain {x: |x| ≥ T^{−1/3}} to this integral is very small. On the other hand, the contribution of the domain {x: |x| ≤ T^{−1/3}} is less than T^{−2/3} which is also very small in this case. To see that for T < η^{−100} the pair (0, T) does not belong to the high temperature domain it is enough to observe that in this case by formula (5.12) the H_{0,T} measure of the ball {x: |x| ≤ η^{100}} is less than const. T^{r/2}η^{100r} ≤ 1/2. Hence in this case D_{0}^{2}(T) ≥ \frac{1}{2}η^{200}. (We get this estimate by restricting the integral expressing D_{0}^{2}(T) to the domain {x: |x| ≥ η^{100}}). This means that T is not in the high temperature region.

Inequality (5.11) together with the fact that if the pair (0, T) does not belong to the high temperature region then T ≤ e^{−1/η^{9}} imply relations (5.14) and (5.15).

Since T > η^{−100} if the pair (0, T) is in the high temperature region, relation (5.13) implies relation (5.16). Lemma 5.2 is proved. □

To formulate Lemmas 5.3 and 5.4 we rewrite formula (2.9) for the functions ĥ_{n}(x, T) defined in (5.1). It has the form

ĥ_{n+1}(x, T) = \frac{1}{Z_{n}(T)} ∫_{ℝ^{r}} exp \{−\frac{c^{(n)}}{c^{(n̄(T))}} u^{2}\} ĥ_{n}(x − u, T)ĥ_{n}(x + u, T) du

(5.18)

with

Z_{n}(T) = ∫_{ℝ^{r}×ℝ^{r}} exp \{−\frac{c^{(n)}}{c^{(n̄(T))}} u^{2}\} ĥ_{n}(x − u, T)ĥ_{n}(x + u, T) du dx (5.19)

for all n ≥ n̄(T).

Let us also introduce the moment generating function of the measures Ĥ_{n,T}, defined in (5.2):

φ_{n,T} (u) = ∫_{ℝ^{r}} e^{ux}ĥ_{n,T} (x) dx, u ∈ ℝ^{r},

where ux denotes scalar product. By studying the properties of the moment generating function φ_{n,T} (u), we get an upper bound for the function Ĥ_{n,T} (R) for large values R. Namely, we have the following result:

Lemma 5.3. There exists some κ_{0} = κ_{0}(N) such that for all 0 < κ < κ_{0} and 0 < η̄ < η (with the numbers N and η defined in (1.12)) the following relations hold. If we have such a positive integer L for which Conditions 1 and 5 (with η̄ and this number L) are satisfied, then for all such temperatures T > 0 for which the number n̄(T) exists, and the pair(n̄(T), T) does not belong to the high temperature region, the inequality

Ĥ_{n̄(T)+l,T} (R) ≤ e^{−2^{l}αR^{2}/5r} if R ≥ D and 0 ≤ l ≤ L  (5.20)

holds with appropriate constants α > 0 and D > 0, and also the norming factor Z_{n}(T) in (5.19) can be estimated as

Z_{n̄(T)+l}(T) ≥ D_{1} for 0 ≤ l ≤ L  (5.21)

with some constant D_{1} > 0. These constants can be chosen as some functions of κ and η̄, i.e. α = α(κ, η̄), D = D(κ, η̄) > 0 and D_{1} = D_{1}(κ, η̄) > 0. This means in particular that they do not depend on the temperature T.

Proof of Lemma 5.3. It follows from formulas (5.4) and (5.14) that

φ_{n̄(T),T} (u) ≤ exp \{K_{0} + \frac{u^{2}}{α} \} for all u ∈ ℝ^{r}

with some K_{0} = K_{0}(η, κ) > 100 and α = α(η) > 0. It can be seen by induction with respect to l that

φ_{n̄(T)+l,T} (u) ≤ exp \{2^{l}K_{l} + \frac{u^{2}}{2^{l}α}\} for all 0 ≤ l ≤ L and u ∈ ℝ^{r} (5.22)

with some K_{0} > 0 and

K_{l} = K_{l−1} − \frac{log Z_{n̄(T)+l−1,T}}{2^{l}} , 1 ≤ l ≤ L.  (5.23)

Indeed, the function ĥ_{n̄(T)+l+1,T} (x, T) is increased if the kernel term

exp \{−\frac{c^{(n)}}{c^{(n̄(T))}} u^{2}\} is omitted from the integral in (5.18), and the integral

turns into the convolution 2ĥ_{n̄(T)+l,T} ∗ ĥ_{n̄(T)+l,T} (2x) after this change. By computing this convolution with the help of the inductive hypothesis and dividing it by Z_{n̄(T)+l+1} we get an upper bound for φ_{n̄(T)+l+1,T} (u). Formulas (5.22) and (5.23) follow from these calculations. We will prove formulas (5.20) and (5.21) from these relations by induction for l together with the inductive hypothesis that

K_{l} ≤ B for all 0 ≤ l ≤ L  (5.24)

with some constants B > 10 depending only on κ and η̄.

Observe that formula (5.22) with the choice of vectors of the form (u, 0), u ∈ ℝ^{1}, u > 0, 0 ∈ ℝ^{r−1}. implies that the function Ĥ_{n̄(T)+l,T} (R) defined in formulas (5.2) and (5.3) satisfies the inequality

Ĥ_{n̄(T)+l,T} (R) ≤ rĤ_{n̄(T)+l,T} (\{x = (x_{1}, x_{2}) ∈ ℝ^{r}, x_{1} > \frac{R}{\sqrt{r}} )
≤ r exp \{−\frac{uR}{\sqrt{r}} + 2^{l}K_{l} + \frac{u^{2}}{2^{l}α}\}

for all real numbers u. In particular,

Ĥ_{n̄(T)+l,T} (R) ≤ r exp \{2^{l} (K_{l} − \frac{R^{2}α}{4r})\}  (5.25)

with the choice u = \frac{2^{l}Rα}{2\sqrt{r}}. Hence

Ĥ_{n̄(T)+l,T} (\sqrt{\frac{4r(r + 1)B}{α}} ) ≤ re^{−2^{l}rB} ≤ \frac{1}{2}  (5.26)

with the number B > 0 appearing in (5.24). Formula (5.26) implies that

Ĥ_{n̄(T)+l,T} (\{x: x ∈ ℝ^{r}, |x| ≤ \sqrt{\frac{4r(r + 1)B}{α}} \}) ≥ \frac{1}{2}.

For z ∈ ℝ^{r} and u > 0 let K(z, u) = {x: x ∈ ℝ^{r}, |x − z| ≤ u} denote the ball

with center z and radius u. Since the ball \{x: x ∈ ℝ^{r}, |x| ≤ \sqrt{\frac{4r(r+1)B}{α}} \}

can be covered by C(r)B(αη̄)^{−1} balls of radius \sqrt{η̄}, where C(r) > 0 depends only on r, there is a ball K (z, \sqrt{η̄}) of radius \sqrt{η̄} whose Ĥ_{n,T} measure (this measure was defined in (5.2)) is greater than \frac{αη̄}{C(r)B}. Hence

Ĥ_{n̄(T)+l,T} × Ĥ_{n̄(T)+l,T} (K(z, \sqrt{η̄}) × K(z, \sqrt{η̄})) ≥ \frac{α^{2}η̄^{2}}{(C(r)B)^{2}} ,

and because of Condition 5 the expression Z_{n}(T) defined in (5.19) can be estimated for n = n̄(T) + l as follows:

Z_{n̄(T)+l}(T) ≥ 2^{−r} ∫_{x∈K(z,\sqrt{η̄}), u∈K(z,\sqrt{η̄})} exp \{−\frac{c^{(n̄(T)+l)}}{c^{n̄(T)}} \frac{(x − u)^{2}}{4} \}

× ĥ_{n̄(T)+l,T} (x)ĥ_{n̄(T)+l,T} (u) dx du

≥ e^{−5}Ĥ_{n̄(T)+l,T} × Ĥ_{n̄(T)+l,T} (K (z, \sqrt{η̄}) × K (z, \sqrt{η̄}))

≥ e^{−5} \frac{α^{2}η̄^{2}}{(C(r)B)^{2}} .

In the above estimation we have exploited that because of Condition 5 and Condition 1 \frac{c^{(n̄(T)+l)}}{c^{n̄(T)}} ≤ \frac{5}{η̄}, hence \frac{c^{(n̄(T)+l)}}{c^{n̄(T)}} \frac{(x−u)^{2}}{4} ≤ 5 if x ∈ K (z, \sqrt{η̄}) and u ∈ K (z, \sqrt{η̄}).

The last relation implies (5.21) with D_{1} = \frac{e^{−5}α^{2}η̄^{2}}{(C(r)B)^{2}} , although we still have to show that the number D_{1} (depending on B) can be bounded by a number which does not depend on the parameter T. We show with the help of (5.21), (5.23) and the inductive hypothesis (5.24) that K_{l} ≤ (1 − 2^{−(l+1)})B if the number B is chosen as B = max(2K_{0}, K^{∗}), where K^{∗} is the larger solution of the equation x = 2 log \frac{x^{2}}{D̄_{1}} with D̄_{1} = \frac{e^{−5}α^{2}η̄^{2}}{C(r)^{2}} . This means that B in (5.24) can be chosen as a number not depending on T.

Indeed, this relation holds for l = 0, and if it holds for l − 1, then

K_{l} ≤ (1 − 2^{−(l+1)})B − 2^{−(l+1} (B − 2 log \frac{B^{2}}{D̄_{1}} ) ≤ (1 − 2^{−(l+1)})B

if B ≥ K^{∗}.

This implies (5.21) (with the constant D_{1} not depending on T and the validity of the inductive hypothesis (5.24) for 0 ≤ l ≤ L. Finally, relation (5.20) follows from (5.24) and (5.25). Lemma 5.3 is proved. □

Formulas (5.18) and (5.19) can be rewritten for the function Ĥ_{n,T} (R) defined in (5.3) as

Ĥ_{n+1,T} (R) = \frac{2^{r}}{Z_{n}(T)} ∫_{|x|≥R} ∫_{u∈ℝ^{r}} exp \{−\frac{c^{(n)}}{c^{(n̄(T))}} u^{2}\}

× ĥ_{n}(x − u, T)ĥ_{n}(x + u, T) du dx

= \frac{1}{Z_{n}(T)} ∫_{|_{\frac{x+u}{2}}|≥R} ∫_{u∈ℝ^{r}} exp \{−\frac{c^{(n)}}{c^{(n̄(T))}} \frac{(x − u)^{2}}{4} \}

× Ĥ_{n,T} ( dx)Ĥ_{n,T} ( du)

(5.27)

with

Z_{n}(T) = ∫_{x∈ℝ^{r}} ∫_{u∈ℝ^{r}} exp \{−\frac{c^{(n)}}{c^{(n̄(T))}} \frac{(x − u)^{2}}{4} \} Ĥ_{n,T} ( dx)Ĥ_{n,T} ( du).

for all R ≥ 0. We apply these formulas in the proof of the following Lemma 5.4. The proof of Lemma 5.4 also exploits the rotational invariance of the measure Ĥ_{n,T}.

Lemma 5.4. Let the conditions of Lemma 5.3 hold. Then there exist some numbers δ = δ (η̄, D_{1}) > 0 and M = M (η̄, D_{1}) > 0 depending only on the numbers D_{1} in formula (5.21) and η̄ in Condition 5 in such a way that

Ĥ_{n̄(T)+l+1,T} ((1 − δ)R) ≤ \frac{1}{2}Ĥ_{n̄(T)+l,T} ((1 − δ)R) + MĤ_{n̄(T)+l,T} (R))
for all R > 0 and 0 ≤ l ≤ L.  (5.28)

Proof of Lemma 5.4. Observe that

\{|\frac{x + u}{2}| ≥ (1 − δ)R\} ⊂ \{|x| ≥ R\} ∪ \{|u| ≥ R\}

∪{|x| ≥ (1 − δ)R, arg(x, u) ≤ α} ∪ {|u| ≤ (1 − δ)R, arg(x, u) ≥ α}

for all R > 0 and 0 < δ < 1 with α = 2 arccos(1 − δ). Indeed, if |\frac{x+u}{2} | ≥ (1 − δ)R, then either |x| > R or |u| > R or both |x| and |u| is less than R, but in this case either |x| > (1 − δ)R or |u| > (1 − δ)R, and the angle between the vectors x and u must be small. On the other hand, because of the rotational invariance of the measure Ĥ_{n,T}

Ĥ_{n̄(T)+l,T} × Ĥ_{n̄(T)+l,T} ({(x, y): |x| ≥ (1 − δ)R, arg(x, u) ≤ α})

≤ \frac{α}{π}Ĥ_{n̄(T)+l,T} (\{x: |x| ≥ (1 − δ)R\}) = \frac{α}{π}Ĥ_{n̄(T)+l,T} ((1 − δ)R).

The last two relations together with (5.27) and the inequality \frac{α}{π} ≤ \sqrt{δ} imply that

Ĥ_{n̄(T)+l+1,T} ((1 − δ)R)  (5.29)

≤ \frac{1}{Z_{n}(T)} (2\sqrt{δ}Ĥ_{n̄(T)+l,T} ((1 − δ)R) + 2Ĥ_{n̄(T)+l,T} (R)) .

Relation (5.28) follows from (5.29) and (5.21) if we choose δ > 0 so small that the inequality \frac{2\sqrt{δ}}{D_{1}} ≤ \frac{1}{2} holds. Lemma 5.4 is proved. □

Next we prove Theorem 3.1 with the help of the previous results.

Proof of Theorem 3.1. First we give a good estimate on H_{n̄(T)+l}(R) if the conditions of Lemma 5.3 hold with a sufficiently large L = L(κ, η̄) and l ≤ L is sufficiently large. For this goal we introduce the following quantities.

Put P (j, l) = P (j, l, T) = Ĥ_{n̄(T)+l}((1−δ)^{j}D), j = 0, 1, …, 0 ≤ l ≤ L with the number D appearing in (5.20) and δ in Lemma 5.4. Clearly, P (j, l) ≤ 1 for all j and l. By Lemma 5.4

P (j, l + 1) ≤ \frac{1}{2}P (j, l) + MP (j − 1, l), j ≥ 1,  (5.30)

and by relation (5.20) P (0, l) ≤ e^{−α2^{l}D^{2}/5r} if l ≤ L. Hence there is a constant k_{0} > 0 in such a way that P (0, k_{0} + l) ≤ (\frac{2}{3})^{l} if k_{0} + l ≤ L. Because of this relation, the inequality P (j, l) ≤ 1 and formula (5.30) there is a constant k_{1} ≥ k_{0} in such a way that P (1, k_{1} + l) ≤ \frac{1}{3M} (\frac{2}{3})^{l} and P (1, k_{1} + l) ≤ (\frac{2}{3})^{l} if k_{1} +l ≤ L. Similarly, there is a constant k_{2} such that P (2, k_{2} +l) ≤ \frac{1}{3M} (\frac{2}{3})^{l}, and P (2, k_{2} + l) ≤ (\frac{2}{3})^{l} if k_{2} + l ≤ L. This procedure can be continued, and we get a sequence k_{0} ≤ k_{1} ≤ k_{2} ≤ ⋯ in such a way that the inequality P (p, k_{p} + l) ≤ (\frac{2}{3})^{l} holds if k_{p} + l ≤ L. The numbers k_{p} depend only on the parameter κ in (1.3) and the number η̄ in Condition 5. The above procedure can be continued till k_{p} ≤ L. In such a way we have proved that for all fixed j ≥ 0

Ĥ_{n̄(T)+l} ((1 − δ)^{p}D) ≤ C(p) (\frac{2}{3})^{l} ,

if 0 ≤ l ≤ L. The above relation together with formula (5.20) imply that if Condition 5 holds with a sufficiently large constant L = L(η̄, t), then an integer k > 0 can be chosen independently of the parameter T in such a way that

Ĥ_{n̄(T)+l,T} (R) ≤ 2 exp \{−\frac{e^{1/η^{3}}R^{2}}{η̄/5} \} for all R > 0 and k ≤ l ≤ L(η̄, t).

(5.31)

Indeed, by (5.20) relation (5.31) holds for R ≥ D if l ≥ k′_{0} with some k′_{0} > 0 and by the previous inequality for all j = 1, 2, … it also holds for (1 − δ)^{j}D ≤ R < (1 − j)^{j−1}D if l ≥ k′j with a sufficiently large k′j ≥. On the other hand, it is enough to demand this inequality for finitely many indices j, since relation (5.31) automatically holds if \frac{e^{1/η^{3}}R^{2}}{η̄/5} ≤ log 2.

Since the measure H_{n,T} defined in (3.5) satisfies the relation

H_{n̄(T)+l,T} \{x: |x| > R\} = Ĥ_{n̄(T)+l,T} (\sqrt{\frac{c^{(n̄(T))}}{c^{(n̄(T)+l)}}} R) ≤ Ĥ_{n̄(T)+l,T} (\sqrt{\frac{η̄}{5}}R)

relation (5.31) implies that

H_{n̄(T)+l,T} (R) ≤ 2 exp {−e^{1/η^{3}}R^{2}} for all R > 0, and l^{∗} ≤ l ≤ L (5.32)

with some appropriate l^{∗} ≥ 0. Relation (5.32) implies in particular that D_{n̄(T)+l}^{2}(T) < e^{−1/η^{2}}, i.e. (n̄(T)) + l, T) is in the high temperature region if l^{∗} ≤ l ≤ L. The inequality D_{n̄(T)}^{2}(T) < K follows from (5.20) with l = 0.

To complete the proof of Theorem 3.1 we have to give a lower bound for D_{n̄(T)+k}^{2}(T). Let us introduce the following notation: Given two positive numbers R_{2} > R_{1} > 0 let K(R_{1}, R_{2}) = {x: x ∈ ℝ^{r}, R_{1} ≤ |x| ≤ R_{2}} denote the annulus between the concentrical balls with center in the origin and radii R_{1} and R_{2}. We claim that for any 0 ≤ l ≤ L there exist some positive numbers R_{1}(l) = R_{1}(l, η̄, κ), R_{2}(l) = R_{2}(l, η̄, κ) and A(l) = A(l, η̄, κ) > 0 such that the measure of the annulus determined by these numbers satisfies the inequality

Ĥ_{n̄(T)+l,T} (K(R_{1}(l), R_{2}(l)) ≥ A(l), 0 ≤ l ≤ L  (5.33)

if the pair (n̄(T), T) does not belong to the high temperature region. Observe that the relation between the measures Ĥ_{n̄(T)+l,T} and H_{n̄(T)+l,T} implies that relation (5.33) also holds for H_{n̄(T)+l,T} (K(R_{1}(k), R_{2}(k)) (i.e. the function Ĥ(·) can be replaced by H(·) in formula (5.33)) if the radii R_{2}(k) and R_{1}(k) > 0 are multiplied with an appropriate number. This implies that the variance D_{n̄(T)+k,T}^{2} can be bounded from below by a positive number which depends only on k and η̄. Hence we can choose e.g. k = L = L(κ, η̄) as the number k satisfying the properties demanded in Theorem 3.1.

We shall prove a slightly stronger statement than relation (5.33) which will be useful in later applications. We shall prove that

Ĥ_{n̄(T)+l,T} (K (\frac{1}{2^{l}} R_{1}, (\frac{\sqrt{3}}{2})^{l} R_{2})) ≥ A(l), 0 ≤ l ≤ L.  (5.34)

with some numbers R_{2} > R_{1} > 0 and A(l) > 0 if the pair (0, T) does not belong to the high temperature region. The numbers R_{j} can be chosen in such a way that R_{j} = R_{j}(η, κ), j = 1, 2.

Let us first observe that relation (5.34) holds for l = 0 if n̄(T) is not in the high temperature region. This follows from relations (5.4) and (5.5) in the case T ≤ c_{0}A_{0}/2 and from (5.11) and (5.15) if T ≥ c_{0}A_{0}/2, but (n̄(T), T) does not belong to the high temperature region. Indeed, formulas (5.5) and (5.15) make possible to choose the number R_{2} in such a way that the H_{n̄(T),T} measure of the ball with center in the origin and radius R_{2} = R_{2}(η) is greater than 1/2. By formulas (5.4) and (5.11) we can choose the number R_{1} = R_{1}(η) in such a way that by cutting out from this ball the ball with radius R_{2} and center in the origin the remaining annulus K(R_{1}, R_{2}) has a measure greater than 1/4. In the case T ≥ c_{0}A/2 we also exploited in the above argument that T cannot be very large if (n̄(T), T) is not in the high temperature region. By Lemma 5.2 T ≤ e^{−1/η^{9}} in this case.

We claim that

Ĥ_{n̄(T)+l+1,T} (K (\frac{R̄_{1}}{2} , \frac{\sqrt{3}}{2}R̄_{2})) ≥ B(R̄_{1}, R̄_{2}, η̄)Ĥ_{n̄(T)+l,T} (K(R̄_{1}, R̄_{2}))^{2}

(5.35)

for all 0 ≤ l ≤ L and R̄_{2} > R̄_{1} > 0 and an appropriate constant

B(R̄_{1}, R̄_{2}, η̄) > 0.

Relation (5.34) follows from (5.35) and the fact that it holds for l = 0.

In the proof of relation (5.35) we exploit the relation

\{(u, x): u ∈ ℝ^{r}, x ∈ ℝ^{r}, \frac{R̄_{1}}{2} ≤ |\frac{x + u}{2}| ≤ \frac{\sqrt{3}}{2}R̄_{2}, \frac{π}{3} ≤ arg (x, u) ≤ \frac{π}{2} \}

⊃ \{(u, x): u ∈ ℝ^{r}, x ∈ ℝ^{r}, R̄_{1} ≤ |x|, |u| ≤ R̄_{2}, \frac{π}{3} ≤ arg (x, u) ≤ \frac{π}{2} \} .

It follows from relation (5.19) that Z_{n̄(T)+l+1}(T) ≤ 1, since we get an upper

bound for it by omitting the kernel term exp \{−\frac{c^{(n)}}{c^{(n̄(T))}} u^{2}\} from the integral in (5.19). Hence the previous relation together with (5.27) and the rotational invariance of the measure Ĥ_{n̄(T)+l,T} yield that

Ĥ_{n̄(T)+l+1,T} (K (\frac{R̄_{1}}{2} , \frac{\sqrt{3}}{2}R̄_{2})) = \frac{1}{Z_{n̄(T)+l+1}(T)} ∫ ∫_{\frac{\sqrt{3}}{2} R̄_{2}≥|\frac{x+u}{2}|≥ \frac{R̄_{1}}{2} , x,u∈ℝ^{r}}

exp \{−\frac{c^{(n̄(T)+l)}}{c̄^{(n̄(T))}} \frac{(x − u)^{2}}{4} \} Ĥ_{n̄(T)+l,T} ( dx)Ĥ_{n̄(T)+l,T} ( du)

≥ e^{−R̄_{2}^{2}/η̄} ∫ ∫\begin{matrix}\frac{\sqrt{3}}{2} R̄_{2}≥|\frac{x+u}{2}|≥ \frac{R̄_{1}}{2} , x,u∈ℝ^{2}, \\ \frac{π}{3} ≤arg(x,u)≤ \frac{π}{2} \end{matrix} Ĥ_{n̄(T)+l,T} ( dx)Ĥ_{n̄(T)+l,T} ( du)

≥ e^{−R̄_{2}^{2}/η̄} ∫ ∫_{R̄_{2}≥|x|,|u|≥R̄_{1}, \frac{π}{3} ≤arg(x,u)≤ \frac{π}{2}} Ĥ_{n̄(T)+l,T} ( dx)Ĥ_{n̄(T)+l,T} ( du)

= C(r)e^{−R̄_{2}^{2}/η̄}Ĥ_{n̄(T)+l,T} (K(R̄_{1}, R̄_{2}))^{2}

with an appropriate constant C(r) > 0. The last estimate implies relation (5.35) with B(R̄_{1}, R̄_{2}, η̄) = C(r)e^{−R̄_{2}^{2}/η̄}. Theorem 3.1 is proved. □

6. Estimates in the High Temperature Region. The proof of Theorem 3.3.

To study the behaviour of the function f_{n}(x, T) in the high temperature region we need a starting index n = ñ(T) for which a good estimate is known about the tail behaviour of the measure H_{ñ(T),T}. We also need a lower bound for the variance D_{n}^{2}(T) defined in (3.4) for n ≥ ñ(T). This requirement will be also taken into consideration in the definition of ñ(T). Let us first define the number

l_{0} = l_{0}(T) = min \{l: (\frac{\sqrt{3}}{2})^{l} R_{2} ≤ \frac{η̄}{10}\}  (6.1)

if the pair (0, T) is not in the high temperature region, where η̄ appeared in Condition 5, and the number R_{2} was introduced in formula (5.34). Now define

ñ(T) = \{ \begin{matrix}0 if (0, T) is in the high temperature region \\ n̄(T) + l with the smallest l satisfying both (5.32) and the \\ inequality l ≥ l_{0} with l_{0} defined in (6.1) \\ if (0, T) is not in the high temperature region. \end{matrix}

(6.2)

It follows from the results of the previous section that for a temperature T which is not in the low temperature region the inequality 0 ≤ ñ(T)−n̄(T) ≤ L(η̄, t) holds if the number L in Condition 5 is chosen sufficiently large. The measure H_{ñ(T),T} introduced in formula (3.5) is strongly concentrated around the origin. Indeed, formulas (5.16) and (5.32) give a good estimate for the H_{ñ(T),T} measure of the sets {x: |x| ≤ R} for all R ≥ 0.

Let us introduce the moments of the functions h_{ñ(T)+l}(x, T) defined in (3.4).

M_{k}(l, T) = ∫_{ℝ^{r}} |x|^{k}h_{ñ(T)+l}(x, T) dx l ≥ 0, k ≥ 1.

We shall estimate the moments M_{2}(l, T) and M_{4}(l, T). It follows from relations (5.16) and (5.32) that

M_{2}(0, T) ≤ η^{∗} and M_{4}(0, T) ≤ η^{∗} with η^{∗} = e^{−1/η^{2}}  (6.3)

for all T > 0 which is not in the low temperature region. To get lower bounds for the second moments M_{2}(l, T) let us introduce the truncated second moments

M_{2,tr.}(l, T) = M_{2,tr.} (\frac{1}{10}, l, T ) = ∫_{|x|≤ \frac{1}{10}} |x|^{2}h_{ñ(T)+l}(x, T) dx.

It follows from (5.13) if (0, T) is in the high temperature region and from (5.34) and the definition of ñ(T) if (0, T) is not in the high temperature region that

M_{2,tr.}(0, T) > 0,  for all T ≥ c_{0}A_{0}/2
M_{2,tr.}(0, T) > η̃,  if T ≥ c_{0}A_{0}/2 and (0, T) is not in the
high temperature region  (6.4)

with some η̃ = η̃(η̄, κ) > 0. First we shall bound M_{2}(l, T) and M_{4}(l, T) from above in Lemma (6.1) for all l ≥ 0. Then the second moment M_{2}(l, T) will be bounded from below in Lemma 6.2. These estimates enable us to prove the central limit theorem for g_{ñ(T)+l}(x, T) by means of the characteristic function technique.

Simple calculation yields that

M_{k}(l + 1, T) = \frac{2^{r}}{Z_{l}(T)} ∫ e^{−u^{2}}|x|^{k}h_{ñ(T)+l} (\frac{x}{\sqrt{c̄_{ñ(T)+l+1}}} − u, T )

h_{ñ(T)+l} (\frac{x}{\sqrt{c̄_{ñ(T)+l+1}}} + u, T ) dx du (6.5)

for all l ≥ 0 and k ≥ 1 with

Z_{l}(T) = 2^{r} ∫ e^{−u^{2}}h_{ñ(T)+l} (\frac{x}{\sqrt{c̄_{ñ(T)+l+1}}} − u, T )

h_{ñ(T)+l} (\frac{x}{\sqrt{c̄_{ñ(T)+l+1}}} + u, T ) dx du  (6.6)

with the constants c̄_{n}, n = 1, 2, … defined in (2.17). These formulas will be used in the proof of the following

Lemma 6.1. Under the conditions of Theorem 3.3 the inequalities

M_{2}(l, T) ≤ η^{∗} (\frac{2}{3})^{l} ,  (6.7)

Z_{l}(T) ≥ \binom{}{c̄_{ñ(T)+l}}^{r/2} (1 − 8\sqrt{η^{∗}} (\frac{5}{6})^{l}) ,  (6.8)

M_{2}(l + 1, T) ≤ \frac{c̄_{ñ(T)+l+1}}{2} (1 + 10\sqrt{η^{∗}} (\frac{5}{6})^{l}) M_{2}(l, T), (6.9)

M_{2}(l, T) ≤ 2 · 2^{−l} \frac{c^{(ñ(T)+l)}}{c^{(ñ(T))}} η^{∗} and M_{4}(l, T) ≤ 5 · 4^{−l} (\frac{c^{(ñ(T)+l)}}{c^{(ñ(T))}} )^{2} η^{∗}

(6.10)

hold for all l ≥ 0 with the same number η^{∗} which appears in (6.3).

Proof of Lemma 6.1. Relation (6.7) holds for l = 0 by relation (6.3). We shall prove that if relation (6.7) holds for an integer l, then relations (6.8) and (6.9) also hold for this l. Then we prove that if relations (6.7) and (6.9) hold for some l, then relation (6.7) holds also for l + 1. These statements imply relations (6.7), (6.8) and (6.9). We prove them with the help of the following calculations.

It follows from formulas (6.5) and (6.6) that

M_{k}(l + 1, T) = \frac{\binom{}{c̄_{ñ(T)+l+1}}^{r/2}}{Z_{l}(T)} ∫ exp \{−\frac{(x − u)^{2}}{4} \} |\frac{x + u}{2}|^{k} (6.11)

\binom{}{c̄_{ñ(T)+l+1}}^{k/2} h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du

≤ \frac{\binom{}{c̄_{ñ(T)+l+1}}^{(k+r)/2}}{2^{k}Z_{l}(T)} ∫ |x + u|^{k}h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du

for all l ≥ 0 and k ≥ 1, and

Z_{l}(T) = (c̄_{ñ(T)+l+1})^{r/2} ∫ exp \{−\frac{(x − u)^{2}}{2} \}
h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du
≥ (c̄_{ñ(T)+l+1})^{r/2}e^{−4\sqrt{M_{2}(l,T)}} ∫_{|x|≤M_{2}(l,T)^{1/4}, |u|≤M_{2}(l,T)^{1/4}}
h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du
≥ (c̄_{ñ(T)+l+1})^{r/2}e^{−4\sqrt{M_{2}(l,T)}}(1 − 2 ∫_{|x|≥M_{2}(l,T)^{1/4}} h_{ñ(T)+l}(x, T)dx)
≥ (c̄_{ñ(T)+l+1})^{r/2}e^{−4\sqrt{M_{2}(l,T)}}

(1 − \frac{2}{\sqrt{M_{2}(l, T}} ∫_{|x|≥M_{2}(l,T)^{1/4}} x^{2}h_{ñ(T)+l}(x, T) dx) .

Hence

Z_{l}(T) ≥ (c̄_{ñ(T)+l+1})^{r/2}e^{−4\sqrt{M_{2}(l,T)}} (1 − 2\sqrt{M_{2}(l, T)}) .

The last relation and formula (6.7) for l together imply that

Z_{l}(T) ≥ (c̄_{ñ(T)+l+1})^{r/2} (1 − 5\sqrt{M_{2}(l, T)}) (1 − 2\sqrt{M_{2}(l, T)})
≥ (c̄_{ñ(T)+l+1})^{r/2} (1 − 8\sqrt{M_{2}(l, T)})
≥ (c̄_{ñ(T)+l+1})^{r/2} (1 − 8\sqrt{η^{∗}} (\frac{5}{6})^{l}) ,

and this is relation (6.8) for the number l. Relation (6.11) for k = 2 and formula (6.8) for l together yield that

M_{2}(l + 1, T) ≤ \frac{(c̄_{ñ(T)+l+1})^{(2+r)/2}}{2Z_{l}(T)} M_{2}(l, T)

≤ \frac{c̄_{ñ(T)+l+1}}{2} (1 + 10\sqrt{η^{∗}} (\frac{5}{6})^{l}) M_{2}(l, T),

and this is formula (6.9) for l. Finally, if η is chosen sufficiently small, then formulas (6.7) and (6.9) for l imply (6.7) for l + 1. Thus formulas (6.7) — (6.9) are proved.

The first relation in (6.10) follows from the first relation in (6.3) and (6.9). Formula (6.11) with the choice k = 4, (6.8) and the first formula in (6.10) imply that

M_{4}(l + 1, T) ≤ \frac{(c̄_{ñ(T)+l+1})^{(r+4)/2}}{8Z_{l}(T)} (3M_{2}(l, T)^{2} + M_{4}(l, T))

≤ \frac{1}{8}c̄_{ñ(T)+l+1}^{2} (1 + 10\sqrt{η^{∗}} (\frac{5}{6})^{l}) (3M_{2}(l, T)^{2} + M_{4}(l, T))

≤ 2 · 4^{−l} (\frac{c̄^{(ñ(T)+l+1)}}{c^{(ñ(T))}} )^{2} η^{∗}^{2} + \frac{M_{4}(l, T)}{6}.

The second relation in (6.10) follows by induction from the last inequality and the second inequality in (6.3). Lemma 6.1 is proved. □

Remark. The Corollary formulated after Theorem 3.1 follows from Theorem 3.1, formula (6.10) and Lemma 4.4. Indeed, if T is not in the low temperature, then by Theorem 3.1 the pair (ñ(T), T) with the definition of ñ(T) given in (6.1) is in the high temperature domain. By formula (6.10) all pairs (n, T), n ≥ ñ(T), are in the high temperature region, i.e. if T > 0 is not in the low temperature region, then it is in the high temperature region. The remaining statements of the Corollary are contained in Lemma 4.4.

In the next lemma we prove an estimate from below for M_{2}(l, T).

Lemma 6.2. Put

σ^{2}(l, T) = 2^{l} \frac{c^{(ñ(T))}}{c^{(ñ(T)+l)}} M_{2}(l, T), l ≥ 0.

Under the conditions of Theorem 3.3 the limit

σ̄^{2}(T) = lim_{l→∞} σ^{2}(l, T) > 0

exists, and it is positive for all T > 0. If ñ(T) ≠ 0, i.e. if (0, T) is not in the high temperature region, then there exist two constants C_{2} > C_{1} > 0 depending only on the parameter η̃ in formula (6.4) in such a way that the inequalities

C_{1} ≤ σ̄^{2}(T) ≤ C_{2}  (6.12)

hold. The upper bound in (6.12) holds for all T > 0 which is not in the low temperature region.

Proof of Lemma 6.2. The hard part of the proof is to show that σ^{2}(l, T) has a non-negative lim inf. It follows simply from formula (6.6) that Z_{l}(T) ≤ (c̄_{ñ(T)+l+1})^{r/2}. A natural lower bound for M_{2}(l, T) can be obtained in the following way. By formula (6.5) and the upper bound for Z_{l}(T)

M_{2}(l + 1, T) ≥ c̄_{ñ(T)+l+1} ∫ e^{−(x−u)^{2}/4} |\frac{x + u}{2}|^{2}
h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du
= \frac{c̄_{ñ(T)+l+1}}{4} (2M_{2}(l, T) − ∫ |x + u|^{2} (1 − e^{−(x−y)^{2}/4})

h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du)

≥ \frac{c̄_{ñ(T)+l+1}}{2} (M_{2}(l, T) − ∫ \frac{1}{4}|x + u|^{2}|x − u|^{2}

h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du)

≥ \frac{c̄_{ñ(T)+l+1}}{2} (M_{2}(l, T) − ∫ \frac{1}{2}(|x|^{4} + |u|^{4})

h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du)

= \frac{c̄_{ñ(T)+l+1}}{2} (M_{2}(l, T) − M_{4}(l, T)) .  (6.13)

However, this estimate is useful only if we know that the right-hand side in it is non-negative. We do not know such an estimate for small l, hence in this case we apply a different argument. Clearly

M_{2}(l, T) ≥ M_{2,tr.}(l, T),

where M_{2,tr.}(l, T) is the truncated moment. On the other hand, we get by using an argument similar to the previous calculation and making the observation

\{(x, u): x ∈ ℝ^{r}, u ∈ ℝ^{r}, c̄_{ñ(T)+l+1} |\frac{x + u}{2}| ≤ \frac{1}{10}\}

⊃ \{(x, u): x ∈ ℝ^{r}, u ∈ ℝ^{2}, |x| ≤ \frac{1}{10}, |u| ≤ \frac{1}{10}, arg (x, u) ⊂ I\}

with I = (\frac{π}{50}, \frac{49π}{50} ) ∪ (\frac{51π}{50} , \frac{99π}{50} ) that

M_{2,tr.}(l + 1, T) ≥ c̄_{ñ(T)+l+1} ∫_{c̄_{ñ(T)+l+1}|_{\frac{x+u}{2}}|≤ \frac{1}{10}} e^{−(x−u)^{2}/4} |\frac{x + u}{2}|^{2}
h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du
≥ c̄_{ñ(T)+l+1}e^{−1/100} ∫_{|x|≤ \frac{1}{10} ,|u|≤ \frac{1}{10} ,arg (x,u)⊂I} |\frac{x + u}{2}|^{2}
h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du
= \frac{c̄_{ñ(T)+l+1}}{4} e^{−1/100} ∫_{|x|≤ \frac{1}{10} ,|u|≤ \frac{1}{10} ,arg (x,u)⊂I}(x^{2} + u^{2})
h_{ñ(T)+l}(x, T)h_{ñ(T)+l} (u, T) dx du
= c̄_{ñ(T)+l+1}e^{−1/100} \frac{12}{25}M_{2,tr.}(l, T) ≥ \frac{1}{3}c̄_{ñ(T)+l+1}M_{2,tr.}(l, T).

The last estimate implies that

σ^{2}(l, T) = 2^{l} \frac{c^{(ñ(T))}}{c^{(ñ(T)+l)}} M_{2}(l, T) ≥ 2^{l} \frac{c^{(ñ(T))}}{c^{(ñ(T)+l)}} M_{2,tr.}(l, T) ≥ (\frac{2}{3})^{l} M_{2,tr.}(0, T).

(6.14)

On the other hand, it follows from (6.13) and the second inequality in (6.10) that

σ^{2}(l + 1, T) ≥ σ_{2}(l, T) − 2^{l+1} \frac{c^{(ñ(T))}}{c^{ñ(T)+l+1}} M_{4}(l, T)  (6.15)
≥ σ^{2}(l, T) − \frac{5η^{∗}}{2^{l}c_{ñ(T)+l+1}} \frac{c^{(ñ(T)+l)}}{c^{(ñ(T))}} ≥ σ^{2}(l, T) − 50η^{∗} (\frac{3}{4})^{l} .

Because of (6.14) and (6.4) an index l̄≥ 0 can be chosen in such a way that

σ^{2}(l̄, T) ≥ 1000η^{∗} (\frac{3}{4})l̄,

and if the pair (0, T) is not in the high temperature region, then we may choose l̄ so that l̄≤ K(η̄, κ) with some appropriate K(η̄, κ). Hence relation (6.15) implies that

\frac{σ^{2}(l̄+ l + 1, T)}{σ^{2}(l̄, T)} ≥ \frac{σ^{2}(l̄+ l, T)}{σ^{2}(l̄, T)} − \frac{1}{20} (\frac{3}{4})^{l} .

This relation and the bound on σ^{2}(l̄, T) imply that lim inf_{l→∞} σ^{2}(l, T) > 0, and this lim inf can be bounded by a positive number which depends only on η̄ and κ if (0, T) is not in the high temperature region. The analogous result for lim sup follows from (6.9). To complete the proof it is enough to show that the lim inf is actually lim. To prove this let us observe that for any ε > 0 and N > 0 there is some m > N such that σ^{2}(m, T) < lim inf_{n→∞} σ^{2}(n, T) + ε.

Then by formula (6.9)

σ^{2}(n, T) ≤ σ^{2}(m, T) ∏_{l=m}^{n} (1 + 10\sqrt{η^{∗}} (\frac{5}{6})^{l})
≤ lim inf_{n→∞} σ^{2}(n, T) + 2ε, n > m

for any ε > 0 if N = N(ε) is chosen sufficiently large. Lemma 6.2 is proven. □

To prove Theorem 3.3 let us introduce the characteristic functions

φ_{n}(s, T) = ∫_{ℝ^{r}} e^{isx}h̃_{n}(x, T) dx, s ∈ ℝ^{r}

and moments

M̃_{k}(n, T) = ∫_{ℝ^{r}} |x|^{k}h̃_{n}(x, T) dx,

where the function h̃(x, T) was defined in (3.17). Clearly,

M̃_{k}(n, T) = (\frac{2^{n}}{c^{(n)}} )^{k/2} M_{k}(n − ñ(T), T) if n ≥ ñ(T).

In particular, M̃_{2}(n, T) = \frac{2^{ñ(T)}}{c^{(ñ(T))}} σ^{2}(n − ñ(T), T). We shall prove Theorem 3.3 by means of the usual characteristic function technique. The following lemma plays a crucial role in the proof. Lemma 6.3. Under the conditions of Theorem 3.3 the relation

lim_{n→∞} \frac{c^{(ñ(T))}}{2^{ñ(T)}} M̃_{2}(n, T) = σ̄^{2}(T)  (6.16)

holds with the constant σ̄^{2}(T) appearing in Lemma 6.2, and

lim_{n→∞} sup_{|s|≤A} |log φ_{n} (s, T) + \frac{2^{ñ(T)}}{c^{(ñ(T))}} σ̄^{2}(T)\frac{s^{2}}{2} | → 0  (6.17)

for all A > 0. Proof of Lemma 6.3. Relation (6.16) follows from Lemma 6.2, and it follows from the second relation in (6.10) that M̃_{4}(n, T) ≤ 5 (\frac{2^{ñ(T)}}{c^{(ñ(T))}} )^{2} η^{∗}. Hence the characteristic function φ can be estimated as

|φ_{n}(s, T) − (1 − M̃_{2}(n, T)\frac{s^{2}}{2} )| ≤ (\frac{2^{ñ(T)}}{c^{(ñ(T))}} )^{2} η^{∗}|s|^{4} (6.18)
for n ≥ ñ(T) and s ∈ ℝ^{r}.

In the proof of formula (6.18) we exploit that ∫ (s, x)h̃_{n}(x, T) dx = 0 and ∫ (s, x)^{3}h̃_{n}(x, T) dx = 0 because of the rotational symmetry of h̃_{n}(x, T). The coefficient of |s|^{4} in (6.18) is bounded by a constant (depending on T), and the coefficient at |s|^{2} converges to the positive constant \frac{2^{ñ(T)}}{c^{(ñ(T))}} σ̄^{2}(T). Hence formula (6.18) implies that for any ε > 0,

|log φ_{n} (s, T) + \frac{2^{ñ(T)}}{c^{(ñ(T))}} σ̄^{2}(T)\frac{s^{2}}{2} | ≤ ε if n > n_{1} and |s| ≤ δ  (6.19)

with some n_{1} = n_{1}(ε, T) and δ = δ(ε, T). By a rescaled version of the recursive formula (2.9) we can write

φ_{n+1}(\sqrt{2}s, T) = \frac{1}{Z_{n}(T)} ∫ exp \{is(x + u) − \frac{c^{(n)}(x − u)^{2}}{4 · 2^{n}} \}

h̃_{n}(x, T)h̃_{n}(u, T) dx du

= \frac{1}{Z_{n}(T)}[φ_{n} (s, T)^{2} − ∫ e^{is(x+u)} (1 − exp \{−\frac{c^{(n)}(x − u)^{2}}{4 · 2^{n}} \})

h̃_{n}(x, T)h̃_{n}(u, T) dx du]

with

Z_{n}(T) = ∫ exp \{−\frac{c^{(n)}(x − u)^{2}}{4 · 2^{n}} \} h̃_{n}(x, T)h̃_{n}(u, T) dx du.

The estimates

|∫ e^{is(x+u)} (1 − exp \{−\frac{c^{(n)}(x − u)^{2}}{4 · 2^{n}} \}) h̃_{n}(x, T)h̃_{n}(u, T) dx du|

≤ ∫ \frac{c^{(n)}(x − u)^{2}}{4 · 2^{n}} h̃_{n}(x, T)h̃_{n}(u, T) dx du = \frac{c^{(n)}}{2 · 2^{n}} M̃_{2}(n, T)

and similarly

1 ≥ Z_{n}(T) ≥ 1 − \frac{c^{(n)}}{2 · 2^{n}} M̃_{2}(n, T)

hold. Hence

φ_{n}^{2} (s, T) − \frac{c^{(n)}}{2 · 2^{n}} M̃_{2}(n, T) ≤ φ_{n+1}(\sqrt{2}s, T) ≤ \frac{φ_{n}^{2} (s, T) + \frac{c^{(n)}}{2·2^{n}} M̃_{2}(n, T)}{1 − \frac{c^{(n)}}{2·2^{n}} M̃_{2}(n, T)}.

The term \frac{c^{(n)}}{2^{n}} M̃_{2}(n, T) is much less than (\frac{2}{3})^{n} for large n. If we have a positive lower bound on φ_{n}(s) then we get by fixing some K > 0 and taking logarithm in the last relation that

|log φ_{n+1} (\sqrt{2}s, T ) − 2 log φ_{n}(s, T)| ≤ (\frac{2}{3})^{n} if n > n_{2} and φ_{n}(s, T) ≥ \frac{1}{K}

(6.20)

with some n_{2} = n_{2}(K, T). Formula (6.17) can be deduced from (6.19) and (6.20). Indeed, define an index k by the relation A ≤ δ2^{k/2} < \sqrt{2}A with the numbers A and δ in (6.17) and (6.18). Put K = 2e^{−2^{ñ(T)}σ̄^{2}(T)A^{2}/c^{(ñ(T))}} and let ε ≤ \frac{1}{8K}. Choose a number n_{3} such that (\frac{2}{3})^{n_{3}} ≤ ε, and let us consider such indices n for which n ≥ max(n_{1}(ε, T), n_{2}(K, T), n_{3}). Then simple induction yields that

|φ_{n+j}(s, T) + \frac{2^{ñ(T)}}{c^{(ñ(T))}} σ̄^{2}(T)\frac{s^{2}}{2} | ≤ ε + 3 (\frac{2}{3})^{n} (1 − (\frac{2}{3})^{j+1}) ≤ 4ε

and |φ_{n+j}(s, T)| ≥ \frac{1}{K}

for j ≤ k and |s| ≤ δ2^{j/2}. Since ε can be chosen arbitrary small in the last relation, it implies (with j = k) relation (6.17). Lemma 6.3 is proved. □

Theorem 3.3 follows from Lemmas 6.2 and 6.3. Indeed, Lemma 6.3 implies that the measures H̃_{n,T} converge in distribution to the normal law with expectation zero and covariance \frac{2^{ñ(T)}}{c^{(ñ(T))}} σ̄^{2}(T)I. The bounds obtained for the variance follow from Lemma 6.2 and the observation that the difference ñ(T) − n̄(T) can be bounded by a number depending only on η̄ and κ.

Let us finally show that Corollary to Theorem 3.3 follows from Theorem 3.3. By formulas (2.7), and (3.17) we can write

2^{−n}p_{n}(2^{−n/2}\sqrt{T} x, T) = C(n) exp \{−\frac{l_{n}A_{n}}{2^{n+1}} x^{2}\} h̃_{n}(x, T)  (6.21)

with an appropriate norming constant C(n). Observe that the expressions at both sides of this identity are density functions, the measures with density function h̃_{n}(x, T) have a limit as n → ∞, the term \{−\frac{l_{n}A_{n}}{2^{n+1}} x^{2}\} is bounded, and it tends to 1 uniformly in any compact set as n → ∞. These facts imply that C(n) → 1 in (6.21), and the measures with density functions 2^{−n}p_{n}(2^{−n/2}\sqrt{T} x, T) have the same limit as the measures with density functions h̃_{n}(x, T). Hence the Corollary of Theorem 3.3 holds. □

7. Estimates in the Low Temperature Region. The proof of Theorem 3.2.

The proof of Theorem 3.2 heavily exploits the results of Section 4. These results show that the replacement of the operator Q_{n} whose application makes possible to compute the function f_{n+1}(x, T) by its linearization T_{n} causes only a negligible error. Formula (4.17) enables one to investigate the operator T_{n} in the Fourier space. In such a way good estimates can be obtained for the Fourier transform of a regularized version of the function f_{n+1}(x, T). The results of Theorem 3.2 can be proved by means of these estimates with the help of inverse Fourier transformation.

It is simpler to work with an appropriately scaled version of the functions f_{n}(x, T). Put

f̄_{n}(x, T) = \frac{1}{M_{n}(T)}f_{n} (\frac{x}{M_{n}(T)}, T )

and

φ̄_{n}(f_{n}(x, T)) = \frac{1}{M_{n}(T)}φ_{n} (f_{n} (\frac{x}{M_{n}(T)}, T )) .

We defined the function φ̄_{n}(f_{n}(x, T)) by means of the definition of the regularization of the function f_{n}(x, T) introduced in Section 4.

Let us also introduce the functions

ψ_{n+1}(f_{n}(x, T)) = \frac{1}{M_{n}(T)}T_{n}φ_{n} (f_{n} (\frac{x}{M_{n}(T)}, T )) .

The estimates of Proposition 4.2 and relation (4.17) can be rewritten for these new functions. We shall rewrite formulas (4.14) and (4.15) only in the case when n > N_{1}(T) with the number N_{1}(T) defined in formula (4.18), i.e. in the case when β_{n}(T) and M_{n}^{−1}(T) have the same order of magnitude. In this case M_{n}(T)\sqrt{β_{n+1}(T)} ≤ 10,

\begin{aligned}| & \frac{∂^{j}}{∂x^{j}} & (f̄_{n+1}(x, T) − ψ_{n+1}(f_{n}(x, T))) & | \\ & & ≤ K_{1} \frac{β_{n}(T)}{c^{(n)}} [exp \{−\frac{1}{10} |2x + \frac{x^{2}}{c^{(n+1)}} |\} + exp \{−\frac{|x|}{5}\}] \\ & & ≤ K_{2} \frac{β_{n}(T)}{c^{(n)}}e^{−|x|/10},  x > −c^{(n+1)}M_{n+1}^{2}(T), j = 0, 1, 2, \end{aligned} (7.1)

and

|\frac{∂^{j}}{∂x^{j}} ψ_{n+1}(f_{n}(x, T))| ≤ K_{3}e^{−|x|/5}, x ∈ ℝ^{1}, j = 0, 1, 2, 3, 4,  (7.2)

with some universal constants K_{1}, K_{2} and K_{3}. Formula (4.17) can be rewritten as

\begin{aligned}ψ̃_{n+1}(f_{n}(ξ, T)) & = T̃_{n}φ̃_{n}(f_{n}(M_{n}(T)ξ, T)) \\ & = \begin{matrix}exp & \{i\frac{(r−1)c̄_{n+1}}{4} ξ\} \\ (1 + i\frac{c̄_{n+1}}{2} ξ) & (r−1)/2 \end{matrix} ˜ φ̄_{n}^{2} (f_{n} (\frac{c̄_{n+1}}{2} ξ, T )) . \end{aligned}  (7.3)

We claim that under the conditions of Theorem 3.2,

\begin{matrix}lim_{n→∞} sup_{x≥−c^{(n)}M_{n}^{2}(T)} & |\frac{∂^{j}}{∂x^{j}} (f̄_{n}(x, T) − φ̄(f_{n}(x, T)))| e^{|x|/20} = 0, \\ j = 0, 1, 2. \end{matrix}  (7.4)

Indeed, by relations (7.1) and (7.2)

|\frac{∂^{j}}{∂x^{j}} f̄_{n}(x, T)| ≤ e^{−|x|/10}, j = 0, 1, 2, if x ≥ −c^{(n)}M_{n}^{2}(T),  (7.5)

and φ̄_{n}(f_{n}(x, T)) is the appropriate scaling of the function

φ (\frac{x}{\sqrt{c^{(n)}}M_{n}(T)}) f_{n} (\frac{x}{\sqrt{c^{(n)}}M_{n}(T)}) .

Under the conditions of Theorem 3.2, formula (4.29) holds, which implies that

lim_{n→∞} \sqrt{c^{(n)}}M_{n}(T) = ∞.

This fact together with (7.5) allow us to give a good bound on the difference between the functions φ̄_{n}(f_{n}(x, T)) and φ (\frac{x}{\sqrt{c^{(n)}}M_{n}(T)}) f_{n} (\frac{x}{\sqrt{c^{(n)}}M_{n}(T)}).

Relation (7.4) can be deduced from this bound and formula (7.5).

It follows from Lemma 4.4 that lim_{n→∞} \frac{M_{n+1}(T)}{M_{n}(T)} = 1. Relations (7.1) and (7.4) together with this fact imply that

\begin{matrix}lim_{n→∞} sup_{|x|<∞} & |\frac{∂^{j}}{∂x^{j}} (ψ_{n}(f_{n−1}(x, T)) − φ̄_{n}(f_{n}(x, T)))| e^{|x|/20} = 0, \\ j = 0, 1, 2. \end{matrix}  (7.6)

Now we prove, using an adaptation of the proof of Lemmas 14 and 15 in [BM3], that the Fourier transforms of the functions ψ_{n+1}(f_{n}(x, T)) converge to the Fourier transform of the function g(x), and this convergence is uniform in all compact domains. First we prove a modified version of this statement, where ψ_{n} is replaced with φ̄_{n} in a small neighbourhood of the origin. We want to work with the functions log ˜ φ̄_{n}(f_{n}(ξ, T))). To do this, observe first that for n > N_{1}(T) there is some constant A > 0 such that all functions ˜φ̄_{n+1}(f_{n}(ξ, T))) are separated from zero in the interval |ξ| ≤ A. Indeed,

|1 − ˜ φ̄_{n}(f_{n}(ξ, T)))| ≤ ∫ |e^{ixξ} − 1|φ̄_{n}(f_{n}(x, T))) dx
≤ ∫ |ξ||x|φ̄_{n}(f_{n}(x, T))) dx ≤ const. |ξ|.

Similarly,

|\frac{∂^{j}}{∂ξ^{j}} ˜ φ̄_{n}(f_{n}(ξ, T))| ≤ C(j) for all j ≥ 0 and n ≥ N_{1}(T).

Hence a constant A > 0 can be chosen in such a way that

sup_{|ξ|≤2A} max (|1 − g̃(ξ)|, sup_{n≥N_{1}(T)} |1 − ˜ φ̄_{n}(f_{n}(ξ, T))|) ≤ \frac{1}{2}.

These estimates imply that

sup sup_{|ξ|≤A} |\frac{∂^{2}}{∂ξ^{2}} log ˜ φ̄_{n}(f_{n}(ξ, T)))| ≤ C(T),  (7.7)

with a constant C(T) < ∞ independent of n. We claim that

sup_{|ξ|≤A} |\frac{∂^{2}}{∂ξ^{2}} log ˜ φ̄_{n}(f_{n}(ξ, T))) − \frac{d^{2}}{d^{2}ξ} log g̃(ξ)| → 0 as n → ∞.  (7.8)

To prove (7.8) let us first observe that lim_{n→∞} c̄_{n} = 1 by Condition 1. By (7.6),

lim_{n→∞} sup_{|ξ|≤A} |\frac{∂^{2}}{∂ξ^{2}} (log ψ̃_{n+1}(f_{n}(ξ, T)) − log ˜ φ̄_{n+1}(f_{n+1}(ξ, T)))| = 0,

and because of the estimates obtained for the derivatives of ˜ φ̄_{n}(ξ, T)

|\frac{∂^{2}}{∂ξ^{2}} log ˜ φ̄_{n}(f_{n}(ξ_{1}, T))) − \frac{∂^{2}}{∂ξ^{2}} log ˜ φ̄_{n}(f_{n}(ξ_{2}, T)))| ≤ const. |ξ_{1} − ξ_{2}|

if |ξ_{1}| ≤ A and |ξ_{2}| ≤ A

for all large indices n with a constant independent of n. Taking logarithm and then differentiating twice in formulas (7.3) and (1.20) we get with the help of the above observations that

sup_{|ξ|≤A} |\frac{∂^{2}}{∂ξ^{2}} log ˜ φ̄_{n+1}(f_{n+1}(ξ, T))) − \frac{d^{2}}{d^{2}ξ} log g̃(ξ)|

≤ \frac{1}{2} sup_{|ξ|≤A} |\frac{∂^{2}}{∂ξ^{2}} log ˜ φ̄_{n}(f_{n}(ξ, T))) − \frac{d^{2}}{d^{2}ξ} log g̃(ξ)| + δ_{n}(T)

with some sequence lim_{n→∞} δ_{n}(T) = 0. This relation together with (7.7) imply (7.8). Since

\frac{∂}{∂ξ} log ˜ φ̄_{n}(f_{n}(ξ, T)))|_{ξ=0} = \frac{d}{dξ} log g̃(ξ)|_{ξ=0} = 0

and log ˜ φ̄_{n}(f_{n}(0, T))) = log g̃(0) = 0,

relation (7.8) also implies that

lim_{n→∞} sup_{|ξ|≤A} | ˜φ̄_{n}(f_{n}(ξ, T))) − g̃(ξ)| = 0.  (7.9)

Moreover, relation (7.9) holds for all A > 0. This can be proved similarly to the argument of Lemma 15 in [BM3]. One has to observe that because of the structure of formulas (7.3) and (1.20), the relation c̄_{n} → 1 as n → ∞, the continuity of the function g̃(ξ) and the relation

lim_{n→∞} sup_{|ξ|<∞} |ψ̃_{n+1}(f_{n}(ξ, T)) − ˜ φ̄_{n+1}(f_{n+1}(ξ, T)))| = 0,

the validity of relation (7.9) in an interval |ξ| ≤ A also implies its validity in the interval |ξ| ≤ (2 − ε)A for any ε > 0. In relation (7.9) the function ˜φ̄_{n}(f_{n}(ξ, T))) can be replaced by ψ̃_{n+1}(f_{n}(ξ, T)), i.e. the relation

lim_{n→∞} sup_{|ξ|≤A} |ψ̃_{n+1}(f_{n}(ξ, T))) − g̃(ξ)| = 0  (7.10)

holds for all A > 0. It can be proved from (7.10), by means of inverse Fourier transformation, that

lim_{n→∞} sup_{|x|<∞} |\frac{∂^{j}}{∂x^{j}} ψ_{n+1}(f_{n}(x, T))) − \frac{d^{j}}{dx^{j}} g(x)| = 0, j = 0, 1, 2. (7.11)

To prove (7.11) we need, besides the estimate (7.10), some bound about the decrease of the functions g̃(ξ) and ψ̃_{n+1}(f_{n}(ξ, T))) as ξ → ±∞. The estimate (1.19) gives a good bound for the Fourier transform of the function g(x). We can get a good estimate for the Fourier transform of the function ψ_{n+1}(f_{n}(x, T)) with the help of the inductive hypothesis J(n) in Section 4 and relation (7.3). Rewriting the inductive hypothesis J(n) for the function φ̄_{n}(f_{n}(x, T)) we get with the help of some standard calculation that the Fourier transform ψ̃_{n+1}(f_{n}(ξ, T)) decreases at infinity faster than |ξ|^{−4}. These estimates are sufficient for the proof of (7.11). Relations (7.11) and (7.1) give an estimate on the function f̄_{n}(x, T) and its derivatives, which is equivalent to (3.8). Theorem 3.2 is proved. □

8. Estimates Near the Critical Point. The proof of Theorems 3.4, 1.3, and 1.5.

Our previous results suggest that M_{n+1}^{2}(T) ∼ M_{n}^{2}(T) − \frac{r−1}{2c^{(n)}} , hence the

derivative \frac{dM_{n}^{2}(T)}{dT}, as a function of n, changes very little if the pair (n, T) is in the low domain region (observe that c^{(n)} does not depend on T). Therefore, it is natural to expect that \frac{dM_{∞}^{2}(T)}{dT} is of constant order below the critical value T_{c}, and M_{∞}^{2}(T) − M_{∞}^{2}(T_{c}) ∼ const. (T_{c} − T) for T < T_{c}. If T_{n} denotes the smallest T for which the pair (n, T) leaves the low temperature region at the n-th step, then the following heuristic argument may suggest the magnitude of T_{n} − T_{n+1} for large n. Since both c^{(n)}M_{n}^{2}(T_{n}) ∼ η^{−1} and c^{(n+1)}M_{n+1}^{2}(T_{n+1}) ∼ η^{−1}, besides this M_{n}^{2}(T_{n+1}) − M_{n+1}^{2}(T_{n+1}) ∼ \frac{r−1}{2c^{(n)}} , M_{n}^{2}(T_{n+1}) − M_{n}^{2}(T_{n}) ∼ \frac{r−1}{2c^{(n)}} . On the other hand, M_{n}^{2}(T_{n+1}) − M_{n}^{2}(T_{n}) ∼ T_{n+1} − T_{n}. This argument suggests that T_{n+1} − T_{n} ∼ \frac{r−1}{2c^{(n)}} and T_{n} − T_{c} ∼

∑_{k=n}^{∞} \frac{r−1}{2c^{(k)}} . In this section we justify these heuristic arguments. The proofs

are based on the following result:

Theorem 8.1. There exists κ_{0} = κ_{0}(N) such that if (i) 0 < κ < κ_{0} in formula (1.7), (ii) Conditions 1—4 are satisfied, (iii) 0 < T̄ < c_{0}A_{0}/2, and (iv) the integer n ≥ 1 has the property that the pair (n, T̄) belongs to the low temperature region, then for all 0 < T < T̄ the pair (n, T) also belongs to the low temperature region, and the following inequalities hold for T ≤ T̄:

a.) If 0 ≤ n ≤ N, then

\frac{C_{1}}{\sqrt{κ}T^{2}} < −\frac{dM_{n+1}(T)}{dT} < \frac{C_{2}}{\sqrt{κ}T^{2}} with some ∞ > C_{2} > C_{1} > 0.

b.) If n ≥ N, then

\frac{dM_{n+1}(T)}{dT} = \frac{dM_{n}(T)}{dT} (1 + \frac{r − 1}{4c^{(n)}M_{n}^{2}(T)} + \frac{δ_{n}(T)}{c^{(n)}}) ,

where |δ_{n}(T)| ≤ C \frac{β_{n+1}(T)}{c^{(n+1)}}β_{n}(T) with some appropriate C > 0.

We will prove Theorem 8.1 in Appendix A below with the help of Proposition A which is proved also there. This result can be interpreted in an informal way as the “differentiation” of the asymptotic identity (4.12). The main difficulty in the proof of Proposition A is to bound the error caused by the linear approximation of the operator Q_{n} by T_{n} when differentiating with respect to T. To overcome this difficulty we need a good control not only on the functions f_{n}(x, T) but also on their derivatives \frac{∂}{∂T} f_{n}(x, T). Hence we have to work out the estimation of these derivatives. In particular, we have to find the inductive hypotheses describing their behaviour. These are the analogs of the inductive hypotheses I(n) and J(n) formulated in Section 4. It demands fairly much work to work out the details, but after the formulation and proof of these inductive hypotheses the proof of Proposition A is simple.

Proof of Theorem 3.4. We prove with the help of Proposition A that if the conditions of Theorem 3.4 hold, 0 < T̄ < c_{0}A_{0}/2 and the pair (n − 1, T̄) belongs to the low temperature region, then there exist some constants 0 < C_{1} < C_{2} independent of T such that

\frac{C_{1}}{κT^{3}} < −\frac{dM_{n}^{2}(T)}{dT} < \frac{C_{2}}{κT^{3}} .  (8.1)

for all 0 < T ≤ T̄.

For 0 ≤ n ≤ N (n, T) is in the low temperature region for 0 < T < c_{0}A_{0}/2. In this case Properties K_{1}(n)—K_{4}(n) hold by Proposition A, and the validity of (8.1) for n = 0 follows from relations (4.1), (4.5) and (A.1). Its validity for 0 ≤ n ≤ N can be proved by induction with the help of Properties K_{1}(n) and K_{3}(n), 1 ≤ n ≤ N.

To prove formula (8.1) for n > N first we show that

−\frac{dM_{n}^{2}(T)}{dT} exp \{−K (\frac{β_{n+1}(T)}{c^{(n)}})^{2}\} ≤ −\frac{dM_{n+1}^{2}(T)}{dT}  (8.2)

≤ −\frac{dM_{n}^{2}(T)}{dT} exp \{K (\frac{β_{n+1}(T)}{c^{(n)}})^{2}\}

for all T < T̄ and n ≥ N with an appropriate K > 0. Relation (8.2) is a consequence of Part b) of Proposition A, formula (4.12), the inequality β_{n+1}(T)M_{n}^{2}(T) ≥ 10 and the relation \frac{β_{n+1}(T)}{c^{(n)}} ≤ η if (n, T) is in the low temperature domain. Indeed, since

−\frac{dM_{n+1}^{2}(T)}{dT} = − (1 − \frac{m_{n}(T)}{c^{(n+1)}M_{n}(T)}) (1 − \frac{\frac{dm_{n}(T)}{dT}}{c^{(n+1)} \frac{dM_{n}(T)}{dT}} ) \frac{dM_{n}^{2}(T)}{dT}

these relations imply that

− (1 − \frac{r − 1}{4c^{(n+1)}M_{n}^{2}(T)} − C_{1} \frac{β_{n+1}^{3/2}(T)}{c^{(n+1)}^{2}M_{n}(T)})

(1 + \frac{r − 1}{4c^{(n+1)}M_{n}^{2}(T)} − C \frac{β_{n+1}^{2}(T)}{(c^{(n+1)})^{2}} ) \frac{dM_{n}^{2}(T)}{dT}

≤ −\frac{dM_{n+1}^{2}(T)}{dT}

≤ − (1 − \frac{r − 1}{4c^{(n+1)}M_{n}^{2}(T)} + C_{1} \frac{β_{n+1}^{3/2}(T)}{c^{(n+1)}^{2}M_{n}(T)})

(1 + \frac{r − 1}{4c^{(n+1)}M_{n}^{2}(T)} + C \frac{β_{n+1}^{2}(T)}{c^{(n+1)}^{2}}) \frac{dM_{n}^{2}(T)}{dT}.

In this calculation we have exploited that c^{(n+1)}M_{n}^{2}(T) ≥ \frac{β_{n+1}(T)}{η}M_{n}^{2}(T) ≫ 1.

The left and right-hand side of this inequality can be bounded by

− (1 ± K \frac{β_{n+1}^{2}(T)}{c^{(n)}^{2}}) \frac{dM_{n}^{2}(T)}{dT},

and formula (8.2) can be deduced from these relations.

For N ≤ n ≤ N_{1}(T) with the number N_{1}(T) defined in relation (4.18) relation (8.1) follows from (8.2) and (4.19). Since by (4.20) β_{n+1}M_{n}^{2}(T) ≤ 100 if n ≥ N_{1}(T) and the pair (n, T) is in the low temperature domain, to prove formula (8.1) with the help of (8.2) for n > N_{1}(T) it is enough to show that

∑_{k=N_{1}(T)}^{n} \frac{1}{(c^{(k)}M_{k}^{2}(T))^{2}} ≤ L if n ≥ N_{1}(T)

and (n, T) is in the low temperature domain

with a constant L > 0 independent of T and n. Since M_{n}^{2}(T) ≥ \frac{1}{10β_{n+1}(T)} ≥

\frac{1}{10ηc^{(n)}} and M_{k}^{2}(T) = M_{n}^{2}(T) + (M_{k}^{2}(T) − M_{n}^{2}(T)) ≥ \frac{1}{10ηc^{(n)}} + ∑_{j=k}^{n−1} \frac{1}{8c^{(j)}} ,

∑_{k=N_{1}(T)}^{n} \frac{1}{(c^{(k)}M_{k}^{2}(T))^{2}} ≤ const. ∑_{k=N_{1}(T)}^{n} \frac{1}{(c^{(k)} ∑_{j=k}^{n} \frac{1}{c^{(j)}} )^{2}} ≤ L

because of Condition 3. Hence formula (8.1) holds.

It follows from (1.2), Condition 4, and the results of Section 4 that all T > c_{0}A_{0}/4 belong to the high temperature region. Indeed, it follows from formulas (4.26), (4.27), (4.1), (4.5) and (4.9), that if T > 0 is in the low temperature region, then

0 ≤ M_{n}^{2}(T) ≤ M_{N}^{2}(T) − 30(M_{N}(T) + 1) − ∑_{n=1}^{∞} \frac{1}{8c^{(n)}} ≤ \frac{3}{κT^{2}} − ∑_{n=1}^{∞} \frac{1}{8c^{(n)}}

for all n ≥ N, and T ≤ ( ∑_{n=1}^{∞} \frac{κ}{24c^{(n)}} )^{−1/2}. Hence Condition 4 implies that

T ≤ c_{0}A_{0}/4.

It follows from (8.2) that for a fixed n the function M_{n}^{2}(T) is strictly monotone decreasing. Hence a simple induction with respect to n yields that the function β_{n}(T) is a monotone increasing, continuous function of T for all n > N. Put

T_{n} = sup{T: (T, n) is in the low temperature region}.  (8.3)

The sequence T_{n} is monotone decreasing, hence the limit T_{c} = lim_{n→∞} T_{n} exists, and by Lemma 4.4 T_{c} > 0 under Dyson’s condition (1.2). We want to show that

C_{1} ∑_{k=n}^{∞} \frac{1}{c^{(k)}} ≤ T_{n} − T_{c} ≤ C_{2} ∑_{k=n}^{∞} \frac{1}{c^{(k)}} .  (8.4)

Since we can handle the sequence M_{n}(T) better than the sequence β_{n}(T) we also introduce besides the sequence T_{n} defined in (8.3) the sequence T (n)

T (n) = sup \{T: M_{n}^{2}(T) ≥ \frac{100}{c^{(n)}η} \} .

We will show that

T_{n+K} ≤ T (n) ≤ T_{n}  (8.5)

for all sufficiently large n with an appropriate K > 0, and

\frac{C_{1}}{c^{(n)}} ≤ T (n) − T (n + 1) ≤ \frac{C_{2}}{c^{(n)}}  (8.6)

with some appropriate C_{2} > C_{1} > 0 for all sufficiently large n. Because of Condition 5 relation (8.4) follows from (8.5) and (8.6) together with the relation lim_{n→∞} T_{n} = T_{c}.

If T ≤ T (n), and m ≤ n then either m ≤ N_{1}(T) with the number N_{1}(T) defined in (4.18) or β_{m+1}(T) ≤ \frac{100}{M_{m}^{2}(T)} ≤ \frac{100}{M_{n}^{2}(T)} ≤ c^{(n)}η. This implies that for T ≤ T (n) the pair (m, T) is in the low temperature region for all m ≤ n, and T (n) ≤ T_{n}. This is the right-hand side of relation (8.5).

To prove its left-hand side observe that because of Condition 5 there is some K such that

∑_{k=n}^{n+K−1} \frac{1}{8c^{(n)}} > \frac{100}{c^{(n)}η}

for all sufficiently large n with appropriate K > 0. We claim that for T ≥ T (n) the pair (n + K, T) is not in the low temperature region. This relation implies the left-hand side of (8.5). If (n + K, T) were in the low temperature region, then we would get with the help of formula (4.26) that

M_{n+K}^{2}(T) ≤ M_{n}^{2}(T)− ∑_{k=n}^{n+K−1} \frac{1}{8c^{(n)}} < \frac{100}{c^{(n)}η} − ∑_{k=n}^{n+K} \frac{1}{8c^{(n)}} < \frac{100}{c^{(n)}η} − \frac{100}{c^{(n)}η} = 0,

and this is a contradiction.

To prove formula (8.6) let us first observe that because of the continuity and strict monotonicity of the function M_{n}^{2}(T), M_{n}^{2}(T (n)) = \frac{100}{c^{(n)}η}. It follows from the last statement of Lemma 4.3 and formula (8.1) that N_{1}(T) ≤ n for all T (n) − ε < T < T (n) with an appropriately small ε > 0. (The number N_{1}(T) was defined in (4.18).). Hence we get with the help of formula (8.1) that for sufficiently large n and T (n) − ε < T < T (n)

\frac{100}{c^{(n)}η} − \frac{2}{c^{(n)}} + C̄_{1}(T (n) − T) ≤ \frac{100}{c^{(n+1)}η} − \frac{1}{c^{(n)}} + C̄_{1}(T (n) − T)

≤ M_{n+1}^{2}(T)

≤ \frac{100}{c^{(n)}η} − \frac{1}{8c^{(n)}} + C̄_{2}(T (n) − T)

≤ \frac{100}{ηc^{(n+1)}} − \frac{1}{9c^{(n)}} + C̄_{2}(T (n) − T)

with some appropriate constants C̄_{2} > C̄_{1} > 0. Hence the solution of the equation M_{n+1}^{2}(T) = \frac{100}{c^{(n+1)}η} satisfies the inequality K_{1} < c^{(n)}(T − T (n)) < K_{2} with appropriate constants K_{2} > K_{1} > 0. Since the solution of this equation is T (n + 1), this fact implies relation (8.6).

It is not difficult to see that T_{c} is in the low temperature region. Since the inequality M_{n}^{2}(T_{c}) = M_{n}^{2}(T (n)) + (M_{n}^{2}(T_{c}) − M_{n}^{2}(T (N))) ≤ \frac{100}{c^{(n)}η} + const. (T (n) − T_{c}) holds for all large n because of (8.1), lim_{n→∞} M_{n}(T_{c}) = 0. Then relation (8.1) implies that

C_{1} (T_{c} − T)) ≤ M_{n}^{2}(T_{c}) − M_{n}^{2}(T) ≤ C_{2} (T_{c} − T)

with some positive constants C_{2} > C_{1} > 0 if T_{c} ≥ T ≥ T_{c} − ε. Letting n tend to infinity in the last relation we get formula (3.21). Since formula (8.4) is equivalent to (3.20) Theorem 3.4 is proved. □

Proof of Theorem 1.3. By Corollary of Theorem 3.1, if the Dyson condition (1.2) is violated then all temperatures T > 0 belong to the high temperature region. By Corollary of Theorem 3.3 relation (1.13) holds, and the measures ν̃_{n,T} (dx) tends to the standard normal distribution as n → ∞. Theorem 1.3 is proved. □

Proof of Theorem 1.5.

Part 1). The convergence of ν̃_{n,T} (dx) to the r-dimensional standard Gaussian distribution and relation (1.26) follow from Corollary of Theorem 3.3. The asymptotics (1.27) follows from (3.19) and (3.20).

Part 2). Formula (1.28) follows from (3.7), and the convergence of ν̃_{n,T_{c}}(dx) to the uniform distribution on the sphere follows from Theorems 3.2 and 3.4. Namely, Theorem 3.4 tells us that the critical temperature T_{c} belongs to the low temperature region. Then formula (3.8) proves that the probability distribution ν̃_{n,T_{c}}(dx) converges to the uniform distribution on the sphere. As a matter of fact, (3.8) proves much more: it proves the convergence at T = T_{c} of the distribution of normalized fluctuations of the mean spin along the radius to a limit. Indeed, by (3.8),

lim_{n→∞} ∥\frac{1}{M_{n}(T_{c})}f_{n} (\frac{t}{M_{n}(T_{c})}, T_{c}) − g (t)∥ = 0,

where

∥f(t)∥ = ∑_{j=0}^{2} sup_{t≥−c^{(n)}M_{n}^{2}(T)} e^{|t|} |\frac{d^{j}f(t)}{d t^{j}} |

and the probability density g(t) is defined as a solution of the fixed point equation (1.18). By (2.13),

f_{n}(t, T_{c}) = \frac{1}{c^{(n)}} q̄_{n} (M_{n}(T_{c}) + \frac{t}{c^{(n)}} , T_{c}) ,

hence

lim_{n→∞} ∥\frac{q̄_{n} (M_{n}(T_{c}) + \frac{t}{c^{(n)}M_{n}(T_{c})}, T_{c})}{c^{(n)}M_{n}(T_{c})} − g (t)∥ = 0.

To obtain a scaling limit of q_{n} near M_{n}(T_{c}), let us rewrite the latter formula as

lim_{n→∞} ∥\frac{q̄_{n} (M_{n}(T_{c}) (1 + \frac{t}{c^{(n)}M_{n}^{2}(T_{c})}) , T_{c})}{c^{(n)}M_{n}(T_{c})} − g (t)∥ = 0.  (8.7)

Let us evaluate the asymptotics of c^{(n)}M_{n}^{2}(T_{c}) as n → ∞. By (3.7),

lim_{n→∞} \frac{M_{n}^{2}(T_{c})}{∑{∞ \atop k=n} \frac{1}{c^{(k)}}} = \frac{r − 1}{2},  (8.8)

since M_{∞}(T_{c}) = 0. Define

λ_{n} = l_{n} ∑_{k=n}^{∞} \frac{1}{l_{k}} .  (8.9)

By Condition 2 on the sequence {l_{n}},

lim_{n→∞} λ_{n} = ∞.  (8.10)

By (2.23)

lim_{n→∞} \frac{c^{(n)}}{l_{n}} = 3,

hence

lim_{n→∞} \begin{matrix}∑ & {∞ \atop k=n} & \frac{1}{c^{(k)}} \\ ∑ & {∞ \atop k=n} & \frac{1}{l_{k}} \end{matrix} = \frac{1}{3},

and by (8.8), (8.9),

lim_{n→∞} \frac{M_{n}^{2}(T_{c})}{∑{∞ \atop k=n} \frac{1}{l_{k}}} = lim_{n→∞} \frac{M_{n}^{2}(T_{c})l_{n}}{λ_{n}} = \frac{r − 1}{6},  (8.11)

which relation is equivalent to (1.28). Therefore,

lim_{n→∞} \frac{c^{(n)}M_{n}^{2}(T_{c})}{λ_{n}} = \frac{r − 1}{2}.

Substituting this limit into (8.7), we obtain that

lim_{n→∞} ∥\frac{2M_{n}(T_{c})}{(r − 1)λ_{n}} q̄_{n} (M_{n}(T_{c}) (1 + \frac{t}{\frac{r−1}{2} λ_{n}} ) , T_{c}) − g (t)∥ = 0. (8.12)

This implies that the probability density Z̄_{n}(T_{c})^{−1} q̄_{n}(M_{n}(T_{c})x, T_{c}) is localized in a neighborhood of order λ_{n}^{−1} of the point 1, and after the proper scaling it converges to g (t) as n → ∞.

Let us consider now the scaling limit of the probability density p̄_{n}(x, T_{c}). By equations (2.7) and (2.11),

p̄_{n}(x, T_{c}) = Z_{n}(T_{c})^{−1} exp (−\frac{A_{n}l_{n}T_{c}^{−1}x^{2}}{2} ) q̄_{n}(T_{c}^{−1/2}x, T_{c}), x ≥ 0,

where Z_{n}(T_{c})^{−1} is a norming factor, hence

\begin{matrix}p̄_{n}(T_{c}^{1/2}M_{n}(T_{c})x, T_{c}) \\ = Z_{n}(T_{c})^{−1} exp (−\frac{A_{n}l_{n}M_{n}^{2}(T_{c})x^{2}}{2} ) q̄_{n}(M_{n}(T_{c})x, T_{c}). \end{matrix}

Applying the same scaling as in (8.12), we obtain that

\begin{aligned}p̄_{n} & (T_{c}^{1/2}M_{n}(T_{c}) (1 + \frac{t}{\frac{r−1}{2} λ_{n}} ) , T_{c}) \\ & = Z_{n}(T_{c})^{−1} exp (−\frac{A_{n}l_{n}M_{n}^{2}(T_{c})}{2} (1 + \frac{t}{\frac{r−1}{2} λ_{n}} )^{2}) \\ & × q̄_{n} (M_{n}(T_{c}) (1 + \frac{t}{\frac{r−1}{2} λ_{n}} ) , T_{c}) . \end{aligned}  (8.13)

Consider the expression in the exponent,

\begin{aligned}\frac{A_{n}l_{n}M_{n}^{2}(T_{c})}{2} (1 + \frac{t}{\frac{r−1}{2} λ_{n}} )^{2} & = \frac{A_{n}l_{n}M_{n}^{2}(T_{c})}{2} + \frac{2A_{n}l_{n}M_{n}^{2}(T_{c})t}{(r − 1)λ_{n}} \\ & + \frac{2A_{n}l_{n}M_{n}^{2}(T_{c})t^{2}}{(r − 1)^{2}λ_{n}^{2}} . \end{aligned} (8.14)

By (8.11), (2.22) and (8.10)

lim_{n→∞} \frac{2A_{n}l_{n}M_{n}^{2}(T_{c})}{(r − 1)λ_{n}} = \frac{2}{3}, lim_{n→∞} \frac{2A_{n}l_{n}M_{n}^{2}(T_{c})}{(r − 1)^{2}λ_{n}^{2}} = 0.  (8.15)

The constant term in (8.14) is not important, because it changes in (8.13) the norming constant only. Therefore, from (8.12)-(8.15) we obtain that

lim_{n→∞}∥Z′_{n}^{−1}e^{(2/3)t}p̄_{n} (T_{c}^{1/2}M_{n}(T_{c}) (1 + \frac{t}{\frac{r−1}{2} λ_{n}} ) , T_{c}) − g (t)∥ = 0 (8.16)

This implies that

\begin{aligned}lim_{n→∞} & ∥ & \frac{2T_{c}^{1/2}M_{n}(T_{c})}{(r − 1)λ_{n}} & p̄_{n} & (T_{c}^{1/2}M_{n}(T_{c}) (1 + \frac{t}{\frac{r−1}{2} λ_{n}} ) , T_{c}) \\ & − Z^{−1}e^{−(2/3)t}g (t)∥′ = 0. \end{aligned}  (8.17)

where Z^{−1}e^{−(2/3)t}g (t) is a probability density (this determines the constant Z), and

∥f(t)∥′ = ∑_{j=0}^{2} sup_{t≥− \frac{r−1}{2} λ_{n}} e^{|t|/3} |\frac{d^{j}f(t)}{d t^{j}} | .  (8.18)

Substituting t for (bt − a) in (8.17), we obtain that

lim_{n→∞} ∥\frac{A_{n}}{d_{n}} p̄_{n} (A_{n} (1 + \frac{t}{d_{n}} )) − π(t)∥′ = 0.  (8.19)

where A_{n} > 0 is some constant. Since

∫_{−∞}^{∞} t p̄_{n} (M̂_{n}(T_{c}) (1 + \frac{t}{d_{n}} )) = 0, ∫_{−∞}^{∞} t π(t) dt = 0,

we can replace A_{n} by M̂_{n}(T_{c}) in (8.19). This proves equation (1.32).

Part 3). The results of Part 3) of Theorem 3 were already proved with the exception of relation (1.35) in the discussion after the formulation of Theorem 3.2. But relation (1.35) is a direct consequence of relation (3.21) proved in Theorem 3.4 and the identity M(T) = \sqrt{T} M_{∞}(T) for T < T_{c} which was also proved in the above discussion. Theorem 1.5 is proved. □

Appendix A. The Proof of Theorem 8.1.

To prove Theorem 8.1 we need good estimates on the partial derivatives

g_{n}(x, T) = \frac{∂}{∂T} f_{n}(x, T),

of a scaled version of the functions q_{n}(x, T). This can be done similarly to the estimation of the functions f_{n}(x, T), done in Section 4. First we give estimates for the starting function g_{0}(x, T), then prove that similar estimates hold for small indices n, more explicitly for n ≤ N with the index N defined in (1.12). Then inductive hypotheses can be formulated and proved for the functions g_{n}(x, T). In Section 4 we have introduced certain operators Q̄_{n}, their normalization Q_{n} and the linearization of these operators denoted by T̄_{n} and T_{n}. The inductive hypotheses formulated there were closely related to the properties of these operators. Now we want to work similarly. To do this we have to introduce some new operators. We introduce certain operators R̄_{n} and R_{n} which are the derivatives of the operators Q̄_{n} and Q_{n} with respect to the variable T. We also need their linear approximation which we shall denote by Ū_{n} and U_{n}. We have to study the action of these operators on the functions g_{n}(x, T) = \frac{∂}{∂T} f_{n}(x, T) and their Fourier transform.

An appropriate description of the asymptotic behaviour of the starting functions f_{0}(x, T) and numbers M_{0}(T) were already given in formulas (4.2) — (4.8). Some more calculation yields, with the help of some formulas in Section 4, the following estimates for the derivatives of the magnetization M_{0}(T) and the norming constant Z_{0}(T) if T < c_{0}A_{0}/2.

|\frac{d}{dT} (M_{0}(T) − M̄_{0}(T))| ≤ const. \sqrt{κ}.

\frac{C_{1}}{\sqrt{κ} T^{2}} < −\frac{dM_{0}(T)}{dT} < \frac{C_{2}}{\sqrt{κ} T^{2}} with some ∞ > C_{2} > C_{1} > 0, (A.1)

and

|\frac{dZ_{0}(T)}{dT} − \frac{\sqrt{π}}{2(A_{0} − T)^{3/2}} | ≤ const. \sqrt{κ}.

The derivatives of the functions q̄_{0}(x, T) and f_{0}(x, T) satisfy the inequalities

|\frac{∂q̄_{0}(x + M_{0}(T), T)}{∂T} − \begin{matrix}√ & \overline{A_{0} − T} \\ √ & \overline{π} \end{matrix} (x^{2} − \frac{1}{2(A_{0} − T)}) e^{−(A_{0}−T)x^{2}} |

≤ const. κ^{1/4}, if |x| < log κ^{−1},

and

|\frac{∂q̄_{0}(x + M_{0}(T), T)}{∂T}| ≤ C exp \{−\frac{(A_{0} − T)}{4} |2x + \frac{x^{2}}{M_{0}^{2}(T)}|\} (A.2)
for x ≥ −M_{0}(T).

We shall apply the notation

g_{n}(x, T) = \frac{∂f_{n}(x, T)}{∂T}, n = 0, 1, ….  (A.3)

Since f_{0}(x, T) = q_{0}(x + M_{0}(T), T) the previous estimates together with the results of Section 4 yield a sufficiently good control on g_{0}(x, T). The functions g_{n}(x, T), n = 1, 2, …, can be estimated inductively with respect to the parameter n.

Put

R̄_{n}f_{n}(x, T) = \frac{∂}{∂T} Q̄_{n,M_{n}(T)}^{c}f_{n}(x, T)

and

R_{n}f_{n}(x, T) = \frac{∂}{∂T} Q_{n,M_{n}(T)}^{c}f_{n}(x, T) = g_{n+1}(x, T).

Then

R̄_{n}f_{n}(x, T) = R̄_{n}^{(1)}f_{n}(x, T) + R̄_{n}^{(2)}f_{n}(x, T)  (A.4)

with

R̄_{n}^{(1)}f_{n}(x, T) = 2 ∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} exp \{−\frac{u^{2}}{c^{(n)}} − v^{2}\} f_{n}(ℓ_{n,M_{n}(T)}^{c,+}(x, u, v), T)
g_{n}(ℓ_{n,M_{n}(T)}^{c,−}(x, u, v), T) du dv,

where the functions g_{n}(x, T) and ℓ_{n,M_{n}(T)}^{c,±}(x, u, v), T) were defined in (A.3) and (2.18), and

R̄_{n}^{(2)}f_{n}(x, T) = −2 ∫_{u∈ℝ^{1}, v∈ℝ^{r−1}}exp \{−\frac{u^{2}}{c^{(n)}} − v^{2}\} f_{n}(ℓ_{n,M_{n}(T)}^{c,+}(x, u, v), T)

h_{n}(x, u, v, T)\frac{∂}{∂x}f_{n}(ℓ_{n,M_{n}(T)}^{c,−}(x, u, v), T) du dv

with

h_{n}(x, u, v, T) = −\frac{∂ℓ_{n,M_{n}(T)}^{c,−}(x, u, v)}{∂T}

= \frac{M′_{n}(T)v^{2}}{\sqrt{(M_{n}(T) + \frac{x}{c^{(n+1)}} − \frac{u}{c^{(n)}} )^{2} + \frac{v^{2}}{c^{(n)}}}}

\frac{1}{\sqrt{(M_{n}(T) + \frac{x}{c^{(n+1)}} − \frac{u}{c^{(n)}} )^{2} + \frac{v^{2}}{c^{(n)}}} + (M_{n}(T) + \frac{x}{c^{(n+1)}} − \frac{u}{c^{(n)}} )}.

The function g_{n+1}(x, T) can be expressed as

g_{n+1}(x, T) = R_{n}f_{n}(x, T)
= \frac{R̄_{n}f_{n}(x + m_{n}(T), T)}{Z_{n}(T)} + \frac{\frac{∂}{∂x} Q̄_{n}f_{n}(x + m_{n}(T), T)}{Z_{n}(T)} \frac{dm_{n}(T)}{dT}
−\frac{Q̄_{n}f_{n}(x + m_{n}(T), T)}{Z_{n}^{2}(T)} \frac{dZ_{n}(T)}{dT}  (A.5)

with

Z_{n}(T) = ∫_{−c^{(n)}M_{n}(T)}^{∞} Q̄_{n}f_{n}(x, T) dx.

If the parameter κ > 0 in formula (2.10) is sufficiently small, then q̄_{0}(x, T) and the functions g_{n}(x, T), n ≤ N, can be estimated similarly to the proof of Proposition 4.1 or Proposition 1 in [BM3]. Relation (A.7) formulated below can be deduced from formula (A.2) similarly to the proof of Lemma 1 of that paper. Then an argument similar to the proof of Lemma 2 in [BM3] enables one to prove formula (A.6) formulated below. In this argument one can observe that a negligible error is committed if in the integrals appearing in the definition of R̄_{n}f_{n}(x, T) the arguments ℓ_{n,M_{n}(T)}^{c,±}(x, u, v) defined in formula (2.18) are replaced by \frac{x}{c_{n+1}} ± u. Some calculation also shows that

we commit a negligible error by replacing R_{n}f_{n}(x, T) with \frac{R̄_{n}^{(1)}f_{n}(x,T)}{Z_{n}(T)}. In such a way we get that

|g_{n}(x, T) − \begin{matrix}√ & \overline{A_{0} − T} \\ √ & \overline{π} \end{matrix} \frac{2^{n/2}}{c^{(n)}} (x^{2} − \frac{1}{2(A_{0} − T)} \frac{c^{(n)}^{2}}{2^{n}} )

exp \{−(A_{0} − T)\frac{2^{n}x^{2}}{c^{(n)}^{2}} \}|

≤ C(n)κ^{1/4} exp \{−\frac{(A_{0} − T)}{4} \frac{2^{n}}{c^{(n)}} |2x + \frac{x^{2}}{M_{n}^{2}(T)}|\}

if |x| < 2^{−n} log κ^{−1},  (A.6)

|g_{n}(x, T)| ≤ C(n) exp \{−\frac{(A_{0} − T)}{4} \frac{2^{n}}{c^{(n)}} |2x + \frac{x^{2}}{M_{n}^{2}(T)}|\} for x ≥ −M_{n}(T),

(A.7)

|M_{n}(T) − M_{0}(T)| ≤ C(n)κ^{1/2},  |Z_{n}(t) − \begin{matrix}√ & \overline{π} \\ √ & \overline{A_{0} − t} \end{matrix}| ≤ C(n)κ^{1/2}

with some constant C(n) which may depend on n but not on the parameter κ of the model.

The previous results are sufficient to handle the functions g_{n}(x, T) for small indices n ≤ N. To work with indices n ≥ N we have to introduce, similarly to the argument in Section 4, the regularization of the functions g_{n}(x, T), the linearization Ū_{n} and U_{n} of the operators R̄_{n} and R_{n} and to describe their action in the Fourier space.

Define the regularization of the function g_{n}(x, T) as

φ_{n}(g_{n}(x, T)) = \frac{∂φ_{n}(f_{n}(x, T))}{∂T}.  (A.8)

We want to approximate the operator R_{n} with a simpler operator U_{n} in analogy with the approximation of Q_{n} by T_{n}. Then we formulate and prove some inductive hypothesis about the behaviour of the operators R_{n} and U_{n}.

A natural approximation of the operators R̄_{n} and R_{n} by some operators Ū_{n} and U_{n} can be obtained by differentiating T̄_{n}φ(f_{n}(x, T)) and T_{n}φ_{n}(f_{n}(x, T)) with respect to the variable T. These considerations suggest the definition of the operators

Ū_{n}φ_{n}(f_{n}(x, T)) = 2 ∫_{u∈ℝ^{1},v∈ℝ^{r−1}} e^{−v^{2}}φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} + u + \frac{v^{2}}{2M_{n}(T)}, T ))

\{φ_{n} (g_{n} (\frac{x}{c̄_{n+1}} − u + \frac{v^{2}}{2M_{n}(T)}, T ))

−v^{2} \frac{M′_{n}(T)}{2M_{n}(T)^{2}} \frac{∂}{∂x}φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} − u + \frac{v^{2}}{2M_{n}(T)}, T ))\} du dv

with the functions g_{n}(x, T) and φ_{n}(g_{n}(x, T)) defined in (A.3) and (A.8) and

U_{n}φ_{n}(f_{n}(x, T)) = U_{n}^{(1)}φ_{n}(f_{n}(x, T)) + U_{n}^{(2)}φ_{n}(f_{n}(x, T))

with

U_{n}^{(1)}φ_{n}(f_{n}(x, T)) = \frac{8}{c̄_{n+1}Γ(\frac{r−1}{2})V (S^{r−2})}

∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} e^{−v^{2}}φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} + u − \frac{r − 1}{4M_{n}(T)} + \frac{v^{2}}{2M_{n}(T)}, T ))

φ_{n} (g_{n} (\frac{x}{c̄_{n+1}} − u − \frac{r − 1}{4M_{n}(T)} + \frac{v^{2}}{2M_{n}(T)}, T )) du dv,

and

U_{n}^{(2)}φ_{n}(f_{n}(x, T)) = \frac{8}{c̄_{n+1}Γ(\frac{r−1}{2})V (S^{r−2})}

∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} e^{−v^{2}} (\frac{(r − 1)M′_{n}(T)}{4M_{n}^{2}(T)} − v^{2} \frac{M′_{n}(T)}{2M_{n}(T)^{2}} )

φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} + u − \frac{r − 1}{4M_{n}(T)} + \frac{v^{2}}{2M_{n}(T)}, T ))

\frac{∂}{∂x}φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} − u − \frac{r − 1}{4M_{n}(T)} + \frac{v^{2}}{2M_{n}(T)}, T )) du dv.

We can calculate the Fourier transform of the functions Ū_{n}φ_{n}(f_{n}(x, T)), U_{n}^{(1)}φ_{n}(f_{n}(x, T)) and U_{n}^{(2)}φ_{n}(f_{n}(x, T)) by expressing them with the help of convolutions. This is similar to the proof of formula (4.17). In the calculations we exploit the following identity. As simple integration by parts shows

_{\frac{∂}{∂x}}φ_{n}(f_{n}(ξ)) = ∫ e^{iξx} \frac{∂}{∂x}φ_{n}(f_{n}(x)) dx = −iξφ̃(f_{n}(ξ)). Hence we get that

˜Ū_{n}φ̃_{n}(f_{n}(ξ, T))

= \frac{c̄_{n+1}Γ(\frac{r−1}{2})V (S^{r−2})}{2} \begin{matrix}φ̃_{n} & (f_{n} (\frac{c̄_{n+1}}{2} ξ, T )) \\ (1 + i\frac{c̄_{n+1}}{2M_{n}(T)}ξ) & (r−1)/2 \end{matrix} φ̃_{n} (g_{n} (\frac{c̄_{n+1}}{2} ξ, T ))

+i\frac{c̄_{n+1}^{2}Γ(\frac{r+1}{2})V (S^{r−2})}{8} \frac{M′_{n}(T)}{M_{n}(T)^{2}} ξ \begin{matrix}φ̃_{n}^{2} & (f_{n} (\frac{c̄_{n+1}}{2} ξ, T )) \\ (1 + i\frac{c̄_{n+1}}{2M_{n}(T)}ξ) & (r+1)/2 \end{matrix} ,

Ũ_{n}^{(1)}φ̃_{n}(f_{n}(ξ, T))  (A.9)

= 2\begin{matrix}exp & \{i\frac{(r−1)c̄_{n+1}ξ}{4M_{n}(T)} \} \\ (1 + i\frac{c̄_{n+1}}{2M_{n}(T)}ξ) & (r−1)/2 \end{matrix} φ̃_{n} (f_{n} (\frac{c̄_{n+1}}{2} ξ, T )) φ̃_{n} (g_{n} (\frac{c̄_{n+1}}{2} ξ, T ))

and

Ũ_{n}^{(2)}φ̃_{n}(f_{n}(ξ, T)) = −i\frac{c̄_{n+1}(r − 1)M′_{n}(T)}{2M_{n}(T)^{2}} \begin{matrix}exp & \{i\frac{(r−1)c̄_{n+1}ξ}{4M_{n}(T)} \} \\ (1 + i\frac{c̄_{n+1}}{2M_{n}(T)}ξ) & (r−1)/2 \end{matrix} ξ

φ̃_{n}^{2} (f_{n} (\frac{c̄_{n+1}}{2} ξ, T )) (1 − \frac{1}{1 + i\frac{c̄_{n+1}}{2M_{n}(T)}ξ} ) .  (A.10)

The above relation can also be extended to a larger set of the variables ξ in the complex plane by means of analytic continuation.

Now we formulate the inductive hypotheses we want to prove in the Appendix.

Property K_{1}(n).

−\frac{dM_{n}(T)}{dT} > 0.

Property K_{2}(n).

|g_{n}(x, T)| = |\frac{∂}{∂T} f_{n}(x, T)|

< K |\frac{dM_{n}(T)}{dT}| exp \{−\frac{1}{\sqrt{β_{n}(T)}} |2x + \frac{x^{2}}{c^{(n)}M_{n}(T)}|\}
if x > −c^{(n)}M_{n}(T)

with a universal constant K.

Property K_{3}(n).

|g_{n}(x, T) − U_{n−1}φ_{n−1}(f_{n−1}(x, T))|

< K |\frac{dM_{n}(T)}{dT}| \frac{β_{n}(T)}{c^{(n)}} exp \{−\frac{1.4}{\sqrt{β_{n}(T)}} |2x + \frac{x^{2}}{c^{(n)}M_{n}(T)}|\}

if x > −c^{(n)}M_{n}(T)

with a universal constant K. The inequality remains valid if the function g_{n}(x, T) is replaced by its regularization φ_{n}(g_{n}(x, T)).

The following property K_{4}(n) which gives a bound on the Fourier transform of φ_{n}(g_{n}(x, T)) is an analog of Property J(n).

Property K_{4}(n).

|φ̃_{n}(g_{n}(−is, T)| = |∫ e^{sx}φ_{n}(g_{n}(x, T) dx| ≤ β_{n}^{3/2}(T)s^{2} |\frac{dM_{n}(T)}{dT}| e^{β_{n}(T)s^{2}}

if |s| < \frac{2}{\sqrt{β_{n+1}(T)}}.

In Property K_{4}(n) we formulated a weaker estimate than in J(n). It is enough to have a good bound on the moment generating function, i.e. on the analytic continuation of the Fourier transform to the imaginary axis together with the trivial estimate |φ̃_{n}(g_{n}(−is + t, T)| ≤ φ̃_{n}(g_{n}(−is, T) for all t.

The main result of the Appendix is the following Proposition A.

Proposition A. Let the properties K_{1}(m), K_{2}(m), K_{3}(m) and K_{4}(m) hold in a neighbourhood of a parameter T together with the property \frac{β_{m}(T)}{c^{m}} ≤ η (with the same small number η > 0 which appeared in the proof of Propositions 4.1 and 4.2) for all N ≤ m ≤ n, and let also the inductive hypotheses I(n) and J(n) be also satisfied. Then the properties K_{1}(n + 1), K_{2}(n + 1), K_{3}(n + 1) and K_{4}(n + 1) also hold for this parameter T. The expression

δ_{n}(T) = \frac{d}{dT} (m_{n}(T) − \frac{r − 1}{4M_{n}(T)}) = \frac{dm_{n}(T)}{dT} + \frac{r − 1}{4M_{n}^{2}(T)} \frac{dM_{n}(T)}{dT}

satisfies the inequality

|δ_{n}(T)| ≤ C |\frac{dM_{n}(T)}{dT}| \frac{β_{n+1}(T)}{c^{(n+1)}}β_{n+1}(T)  (A.11)

with an appropriate C > 0, where m_{n}(T) was defined in (2.15).

If we want to apply Proposition A, then first we have to show that properties K_{1}(n), K_{2}(n), K_{3}(n) and K_{4}(n) hold for n = N if T < c_{0}A_{0}/2. This can be done with the help of an argument similar to the proof in the Corollary of Lemma 1 in [BM3]. Property K_{1}(N) holds since \frac{dM_{N} (T)}{dT} hardly differs from \frac{dM_{0}(T)}{dT}. Property K_{2}(N) can be proved by means of relations (A.6) and (A.7). In the proof of Property K_{3}(N) still the following additional observation is needed. Relation (A.6) remains valid if the function g_{N}(x, T) = R_{N}f_{N−1}(x, T) is replaced by U_{N}φ_{N−1}(f_{N−1}(x, T)) in this formula. (The term \frac{dM_{n}(T)}{dT} on the right-hand side of the inductive hypotheses do not play an important role for n = N. It is strongly separated from zero if T ≤ c_{0}A_{0}/2.)

Relation K_{4}(N) can be proved again with the help of formulas (A.6), (A.7) and the relations

∫ φ_{n}(g_{n}(x, T)) dx = ∫ xφ_{n}(g_{n}(x, T)) dx = 0.

These relations imply that the value of the function φ̃_{n}(g_{n}(s, T) and of its first derivative is zero in the point s = 0. Hence it is enough to give a good estimate of the second derivative of φ̃_{n}(g_{n}(s, T).

Let us formulate the following Corollary of Proposition A.

Corollary. Under the Conditions of Theorem 3.4 the set of the points T for which (n, T) is in the low temperature region is an interval (0, T_{n}) for all n ≥ 0. The inductive hypotheses K_{1}(n), K_{2}(n), K_{3}(n) and K_{4}(n) hold for all T ∈ (0, T_{n}).

Proof of the Corollary. The Corollary simply follows from Proposition A by means of induction with respect to n. In this induction we assume the statement of the Corollary for a fixed n together with the assumption that β_{n}(T) is monotone increasing in the variable T for 0 < T < T_{n}. The Corollary and the additional assumption hold for n = N with T_{N} = c_{0}A_{0}/2. If properties K_{1}(n), K_{2}(n), K_{3}(n) and K_{4}(n) hold for n, then because of Property K_{1}(n) the function M_{n}(T) is monotone decreasing and β_{n+1}(T) is monotone increasing in the variable T. Then T_{n+1} = min(T_{n}, max(T: β_{n+1}(T) < η)), and by Proposition A the statements of the Corollary hold for n + 1. □

Before turning to the proof of Proposition A we prove Theorem 8.1 with its help.

Proof of Theorem 8.1. The proof of Part a.) is contained in the previous estimates of the Appendix. Part b.) can be obtained by differentiating the second formula in (2.16), and applying formula (A.11). □

Proof of Proposition A. Some calculation yields that because of properties K_{4}(n), J(n) relations (A.9) and (A.10) the Fourier transforms

Ũ_{n}^{(1)}φ̃_{n}(f_{n}(ξ, T)), Ũ_{n}^{(2)}φ̃_{n}(f_{n}(ξ, T))

satisfy the inequalities

|Ũ_{n}^{(1)}φ̃_{n}(f_{n}(t + is, T))|

≤ 2 |\frac{dM_{n}(T)}{dT}| (\frac{c̄_{n+1}}{2} s)^{2} β_{n}(T)^{3/2}

exp \{(\frac{c̄_{n+1}^{2}β_{n}(T)}{2} + \frac{1}{M_{n}^{2}(T)}) s^{2}\} \frac{1}{1 + α_{n}(T)t^{2}}

and

|Ũ_{n}^{(2)}φ̃_{n}(f_{n}(t + is, T))|

≤ \frac{c̄_{n+1}^{2}|M′_{n}(T)|}{8M_{n}(T)^{3}}(s^{2} + t^{2}) exp \{(\frac{c̄_{n+1}^{2}β_{n}(T)}{2} + \frac{1}{M_{n}^{2}(T)}) s^{2}\}

\frac{1}{(1 + α_{n}(T)t^{2})^{2} (1 − \frac{c̄_{n+1}}{2M_{n}(T)}s)}

for |s| < \frac{4}{c̄_{n+1}\sqrt{β_{n+1}(T)}}.

The function φ_{n}(g_{n}(x, T)) can be computed by means of the application of the inverse Fourier transformation and by replacement of the domain of integration from the real line to the line

\{z = i sign x \frac{2}{\sqrt{β_{n+1}(T)}} + t, t ∈ ℝ^{1}\} .

We get, by applying the above estimates for the Fourier transforms Ũ_{n}^{(1)} and Ũ_{n}^{(2)} and exploiting the relation \frac{M′_{n}(T)}{2M_{n}(T)^{3}} ≤ \frac{1}{200}β_{n+1}(T)^{2} \frac{dM_{n}(T)^{2}}{dT} together with the fact that the constants α_{n}(T) and β_{n}(T) introduced in the definition of Properties I(n) and J(n) have the same order of magnitude that

|U_{n}φ_{n}(f_{n}(x, T))| ≤ −K_{1} \frac{dM_{n}(T)}{dT}e^{−2|x|β_{n+1}(T)^{−1/2}}  (A.12)

≤ −K_{2} \frac{dM_{n}(T)}{dT} exp \{−\frac{1}{\sqrt{β_{n+1}(T)}} |2x + \frac{x^{2}}{c^{(n+1)}M_{n+1}(T)}|\} .

The estimates obtained for Ũ_{n}^{(1)} and Ũ_{n}^{(2)} yield, with the choice t = 0 and some calculation that

∥Ũ_{n}φ̃_{n}(f_{n}(−is, T))| ≤ −\frac{9}{10} \frac{dM_{n}(T)}{dT}β_{n+1}(T)^{3/2}s^{2}e^{β_{n+1}(T)s^{2}}
if |s| < \frac{2}{\sqrt{β_{n+2}(T)}}. (A.13)

(In the proof of Property K_{4}(n + 1) it will be important that the right-hand side of (A.13) is less than the expression at the right-hand side of the formula which defines Property K_{4}(n + 1).)

We need a good estimate on the difference of R_{n}f_{n}(x, T)−U_{n}φ_{n}(f_{n}(x, T)) and its Fourier transform. These expressions can be bounded similarly to the proof of the corresponding inequalities in the proof of Proposition 3 in paper [BM3]. One has to compare the difference of the corresponding terms in the expressions Q_{n}φ_{n}(f_{n}(x, T)) and R_{n}φ_{n}(f_{n}(x, T)). Some calculation yields that

|Z_{n}(T) − \frac{c̄_{n+1}\sqrt{π}}{2} | ≤ \frac{β_{n}(T)}{c^{(n)}}, |m_{n}(T) + \frac{1}{4M_{n}(T)}| ≤ \frac{β_{n}(T)}{c^{(n)}} \sqrt{β_{n}(T)},

(A.14)

|\frac{dZ_{n}(T)}{dT}| ≤ −K \frac{β_{n}(T)}{c^{(n)}}β_{n}^{1/2}(T)\frac{dM_{n}(T)}{dT}, (A.15)
|\frac{d}{dT} (m_{n}(T) + \frac{1}{4M_{n}(T)})| ≤ −K \frac{β_{n+1}(T)}{c^{(n+1)}}β_{n+1}(T)\frac{dM_{n}(T)}{dT}.

Relation (A.11) is a consequence of (A.15). Property K_{1}(n + 1) can be deduced from the above inequalities, since

−\frac{dM_{n+1}(T)}{dT} = −\frac{dM_{n}(T)}{dT} + \frac{1}{c^{(n+1)}} \frac{dm_{n}(T)}{dT}
≥ −\frac{dM_{n}(T)}{dT} (1 − \frac{1}{c^{(n+1)}} (\frac{1}{4M_{n}^{2}(T)} + K \frac{β_{n}^{2}(T)}{c^{(n)}}))
≥ −\frac{1}{2} \frac{dM_{n}(T)}{dT}.

Now we turn to the proof of Property K_{3}(n + 1). We do it by estimating the errors we make by replacing the terms in the sum at the right-hand side of (A.5) by their natural approximation if we replace R_{n}f(x, T) by U_{n}φ_{n}(f_{n}(x, T)). (We also use formula (A.4) in that calculation.) We get, by applying again inequalities (A.14) and (A.15) together with the estimates obtained for f_{n}(x, T), similarly to the proof of the estimates in the lemmas needed for the proof of Lemma 3 in [BM1] that

|\frac{Q̄_{n}f_{n}(x + m_{n}(T), T)}{Z_{n}^{2}(T)} \frac{dZ_{n}(T)}{dT}|

≤ K \frac{β_{n}(T)}{c^{(n)}} |\frac{dM_{n}(T)}{dT}| exp \{\frac{−1.5}{\sqrt{β_{n}(T)}} |2x + \frac{x^{2}}{c^{(n)}M_{n}(T)}|\}

if x ≥ −c^{(n+1)}M_{n+1}(T),

| \frac{\frac{∂}{∂x} Q̄_{n}f_{n}(x + m_{n}(T), T)}{Z_{n}(T)} \frac{dm_{n}(T)}{dT} − \frac{8(r − 1)}{c̄_{n+1}Γ(\frac{r−1}{2})V (S^{r−2})} \frac{M′_{n}(T)}{4M_{n}^{2}(T)} ×

∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} e^{−v^{2}}φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} + u − \frac{r − 1}{4M_{n}(T)} + \frac{v^{2}}{2M_{n}(T)}, T ))

\frac{∂}{∂x}φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} − u − \frac{r − 1}{4M_{n}(T)} + \frac{v^{2}}{2M_{n}(T)}, T )) du dv|

≤ K \frac{β_{n}(T)}{c^{(n)}} |\frac{dM_{n}(T)}{dT}| exp \{\frac{−1.5}{\sqrt{β_{n}(T)}} |2x + \frac{x^{2}}{c^{(n)}M_{n}(T)}|\}

if x ≥ −c^{(n+1)}M_{n+1}(T)

and

|\frac{R̄_{n}^{(2)}f_{n}(x + m_{n}(T), T)}{Z_{n}(T)} + \frac{8}{c̄_{n+1}Γ(\frac{r−1}{2})V (S^{r−2})} \frac{M′_{n}(T)}{2M_{n}^{2}(T)} ×

∫_{u∈ℝ^{1}, v∈ℝ^{r−1}} v^{2}e^{−v^{2}}φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} + u − \frac{r − 1}{4M_{n}(T)} + \frac{v^{2}}{2M_{n}(T)}, T ))

\frac{∂}{∂x}φ_{n} (f_{n} (\frac{x}{c̄_{n+1}} − u − \frac{r − 1}{4M_{n}(T)} + \frac{v^{2}}{2M_{n}(T)}, T )) du dv|

≤ K \frac{β_{n}(T)}{c^{(n)}} |\frac{dM_{n}(T)}{dT}| exp \{−\frac{1.5}{\sqrt{β_{n}(T)}} |2x + \frac{x^{2}}{c^{(n)}M_{n}(T)}|\}

if x ≥ −c^{(n+1)}M_{n+1}(T).

To prove Property K_{3}(n + 1) we still need an estimate which compares the terms

\frac{R̄_{n}^{(1)}f_{n}(x + m_{n}(T), T)}{Z_{n}(T)} and U_{n}^{(1)}φ_{n}(f_{n}(x, T)).

We claim that

|\frac{R̄_{n}^{(1)}f_{n}(x + m_{n}(T), T)}{Z_{n}(T)} − U_{n}^{(1)}φ_{n}(f_{n}(x, T))| ≤ K \frac{β_{n}(T)}{c^{(n)}} |\frac{dM_{n}(T)}{dT}|

exp \{−\frac{1.5}{\sqrt{β_{n}(T)}} |2x + \frac{x^{2}}{c^{(n)}M_{n}(T)}|\} if x ≥ −c^{(n+1)}M_{n+1}(T).

This estimate can be proved by means of Property K_{3}(n). With the help of this relation it can be shown that a negligible error is committed if in the integrals defining R̄_{n}^{(1)}f_{n}(x + m_{n}(T), T) and U_{n}^{1}φ_{n}(f_{n}(x, T)) the functions g_{n} and φ_{n}(g_{n}) are replaced by the function U_{n}φ_{n−1}(f_{n−1}). After this replacement the proof of Theorem 3.2 can be adapted, since we can bound not only the function U_{n}φ_{n−1}(f_{n−1}), but also its partial derivative with respect to the variable x.

These estimates together imply Property K_{3}(n+1), and some calculation shows that a version of Property K_{3}(n+1), where the function g_{n+1}(x, T) is replaced by its regularization φ_{n+1}(g_{n+1}(x, T)) is also valid. Since we gave a good estimate on U_{n}φ_{n}(f_{n}(x, T)) in (A.12), some calculation yields the proof of Property K_{2}(n + 1). It remained to prove Property K_{4}(n + 1).

Because of (A.13) and (A.15) (The latter formula together with (2.15) and (2.16) imply that formula (A.13) remain valid with a slightly bigger coefficient if the term \frac{dM_{n}(T)}{dT} is replaced by \frac{dM_{n+1}(T)}{dT} in it), it is enough to give a good bound on the difference φ̃_{n+1}(g_{n+1}(−is)) − Ũ_{n}φ̃_{n}(f_{n}(−is)) to prove property K_{4}(n + 1). This can be done in the following way:

By applying the modified property of K_{3}(n + 1), where the function g_{n+1}(x) is replaced by φ_{n+1}g_{n+1}(x) we get that

|\frac{∂^{2}}{∂s^{2}} [φ̃_{n+1}(g_{n+1}(−is, T)) − Ũ_{n}φ̃_{n}(f_{n}(−is, T))]|

≤ − ∫ x^{2} \frac{dM_{n}}{dT} \frac{β_{n+1}(T)}{c^{(n+1)}} exp \{(|t| − \frac{2.8}{\sqrt{β_{n+1}(T)}}) x\} dx

≤ K \frac{β_{n+1}^{5/2}(T)}{c^{(n+1)}} \frac{dM_{n}^{2}}{dT}

if |s| ≤ \frac{2}{\sqrt{β_{n+2}(T)}}.

Since

φ̃_{n+1}(g_{n+1}(0, T)) − Ũ_{n}φ̃_{n}(f_{n}(0, T))

= \frac{∂}{∂s} (φ̃_{n+1}(g̃_{n+1}(−is, T) − Ũ_{n}φ̃_{n}(f_{n}(−is, T))|_{s=0} = 0,

the last relation implies that

|φ̃_{n+1}(g̃_{n+1}(−is, T) − Ũ_{n}φ̃_{n}(f_{n}(−is, T)| ≤ −K \frac{β_{n+1}(T)}{c^{(n+1)}}β_{n+1}^{3/2} \frac{dM_{n}(T)}{dT}s^{2}

if |s| ≤ \frac{2}{\sqrt{β_{n+2}(T)}}. This estimate together with relation (A.13) imply Prop-

erty K_{4}(n + 1) if the number η which is an upper bound for β_{n+1}(T)/c^{(n+1)} is chosen sufficiently small. Proposition A is proved. □

Appendix B. The Proof of Proposition 1.2.

Condition 1. We have that for n ≥ 1,

1 < c_{n} = (\frac{1 + an}{1 + a(n − 1)})^{λ}

Observe that c_{n} is decreasing and

lim_{n→∞} c_{n} = 1, c_{n} ≤ c_{1} = (1 + a)^{λ}

This implies Condition 1.

Condition 2. We have that

(1 + an)^{λ} ∑_{j=n}^{n+K}(1 + aj)^{−λ} ≥ \frac{K(1 + an)^{λ}}{(1 + a(n + K))^{λ}} → K

as n → ∞. This implies Condition 2.

Condition 3. For k ≤ n/2 we estimate

l_{k} ∑_{j=k}^{n} l_{j}^{−1} = (1+ak)^{λ} ∑_{j=k}^{n}(1+aj)^{−λ} ≥ C(1+ak)^{λ}(1+ak)^{−λ+1} = C(1+ak)^{−1}

and for k > n/2 and n ≥ j ≥ k we estimate

l_{k}l_{j}^{−1} ≥ C_{0} > 0

hence

l_{k} ∑_{j=k}^{n} l_{j}^{−1} ≥ C_{0}(n − k + 1)

Thus,

∑_{k=1}^{n} (l_{k} ∑_{j=k}^{n} l_{j}^{−1})^{−2} ≤ C^{−2} ∑_{k=1}^{n/2}(1 + ak)^{−2} + C_{0}^{−2} ∑_{k=n/2}^{n}(n − k + 1)^{−2} ≤ C_{1}

Condition 3 is checked.

Conditions 4 and 5 are obvious. □

Acknowledgements. An essential part of this work was done at the Mathematisches Forschungsinstitut Oberwolfach, where the authors enjoyed their participation in the program “Research in Pairs”. They are thankful to the Mathematisches Forschungsinstitut for kind hospitality and the Volkswagen– Stiftung for support of their stay at Oberwolfach. The research of the first author (P.B.) was supported in part by the National Science Foundation, Grant No. DMS–1565602, the research of the second author was supported by the Hungarian Foundation NKFI–EPR No. K-125569. These supports are gratefully acknowledged.

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(Pavel Bleher) University – Purdue University at Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202, USA

E-mail address: pbleher@iupui.edu

(Péter Major) Alfréd Rényi Institute of Mathematics, Budapest, P.O.B. 127 H–1364, Hungary

E-mail address: major@renyi.hu