THE LARGE-SCALE LIMIT OF DYSON’S HIERARCHICAL

VECTOR-VALUED MODEL AT LOW TEMPERATURES.

THE NON-GAUSSIAN CASE.

PART II. DESCRIPTION OF THE LARGE-SCALE LIMIT.

1Bleher, P. M. and ^{2}Major, P.

1Keldysh Institute of Applied Mathematics, Moscow A–47 Miusskaya Square 4, U.S.S.R. 2Mathematical Institute of the Hungarian Academy of Sciences, Budapest H–1364 P.O.B. 127, Hungary

1. Introduction. In this paper we investigate the large-scale limit of the equilibrium state of Dyson’s hierarchical vector valued p dimensional, p ≥ 2, model with parameter c, 1 < c < \sqrt{2}, at low temperatures. More precisely, in Theorem 1 we construct a probability measure μ̄ = μ̄(T) on (R^{p})^{Z} with Z = { 1, 2, ... } which is an equilibrium state of the model. In Theorem 2 we determine the large-scale limit of a μ̄ distributed random field together with the right scaling, i.e. we prove that if

σ = { σ(j) = (σ^{(1)}(j), …, σ^{(p)}(j)) ∈ R^{p}, j ∈ Z}

is a μ̄ distributed random field then the finite dimensional distributions of the random fields

(1.1)  R_{n}σ = {(R_{n}σ^{(1)}(j), …, R_{n}σ^{(p)}(j)) ∈ R^{p}, j ∈ Z } ,

(1.2)  R_{n}σ^{(1)}(j) = c^{n} 2^{−n} ∑_{k=(j−1)2^{n}+1}^{j2^{n}} [σ^{(1)}(k) − Eσ^{(1)}(k)] , j ∈ Z ,

(1.3)  R_{n}σ^{(s)}(j) = c^{n/2}2^{−n} ∑_{k=(j−1)2^{n}+1}^{j2^{n}} σ^{(s)}(k) , j ∈ Z, s = 2, …, p

tend to those of a limit random field, and describe the finite dimensional distributions of this limit field.

The distributions of the fields R_{n}σ defined in (1.1)–(1.3) are called the renormalizations of the distribution μ̄ of the underlying field σ. More precisely, they are its renormalization with parameters α = 1 − \frac{log c}{log 2} in the first coordinate and ᾱ = 1 − \frac{1}{2} \frac{log c}{log 2} in the coordinates s = 2, …, p, because we multiplied by 2^{−nα} in

Typeset by AMS-TEX

(1.2), by 2^{−nᾱ} in (1.3), and the number of summands is 2^{n} in these formulas. If the finite dimensional distributions of the fields R_{n}σ converge to those of a limit field then this limit field, more precisely its distribution, is called the large-scale limit of the measure μ̄.

Given some h ∈ R^{1}, h ≥ 0, and positive integer N let the Gibbs measure μ_{N}^{h} = μ_{N}^{h}(T, t) be defined on (R^{p})^{2^{N}} with density function

p_{N}^{h}(x_{1}, …, x_{2^{N}}) = p_{N}^{h}(x_{1}, …, x_{2^{N}} , t, T) ,
x_{j} = (x_{j}^{(1)}, …, x_{j}^{(p)}) ∈ R^{p} , j = 1, …, 2^{N} ,

(1.4)

\begin{matrix}p_{N}^{h}(x_{1}, …, x_{2^{N}}) \\ = Z_{N}^{−1}(T, t, h) exp\{−\frac{1}{T} (− ∑_{i=1}^{2^{N} −1} ∑_{j=i+1}^{2^{N}} U(i, j)x_{i}x_{j} − h ∑_{j=1}^{2^{N}} x_{j}^{(1)})\} ∏_{j=1}^{2^{N}} p(x_{j}, t) , \end{matrix}

where

Z_{N} = ∫ exp\{−\frac{1}{T} (− ∑_{i=1}^{2^{N} −1} ∑_{j=i+1}^{2^{N}} U(i, j)x_{i}x_{j} − h ∑_{j=1}^{2^{N}} x_{j}^{(1)})\} ∏_{j=1}^{2^{N}} p(x_{j}, t) dx_{j}

is the grand partition function, and p(x, t) is defined in (1.3) of Part I. Let p_{N}^{h}(x) denote the density function of the average 2^{−N} ∑_{j=1}^{2^{N}} σ(j) of the μ_{N}^{h} distributed random vector (σ(1), …, σ(2^{n})). Put μ_{N} = μ_{N}^{h}, p(x_{1}, …, x_{2^{N}}) = p_{N}^{h}(x_{1}, …, x_{2^{N}}) and p_{N}(x) = p_{N}^{h}(x) in the case h = 0.

In Part I we have described the asymptotic behaviour of the above defined density function p_{N}(x). The result of Theorem A formulated below are contained in Theorems 1, 2 and Lemma 13 of Part I. Let us consider the integral equation

(1.5) g(x) = (\frac{2}{c\sqrt{π}} )^{p−1} ∫_{R^{p}} exp(−v^{2})g(\frac{x}{c} + u + \frac{v^{2}}{2} )g(\frac{x}{c} − u + \frac{v^{2}}{2} ) dudv ,
x, u ∈ R^{1} , v ∈ R^{p−1} ,

where v^{2} denotes scalar product. In Part I we have proved that equation (1.5) has a unique non-trivial (i.e. not identically zero) solution in the class of functions A = {g, ∫ e^{tx}|g(x)|dx < ∞ if t < t_{0}(g), t_{0}(g) > 0}. In this work we consider this function as the solution of equation (1.5). It is a density function which is positive for all x. Since the function p_{n}(x, t) depends on x only through |x| we can define a function p̄_{n}(z) = p̄_{n}(z, t, T), z ∈ R^{1}, such that p_{n}(x) = p̄_{n}(|x|) for all x ∈ R^{p}. Now we formulate the following

Theorem A. If 1 < c < \sqrt{2} then there exist some thresholds T_{0} = T_{0}(c) > 0 such that for all 0 < T < T_{0}, 0 < t < t_{0}, t_{0} = t_{0}(c), (t is the parameter of p(x, t) in formula (1.3) of Part I) the following relations hold:

There are some M = M(c, T, t) > 0 and n_{0} = n_{0}(c, T, t) > 0 such that for n > n_{0}

(1.6) c^{−n}p_{n}(x, T) = c^{−n}p̄_{n}(|x|, T)

= B exp \{−\frac{a_{0}c^{n}}{T} M(|x| − M)\} g (\frac{a_{1}c^{n}}{T} M(|x| − M)) (1 + r_{n}(x))

for −ηnc^{−n} < |x|−M < ηn^{1/α}c^{−n} with some B = B(c, T, t) > 0, η = η(c, T, t) > 0, and the error term r_{n}(x) satisfies the inequality

(1.7)  |r_{n}(x)| ≤ Kq^{n}

with some K > 0 and 0 < q < 1 depending on c, T and t. In formula (1.6) g(x) is the solution of the equation (1.5), a_{0} = \frac{2}{2−c}, a_{1} = a_{0} + 1 and α = \frac{log 2}{log c}.

(1.8) c^{−n}p̄_{n}(x, T) ≤ Kq^{n} exp {−L(c^{n}|x − M|)^{2+δ}} if x > M + ηn^{1/α}c^{−n}

with some δ > 0, K > 0 and L > 0 which depend on c, T and t. The solution of the equation (1.6) satisfies the inequality

(1.9)  0 < g(x) < C exp(−Ax^{α}) for x > 0

with some C > 0, A > 0. We have

(1.10)  c^{−n}p̄_{n}(x, T) ≤ C_{1} exp{−C_{2}c^{n}|x − M|} for 0 < x < M

with some C_{1} > 0, C_{2} > 0 depending on c, T and t. We also have

M^{2} = \frac{a_{0} − T}{tT} + R(t, T)

with some |R(t, T)| ≤ const., and such that R(t, T) → 0 and T → 0.

Given some integers N ≥ k ≥ 0 we define the probability measure μ_{k,N}^{h} as the projection of the measure μ_{N}^{h} to the first 2^{k} coordinates, i.e. μ_{k,N}^{h} is a probability measure on (R^{p})^{2^{k}}, and for all measurable A ⊂ (R^{p})^{2^{k}} μ_{k,N}^{h}(A) = μ_{N}^{h}(A) =

μ_{N}^{h} (A × (R^{p})^{2^{N} −2^{k}}). Our first result is the following

Theorem 1. Let the conditions of Theorem A be satisfied. Consider an arbitrary sequence of real numbers h_{N}, N = 0, 1, 2, … such that

(1.11)  \frac{2}{2 − c} \frac{M}{T} (\frac{c}{2})^{N} ≤ \frac{h_{N}}{T} ≤ D (\frac{c}{2})^{N}

with some ∞ > D > \frac{2}{2−c} \frac{M}{T}, where M and T are the same as in Theorem A. Then the measures μ_{N}^{h_{N}} tend to a probability measure μ̄ = μ̄(t, T, c) on (R^{p})^{Z}. More precisely, for all k ≥ 0 the measures μ_{k,N}^{h_{N}} converge to the projection of μ̄ to the first 2^{k} coordinates in variational metric as N → ∞. The measure μ does not depend on the choice of sequences h_{N}.

Then we prove the following Theorem 2. Let σ = {σ(n) = (σ^{(1)}(n), …, σ^{(p)}(n)) ∈ R^{p}, n ∈ Z} be a μ̄ distributed random field with the distribution μ̄ defined in Theorem 1. Then the finite dimensional distributions of the random fields R_{n}σ defined in (1.1), (1.2), (1.3) tend to those of a random field Y = (Y (n) = (Y^{(1)}(n), …, Y^{(p)}(n)) ∈ R^{p}, n ∈ Z).

For all k ≥ 0 the density function h_{k} (x_{1}, …, x_{2^{k}}), x_{j} = (x_{j}^{(1)}, …, x_{j}^{(p)}) ∈ R^{p}, of

the random vector (Y (1), …, Y (2^{k})) is given by the formula (1.12)

\begin{matrix}h_{k} (x_{1}, …, x_{2^{k}}) = C(k) exp\{−\frac{1}{T} ∑_{s=2}^{p}(\frac{1}{2 − c} ∑_{j=1}^{2^{k}} x_{j}^{(s)2} − \frac{2 − c}{c}(\frac{c}{4})^{k}(∑_{j=1}^{2^{k}} x_{j}^{(s)})^{2} \\ + ∑_{i=1}^{2^{k}−1} ∑_{j=i+1}^{2^{k}} U(i, j)x_{i}^{(s)}x_{j}^{(s)})\} ∏_{j=1}^{2^{k}} g(\frac{4 − c}{(2 − c)T} (Mx_{j}^{(1)} + \frac{1}{2} ∑_{s=2}^{p} x_{j}^{(s)2})) , \end{matrix}

where the function g is defined in (1.5), the constant M is the same as in Theorem A, and C(k) is an appropriate norming constant.

In Appendix E we prove the following

Theorem B. The measure μ̄ = μ̄(T, t, c) constructed in Theorem 1 is a Gibbs state with Hamiltonian H and free measure ν defined in formulas (1.1)–(1.3) of Part I at temperature T.

Theorem B is very plausible. Its proof depends on a rather standard limiting procedure in statistical physics literature. Nevertheless, we have found no result which could have been directly applied in our case. We present the proof of Theorem B in Appendix E.

Let us discuss the role of condition (1.11) in Theorem 1. The lower bound

(1.11′)  h_{N} > \frac{2}{2 − c}M (\frac{2}{c})^{N}

is essential in Theorem 1, it is needed to get a pure state with magnetization in the direction e_{1} = (1, 0, …, 0) for the limit measure μ̄. If it were violated we would get a Gibbs state with Hamiltonian H again for the limit, but this Gibbs state would be a mixture of Gibbs states with different directions of magnetization, and it is not natural to renormalize such a mixture. On the other hand the upper bound for h_{N} in (1.11) seems not to be essential. We believe that the same limit measure μ̄ would be obtained for any sequence h_{N}, h_{N} > 0 satisfying (1.11′) or with the help of the double limiting procedure μ^{h} = lim_{N→∞} μ_{N}^{h}, h > 0, μ̄ = lim_{h→0} μ^{h}. This second way was chosen to construct the equilibrium state in the case \sqrt{2} < c < 2 in paper [5]. However, to prove these statements we would need a large deviation result on the behaviour of p_{n}(x) which is stronger than Theorem A. Since we are not able to prove such a result we have proved Theorem 1 under the condition (1.11), but we think that this is not an essential restriction.

In formula (1.12) we have a quadratic form inside the exponent. This means that the random variables Y^{(ℓ)}(j), ℓ = 2, …, p appearing in Theorem 2 are jointly Gaussian. We describe the structure of this limit field in more detail. The random

fields \{R_{n}σ^{(s)}(j), j ∈ Z\}, s = 2, …, p, and \{MR_{n}σ^{(1)}(j) + \frac{1}{2} ∑_{s=2}^{p} R_{n}σ^{(s)}(j)^{2}, j ∈

Z} tend to independent random fields as n → ∞. The limit of the random fields {R_{n}(σ^{(s)}(j)), j ∈ Z} is the (disregarding a multiplying factor) unique Gaussian self-similar field with self-similarity parameter 1 − \frac{1}{2} \frac{log c}{log 2}, whose distribution is invariant under all permutations of the index set Z which preserves the hierarchical distance d(i, j). The random fields \{MR_{n}σ^{(1)}(j) + \frac{1}{2} ∑_{s=2}^{p} R_{n}σ^{(s)}(j) , j ∈ Z\} tend to a random field consisting of independent random variables with the density function \frac{2−c}{(4−c)T} g(\frac{4−c}{(2−c)T} x). This is a quadratic functional of a Gaussian field (see Lemma 12 in Part I).

The above result can also be interpreted in the following way: Given a μ̄ distributed random field σ(n), n ∈ Z, define the absolute value of the appropriately normalized partial sums |R_{n}|σ(j) = c^{n}2^{−n} (|R_{n} ∑_{k=(j−1)2^{n}+1}^{j2^{n}} σ(k)| − M),

j ∈ Z. Then the random fields R_{n}σ^{(s)}(j), j = 2, …, p, and the random fields |R_{n}|σ(j) tend in distribution to independent random fields. The limit of R_{n}σ^{(s)}(j) j = 2, …, p is Gaussian, and the limit of |R_{n}|σ(j) consists of independent random variables. This follows immediately from the above description of the limit behaviour of the fields R_{n}σ together with the observation that |R_{n}|σ(j)−(R_{n}σ^{(1)}(j)+ \frac{1}{2M} ∑_{s=2}^{p} R_{n}σ^{(s)}(j)^{2}) ⇒ 0 stochastically as n → ∞.

We believe that the above property is a special case of a more general law. Let us remark that an analogous statement also holds in the case \sqrt{2} < c < 2, but this is a degenerate case. It follows from the results of [5] that if {σ(j), j ∈ Z} is a μ̄ = μ̄(c), \sqrt{2} < c < 2 distributed random field with the equilibrium state μ̄ constructed in [5] then the random fields

|R_{n}|(σ)(j) = 2^{−n/2}(| ∑_{k=(j−1)2^{n}+1}^{j2^{n}} σ(k)| − M) , j ∈ Z,

have the same limit as the random fields

R_{n}σ^{(1)}(j) = 2^{−n/2} ∑_{k=(j−1)2^{n}+1}^{j2^{n}} (σ^{(1)}(k) − M)

as n → ∞, since in this case |R_{n}|σ(j) − R_{n}σ^{(1)}(j) ⇒ 0. This limit consists of independent (Gaussian) random variables which is also independent of the limit of the random fields R_{n}σ^{(j)}, s = 2, …, p.

The method of this paper is very similar to that of [5]. The two main steps in the proofs consist of the description of the limit behaviour of the function p_{n}(x) done in Part I, and a good asymptotic formula for the Radon–Nikodym derivatives

\frac{dμ_{n,N}^{h_{N}}}{dμ_{n}}. Then an appropriate limiting procedure supplies the proof of Theorems 1 and 2. The investigation of the Radon–Nikodym derivatives can be considered as an adaptation of the method of [5] to the present case. The main difference between the two cases is that now p_{n}(x) is not asymptotically Gaussian. But although the

Radon–Nikodym derivative \frac{dμ_{n,N}^{h_{N}}}{dμ_{n}} depends on p_{n}(x), its asymptotic behaviour does not. As we shall see, in the investigation of the asymptotic behaviour of the above Radon–Nikodym derivative we only need some estimates on the tail behaviour of p_{n}(x), but not its explicit form. This is the reason why we can adapt the method of [5].

2. On the basic estimates needed in the proof. Reduction to integral equations. We need a good asymptotic formula for the Radon–Nikodym derivative

\frac{dμ_{n,N}^{h_{N}}}{dμ_{n}}. It can be expressed exactly with the help of the following formulas:

(2.1)  \frac{dμ_{n,N}^{h_{N}}}{dμ_{n}} (x_{1}, …, x_{2^{n}}) = f_{n,N}^{h_{N}}(2^{−n} ∑_{j=1}^{2^{n}} x_{j}), n ≤ N

(2.2)  f_{N,N}^{h_{N}}(x) = K(N, h_{N}) exp (\frac{2^{N}h_{N}x^{(1)}}{T} )
(2.3)  f_{n,N}^{h_{N}}(x) = K(n, N, h_{N})S_{n}f_{n+1,N}^{h_{N}}(x)

with

(2.3′)  S_{n}f(x) = ∫_{R^{p}} exp(\frac{c^{n}}{T} xy)f(\frac{x + y}{2} )p_{n}(y) dy

where K(n, N, h_{N}) are appropriate norming factors, xy denotes scalar product, and p_{n} is the density function appearing in Theorem A. For scalar valued models formulas (2.1)–(2.3′) are proved in the main formula in [4]. The proof for the vector valued case is the same, but since the proof in [4] is a bit complicated we present it in Appendix C.

Let us define the sequences g_{n} = g_{n}(N, h_{N}) and A_{n} = A_{n}(N, h_{N}) by the recursive relations

(2.4)  g_{N} = g_{N}(N, h_{N}) = \frac{2^{N}h_{N}}{T}

(2.4′)  g_{n} = g_{n}(N, h_{N}) = \frac{g_{n+1}}{2} + \frac{c^{n}}{T} M  for n < N

(2.5) A_{N} = A_{N}(N, h_{N}) = 0

(2.5′) A_{n} = A_{n}(N, h_{N}) = \frac{A_{n+1}}{4} + \frac{(\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )^{2}}{\frac{2c^{n}}{T} + \frac{g_{n+1}}{M} − A_{n+1}}  for n < N ,

where M and T are the same as in Theorem A. In Section 7 of [6] we have claimed that

f_{n}(x) = f_{n,N}^{h}(x) ∼ K_{n} exp {g_{n}(x^{(1)} − M) + A_{n} ∑_{s=2}^{p} x^{(s)2}} ,

and have given a heuristic explanation. In the following Proposition 1 we formulate this result in a more precise form. For the sake of simpler notation we assume that R^{p} = R^{2}. From now on C, C_{1}, K, L etc. denote appropriate constants. The same letter may denote different constants in different formulas. Let us define the domains

(2.6)  Ω_{n}^{1} = {x ∈ R^{2}, ||x| − M| < c^{−0.4n} , |x^{(2)}| < c^{−0.4n} , x^{(1)} > 0}
(2.6′)  Ω_{n}^{2} = {x ∈ R^{2}, ||x| − M| < c^{−0.4n}} − Ω_{n}^{1}
(2.6′′)  Ω_{n}^{3} = {x ∈ R^{2}, ||x| − M| ≥ c^{−0.4n}} .

Clearly Ω_{n}^{1} ∪ Ω_{n}^{2} ∪ Ω_{n}^{3} = R^{2}. Now we formulate the following

Proposition 1. For all q, c^{−0,2} < q < 1, there is some n_{0} = n_{0}(T, M, c, D, q) such that if (1.11) holds and N ≥ n ≥ n_{0} then the Radon–Nikodym derivative f_{n}(x) = f_{n,N}^{h_{N}}(x) appearing in (2.1) satisfies the following relations:

a) In the domain Ω_{n}^{1}

(2.7)  f_{n}(x) = L_{n} exp {g_{n} (x^{(1)} − M) + A_{n}x^{(2)2} + ε_{n}(x)}

with

sup_{x∈Ω_{n}^{1}} |ε_{n}(x)| ≤ q^{n} .

b) In the domain Ω_{n}^{2}

(2.8)  0 ≤ f_{n}(x) ≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n})c^{−0,8n} + q^{n}\} .

c) In the domain Ω_{n}^{3}

(2.9)  0 ≤ f_{n}(x) ≤ L_{n} exp \{\frac{g_{n}}{2M} (|x|^{2} − M^{2})\} ,

where the numbers A_{n} and g_{n} are defined in (2.4)–(2.5′), and L_{n} = L_{n}(N, h_{N}) is an appropriate norming constant.

We also prove the following result which is a slight modification of Lemma 1 in [5].

Lemma 1. Let us choose some integer N and h_{N} > 0. Define the sequences g_{n} and A_{n}, 0 ≤ n ≤ N, by formulas (2.4)–(2.5′) and put ḡ_{n} = c^{−n}g_{n}, Ā_{n} = c^{−n}A_{n}. If h_{N} satisfies relation (1.11) then ḡ_{N} ≥ ḡ_{N−1} ≥ ⋯ ≥ ḡ_{0} ≥ ḡ and 0 = Ā_{N} ≤ Ā_{N−1} ≤ ⋯ ≤ Ā_{0} ≤ Ā with ḡ = \frac{2}{2−c} \frac{M}{T}, and Ā = \frac{2−c}{cT}. If the relations N > N_{0} and N > n^{B} also hold with some appropriate N_{0} = N_{0}(c, M, T, D) and B = B(c, M, T, D) then |ḡ_{n} − ḡ| < 4^{−n}, |Ā_{n} − Ā| < 4^{−n}.

Proposition 1 together with the characterization of the asymptotic behaviour of the sequences g_{n} and A_{n} made in Lemma 1 supplies a good asymptotic formula for the Radon–Nikodym derivative f_{n}. Here Ω_{n}^{1} is the typical region, where we have a good asymptotic formula, in Ω_{n}^{2} and Ω_{n}^{3} we have only given an upper bound. Actually we are interested in the density function

p_{n}(2^{−n} ∑_{j=1}^{2^{n}} x_{j})f_{n,N}^{h_{N}}(2^{−n} ∑_{j=1}^{2^{n}} x_{j})

of the measure μ_{n,N}^{h_{N}}. The tail behaviour of the functions p_{n}(x) and f_{n}(x) together show that 2^{−n} ∑_{j=1}^{2^{n}} x_{j} is contained in Ω_{n}^{3} with a negligible small μ_{n,N}^{h_{N}} probability. It is contained in Ω_{n}^{2} also with a small probability, since in this domain f_{n}(x) is small. To see it, let us observe that by Lemma 1

\frac{g_{n}}{2M} − A_{n} ≥ c^{n} (\frac{ḡ}{2M} − Ā) = \frac{c^{n}}{T} (\frac{1}{2 − c} − \frac{2 − c}{c}) = \frac{c^{n}}{T} \frac{(4 − c)(c − 1)}{2 − c} > 0 ,

hence the term − (\frac{g_{n}}{2M} − A_{n}) c^{−0.8n} in the exponent of (2.8), makes this upper bound (2.8) sufficiently small for our purposes.

In Section 7 of [6] we have given a heuristic argument for formula (2.7). The following remark explains the content of the estimate (2.8).

Remark. If x ∈ Ω_{n}^{1} then

|x| = (x^{(1)2} + x^{(2)2})^{1/2} = x^{(1)} (1 + \frac{x^{(2)2}}{x^{(1)2}} )^{1/2} = x^{(1)} + \frac{x^{(2)2}}{2x^{(1)}} + O(x^{(2)4})

= x^{(1)} + \frac{x^{(2)2}}{2M} + O (x^{(2)4} + x^{(2)2}|x^{(1)} − M|) = x^{(1)} + \frac{x^{(2)2}}{2M} + O(c^{−1.2n}) ,

hence

exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) x^{(2)2}\}
= exp {g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} + O(c^{−0.2n})} .

The above calculation shows that on the boundary of the domains Ω_{n}^{1} and Ω_{n}^{2} the right-hand side of formulas (2.7) and (2.8) have the same magnitude. The estimate (2.8) expresses the fact that this is the worst region, where the weakest upper bound can be given for f_{n}(x) in Ω_{n}^{2}.

With the help of Proposition 1, Lemma 1 and Theorem A we are able to carry out a limiting procedure which supplies Theorem 1. Moreover, it yields the following Proposition 2. Let μ̄_{n} denote the projection of the measure μ̄ constructed in Theorem 1 to the first 2^{n} coordinates, i.e. let μ̄_{n} be the measure on (R^{2})^{2^{n}} defined by the relation μ̄_{n}(A) = μ̄ (A × (R^{2})^{∞}) for all measurable sets A ⊂ (R^{2})^{2^{n}}. The following result holds true:

Proposition 2. For all q, c^{−0.2} < q < 1, there is some n_{0} = n_{0}(c, T, M, q) such that for all n ≥ n_{0} the measure μ̄_{n} is absolute continuous with respect to the measure μ_{n}, and its Radon–Nikodym derivative satisfies the relations:

(2.10)  \frac{dμ̄_{n}}{dμ_{n}} (x_{1}, …, x_{2^{n}}) = f̄_{n} (2^{−n} ∑_{i=1}^{2^{n}} x_{i})

and

a) For x ∈ Ω_{n}^{1}

(2.11)  f̄_{n}(x) = L_{n} exp {c^{n}ḡ(x^{(1)} − M) + c^{n}Āx^{(2)2} + ε_{n}(x)}

with

(2.11′)  sup_{x∈Ω_{n}^{1}} |ε_{n}(x)| ≤ q^{n} .

b) For x ∈ Ω_{n}^{2}

(2.12)  0 ≤ f̄_{n}(x) ≤ L_{n} exp \{c^{n}ḡ(|x| − M) − c^{0.2n} (\frac{ḡ}{2M} − Ā) + q^{n}\} .

c) For x ∈ Ω_{n}^{3}

(2.13) 0 ≤ f̄_{n}(x) ≤ L_{n} exp \{\frac{ḡc^{n}}{4M} (x^{2} − M^{2})\}  if 0 < x < M − c^{−0.4n} ,

(2.13′) 0 ≤ f̄_{n}(x) ≤ L_{n} exp \{\frac{ḡc^{n}}{M} (x^{2} − M^{2})\}  if x > M + c^{−0.4n} ,

with ḡ = \frac{2}{2−c} \frac{M}{T}, Ā = \frac{2−c}{cT} and an appropriate norming constant L_{n}. This norming constant satisfies the relation

C_{1} < c^{−n/2}L_{n} < C_{2} with some 0 < C_{1} < C_{2} < ∞ .

Theorem 2 can be deduced from Proposition 2 and Theorem A.

Let us finally remark that the function f_{n}(x) = f_{n,N}^{h_{N}}(x) clearly satisfies Proposition 1 for n = N, since in this case f_{N}(x) = L_{N} exp{g_{N}(x^{(1)} − M)}. Hence Proposition 1 follows from Lemma 1 and the following

Proposition 1′. For all q, c^{−0.2} < q < 1, there exists some n_{0} = n_{0}(T, M, c, D, q) such that if for n ≥ n_{0} the function f(x) satisfies the following relations with some ḡc^{n+1} < g_{n+1} ≤ Dc^{n+1}, 0 ≤ A_{n+1} ≤ Āc^{n+1}, ḡ = \frac{2}{2−c} \frac{M}{T}, Ā = \frac{2−c}{cT}, D > ḡ:

a) For x ∈ Ω_{n+1}^{1}

(2.15)  f(x) = exp {g_{n+1}(x^{(1)} − M) + A_{n+1}x^{(2)2} + ε_{n+1}(x)}

sup_{x∈Ω_{n+1}^{1}} |ε_{n+1}(x)| ≤ q^{n+1} . (2.15′)

b) For x ∈ Ω_{n+1}^{2}

(2.16) 0 ≤ f(x) ≤ exp \{g_{n+1}(|x| − M) + (\frac{g_{n+1}}{2M} − A_{n+1}) c^{−0.8(n+1)} + q^{n+1}\} ;

c) For x ∈ Ω_{n+1}^{3}

(2.17)  0 ≤ f(x) ≤ exp \{\frac{g_{n+1}}{2M} (|x|^{2} − M^{2})\}

then the function S_{n}f(x) defined by (2.3′) satisfies, with the constants g_{n} and A_{n} defined by (2.4′) and (2.5′) with the above g_{n+1} and A_{n+1}, the following relations with some appropriate norming constant L_{n}:

a) In the domain Ω_{n}^{1}

(2.18)  S_{n}f(x) = L_{n} exp {g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} + ε_{n}(x)}

with

(2.18′)  sup_{x∈Ω_{n}^{1}} |ε_{n}(x)| ≤ q^{n} .

b) In the domain Ω_{n}^{2}

(2.19)  0 ≤ S_{n}f(x) ≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) c^{−0.8n} + q^{n}\} .

c) In the domain Ω_{n}^{3}

(2.20)  0 ≤ S_{n}f(x) ≤ L_{n} exp \{\frac{g_{n}}{2M} (|x|^{2} − M^{2})\} .

3. The proof of Lemma 1. The proof is a modification of that given for Lemma 1 in [5]. Simple calculation shows that ḡ_{n} − ḡ = (\frac{c}{2})^{N−n}(ḡ_{N} − ḡ). The statements of Lemma 1 about the sequence g_{n} follow from this identity. To investigate Ā_{n} let us introduce the function

T (x, g) = \frac{c}{4}x + \frac{(\frac{1}{T} + \frac{c}{2}x)^{2}}{\frac{2}{T} + \frac{cg}{M} − cx} , x ∈ R^{1} , g ∈ R^{1} .

Clearly, Ā_{n} = T (Ā_{n+1}, ḡ_{n+1}). On the other hand T (Ā, ḡ) = Ā, and some calculation shows that T has the following monotonicity properties: T (x, g) < T (x, g′) if 0 < x < Ā and g > g′ > ḡ; and T (x′, g) > T (x, g) if 0 < x < x′ < A and g > ḡ. (These properties follow e.g. from the relations

\frac{∂}{∂g} T (x, g) = −\frac{c}{M} \begin{matrix}(\frac{1}{T} + \frac{c}{2}x) & 2 \\ (\frac{2}{T} + \frac{c}{M} g − cx) & 2 \end{matrix} < 0 ,

and

\frac{∂}{∂x}T (x, g) = \frac{c (\frac{2}{T} + \frac{cg}{2M} )^{2}}{(\frac{2}{T} + \frac{c}{M} g − cx)^{2}} > 0 ,

and the fact that T (x, g) has no singularity in the domain {(x, g), 0 < x < Ā, g > ḡ}.) We have 0 < Ā_{N−1} < Ā, since Ā_{N−1} = T (0, ḡ_{N}) > 0 and Ā_{N−1} = T (0, ḡ_{N}) < T (Ā, ḡ) = Ā. Then we get by induction that 0 ≤ Ā_{n+1} < Ā_{n} < Ā implies that 0 < Ā_{n} < Ā_{n−1} < Ā. Indeed, Ā_{n−1} = T (Ā_{n}, ḡ_{n}) < T (Ā, ḡ) = Ā, and Ā_{n−1} = T (Ā_{n}, ḡ_{n}) > T (Ā_{n+1}, ḡ_{n+1}) = Ā_{n}, as we have claimed.

The conditions N > N_{0} and N > n^{B} with sufficiently large N_{0} and B imply that |ḡ_{ℓ}−ḡ| < 10^{−n} for all 0 < ℓ ≤ \sqrt{N}, and Ā−5^{−n} < A^{∗} < Ā, where A^{∗} is the smaller solution of the equation T (x, g^{∗}) = x with g^{∗} = ḡ\sqrt{N}. Indeed, the last equation is a small perturbation of the equation T (x, ḡ) = x, which has two solutions A_{1} = Ā and A_{2} = \frac{1}{(2−c)T} > Ā. Hence the solutions of the equation T (x, g^{∗}) = x are very close to A_{1} and A_{2}. We claim that the monotonicity properties of the sequences ḡ_{n} and Ā_{n} and the function T (x, g) imply that Ā > Ā_{n} ≥ T_{g^{∗}}^{\sqrt{N}−n}(0), where T_{g^{∗}}^{k} denotes the k-th iteration of the function T (x, g^{∗}) with fixed g^{∗} in the variable x. Indeed, Ā > Ā_{n}, 0 < Ā\sqrt{N} < Ā, and we get by induction that for all ℓ ≥ 0 Ā_{\sqrt{N}−ℓ} ≥ T_{g^{∗}}^{ℓ}(0), which implies the required statement with ℓ = \sqrt{N} − n.

To complete the proof of Lemma 1 it is enough to show that T_{g^{∗}}^{n}(0) tends exponentially fast in n to the smaller solution A^{∗} of the equation T_{g^{∗}}(x) = x. Since T (x, g^{∗}) is a convex increasing function (in the variable x) it is enough to show that \frac{∂T (x,g^{∗})}{∂x} ≤ α < 1 for x = A^{∗} if (A^{∗}, g^{∗}) is in a small neighbourhood of the point (Ā, ḡ). But this follows from the continuity of the function \frac{∂T (x,g)}{∂x}, and the fact that its value in the point (Ā, ḡ) equals c^{−1} < 1. Lemma 1 is proved.

4. Some preparatory remarks to the proof of Proposition 1′. We shall prove the following estimates under the conditions of Proposition 1′.

Put

S_{n}^{i}f(x) = ∫_{\{y, \frac{x+y}{2} ∈Ω_{n+1}^{i}\}} exp (\frac{c^{n}}{T} xy) f (\frac{x + y}{2} ) p_{n}(y) dy , i = 1, 2, 3 .

Then we have with some appropriate ε′ = ε′(c), ε′ > 0, and the same q as in Proposition 1′:

In the domain x ∈ Ω_{n}^{1}

(4.1)  S_{n}^{1}f(x) = L_{n} exp {g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} + ε̄_{n}(x) + ε̂_{n}(x)}

with

(4.1′)  sup_{x∈Ω_{n}^{1}} |ε̄_{n}(x)| ≤ q^{n+1} , sup_{x∈Ω_{n}^{1}} |ε̂_{n}(x)| ≤ Kc^{−0.2n} ,

12  P. M. Bleher and P. Major

where K does not depend on n,

L_{n} ≥ c^{−n} exp (\frac{c^{n}}{T} M^{2}) , (4.1′′)

S_{n}^{2}f(x) ≤ L_{n} exp {g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} − ε′c^{0.2n}} , (4.2)

S_{n}^{3}f(x) ≤ L_{n} exp \{g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} − \frac{1}{6}c^{n/2}\} . (4.3)

In the domain x ∈ Ω_{n}^{2}

S_{n}^{1}f(x) ≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) c^{−0.8n} + q^{n+1} + Kc^{−0.2n}\} , (4.4)
S_{n}^{2}f(x) ≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) c^{−0.8n} − ε′c^{0.2n}\} , (4.5)
S_{n}^{3}f(x) ≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) c^{−0.8n} − \frac{1}{6}c^{n/2}\} . (4.6)

In the domain x ∈ Ω_{n}^{3}

(4.7)  S_{n}f(x) ≤ L_{n} exp \{\frac{g_{n}}{2M} (|x|^{2} − M^{2})\} .

We show that these estimates imply Proposition 1′. Indeed, for x ∈ Ω_{n}^{1} S_{n}f(x) = S_{n}^{(1)}f(x)+S_{n}^{(2)}f(x)+S_{n}^{(3)}f(x) = L_{n} exp{g_{n}(x^{(1)} −M)+A_{n}x^{(2)2} +ε_{n}(x)} with

sup_{x∈Ω_{n}^{1}} |ε_{n}(x)| ≤ sup_{x∈Ω_{n}^{1}} |ε̄_{n}(x)| + sup_{x∈Ω_{n}^{1}} |ε̂_{n}(x)| + 2 exp(−ε′c^{0.2n}) + 2 exp (−\frac{1}{6}c^{n/2}) .

(Here we have exploited that e^{t} < 1 + 2|t| for small t.) Hence

sup_{x∈Ω_{n}^{1}} |ε_{n}(x)| ≤ q^{n+1} + Kc^{−0.2n} + 2 exp(−ε′c^{0.2n}) + 2 exp (−\frac{1}{6}c^{n/2}) ≤ q^{n}

if c^{−0.2} < q < 1, and n ≥ n_{0}(q, D, ε′).

For x ∈ Ω_{n}^{2} we have analogously

S_{n}f(x) = S_{n}^{1}f(x) + S_{n}^{2}f(x) + S_{n}^{3}f(x)

≤ L_{n} exp\{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) c^{−0.8n} + q^{n+1} + Kc^{−0.2n}

+ 2 exp(−ε′c^{0.2n}) + 2 exp (−\frac{1}{6}c^{n/2})\}

≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) c^{−0.8n} + q^{n}\} ,

as we have claimed in Proposition 1′. For x ∈ Ω_{n}^{3} (4.7) contains the needed estimate.

The above estimates will be proved in the next Section. In this Section we prove two lemmas which we need during the proof. Put

(4.8)  S_{n}^{ε}f(x) = ∫_{R^{2}−V_{n}^{ε}(x)} exp(\frac{c^{n}}{T} xy)f_{n}(\frac{x + y}{2} ) p_{n}(y) dy

with

(4.8′)  V_{n}^{ε}(x) = {y ∈ R^{2} , ||y| − M| ≤ εc^{−0.4n} , xy ≥ |x||y| − εc^{−0.4n}} ,

where xy denotes scalar product.

Lemma 2. There is some ε_{0} = ε_{0}(c) and n_{0} = n_{0}(T, M, D, c, ε) such that if n ≥ n_{0}, 0 < ε < ε_{0}, 0 ≤ g_{n+1} < Dc^{n+1} and

(4.9)  0 ≤ f(x) ≤ exp \{\frac{g_{n+1}}{2M} (|x|^{2} − M^{2})\}  for all x ∈ R^{2}

then

a) 0 ≤ S_{n}f(x) ≤ c^{n} exp \{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − \frac{c^{n}}{3T} (|x| − M)^{2}\} , x ∈ R^{2}

and

b) 0 ≤ S_{n}^{ε}f(x) ≤ exp \{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M) − c^{n/2}\}  if ||x| − M| < c^{−0.4n}

with g_{n} = \frac{g_{n+1}}{2} + \frac{c^{n}}{T} M.

Lemma 3. There is some n_{0} = n_{0}(T, M, D, c) such that if for n ≥ n_{0}

0 ≤ f(x) ≤ exp{g_{n+1}(|x| − M)}  for ||x| − M| < c^{−0.4(n+1)}
f(x) = 0  for ||x| − M| ≥ c^{−0.4(n+1)}

then

0 ≤ S_{n}f(x) ≤ K exp \{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M)\} for ||x| − M| ≤ c^{−0.4n}

with

g_{n} = \frac{g_{n+1}}{2} + \frac{c^{n}}{T} M

and some K = K(T, M, D, c) > 0.

Proof of Lemma 2. Part a). We have

0 ≤ S_{n}f(x) ≤ ∫ exp \{\frac{c^{n}}{T} xy + \frac{g_{n+1}}{2M} (|\frac{x + y}{2} |^{2} − M^{2})\} p_{n}(y) dy .

Clearly, max_{|y|=r} xy is taken in the point y = \frac{x}{|x|}r, and it equals |x|r. Similarly max_{|y|=r}(x + y)^{2} = (|x| + r)^{2}. Hence (4.10)

0 ≤ S_{n}f(x) ≤ 2π ∫_{0}^{∞} exp \{\frac{c^{n}}{T} |x|r + \frac{g_{n+1}}{2M} [(\frac{|x| + r}{2} )^{2} − M^{2}]\} rp̄_{n}(r) dr .

Let us split up the integral into two parts, ∫_{0}^{M} and ∫_{M}^{∞}. Put

(4.11)  f_{n}(t) = c^{−n}p̄_{n}(M + c^{−n}t) .

It follows from (1.10) that

(4.11′)  f_{n}(t) ≤ C_{1} exp(−C_{2}|t|)  for − c^{n}M < t < 0

and from (1.8), (1.7) and (1.9) that

(4.11′′)  f_{n}(t) ≤ C_{3} exp(−t^{2})  for t > 0

with some appropriate C_{1} > 0, C_{2} > 0 and C_{3} > 0. (Relations (1.7) and (1.9) are needed in the domain 0 < |x| < ηn^{1/α}c^{−n}.)

First we estimate the integral ∫_{0}^{M}. For 0 ≤ r ≤ M we have

\frac{c^{n}}{T} |x|r + \frac{g_{n+1}}{2M} ((\frac{|x| + r}{2} )^{2} − M^{2}) ≤ \frac{c^{n}}{T} |x|M + \frac{g_{n+1}}{2M} ((\frac{|x| + M}{2} )^{2} − M^{2}) .

Hence (4.11) and (4.11′) imply that

∫_{0}^{M} … dr ≤ C_{1} exp \{\frac{c^{n}}{T} |x|M + \frac{g_{n+1}}{2} [(\frac{|x| + M}{2} )^{2} − M^{2}]\}

∫_{−c^{n}M}^{0}(M + c^{−n}t)e^{−C_{2}|t|} dt

≤ C_{4}M exp \{\frac{c^{n}}{T} |x|M + \frac{g_{n+1}}{2M} [(\frac{|x| + M}{2} )^{2} − M^{2}]\} .

Simple calculation yields the identity

\frac{c^{n}}{T} |x|M + \frac{g_{n+1}}{2M} [(\frac{|x| + M}{2} )^{2} − M^{2}]

= \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − (\frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} ) (|x| − M)^{2} .

Then, since C_{4}M < c^{n/2} we get that (4.12)

∫_{0}^{M} … dr ≤ c^{n/2} exp \{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) + (\frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} ) (|x| − M)^{2}\} .

Let us estimate ∫_{M}^{∞} … dr. We make the change of variables r = M + c^{−n}t, introduce f_{n}(t) = c^{−n}p̄_{n}(M + c^{−n}t) , ḡ_{n} = c^{−n}g_{n} and ḡ_{n+1} = c^{−(n+1)}g_{n+1}. Since

(4.13)  \begin{matrix}\frac{c^{n}}{T} |x|r + \frac{g_{n+1}}{2M} [(\frac{|x| + r}{2} )^{2} − M^{2}] \\ = \frac{c^{n}}{T} |x|M + \frac{g_{n+1}}{2M} [(\frac{|x| + M}{2} )^{2} − M^{2}] \\ \frac{c^{n}}{T} |x|(r − M) + \frac{g_{n+1}}{8M} (r − M)(2|x| + r + M) \\ = \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − (\frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} ) (|x| − M)^{2} \\ + (\frac{c^{n}}{T} + \frac{g_{n+1}}{4M} ) |x|(r − M) + \frac{g_{n+1}}{4} (r − M) + \frac{g_{n+1}}{8M} (r − M)^{2} \end{matrix}

hence (4.14)

∫_{M}^{∞} … dr = exp \{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − (\frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} ) (|x| − M)^{2}\} J_{n}(|x|)

with

(4.14′) \begin{aligned}J_{n}(|x|) & = ∫_{0}^{∞} exp\{(\frac{1}{T} + \frac{c}{4M} ḡ_{n+1}) |x|t \\ & + \frac{c}{4}ḡ_{n+1}t + c\frac{c^{−n}}{8M} ḡ_{n+1}t^{2}\}(M + c^{−n}t)f_{n}(t) dt \\ & = ∫_{0}^{∞} exp\{(\frac{1}{T} + \frac{c}{4M} ḡ_{n+1}) (|x| − M)t \\ & + (\frac{M}{T} + \frac{c}{2}ḡ_{n+1}) t + \frac{c^{−n+1}}{8M} ḡ_{n+1}t^{2}\}(M + c^{−n}t)f_{n}(t) dt . \end{aligned}

Relation (4.11′′) implies that

J_{n}(|x|) ≤ C_{5} ∫_{0}^{∞} exp \{(\frac{1}{T} + \frac{c}{4M} ḡ_{n+1}) (|x| − M)t − \frac{t^{2}}{2} \} dt
≤ C_{6}{exp C_{7}(|x| − M)^{2}} .

Thus

∫_{M}^{∞} … dr ≤ C_{6} exp\{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2})

− (\frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} )(|x| − M)^{2} + C_{7}(|x| − M)^{2}\} .

16  P. M. Bleher and P. Major

Since \frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} − C_{7} ≥ \frac{c^{n}}{3T} and C_{6} ≤ c^{n/2} for large n, hence

∫_{M}^{∞} … dr ≤ c^{n/2} exp\{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − \frac{c^{n}}{3T} (|x| − M)^{2}\} .

This inequality together with (4.12) imply that

(4.15) S_{n}f(x) ≤ 4πc^{n/2} exp\{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − \frac{c^{n}}{3T} (|x| − M)^{2}\} ≤

≤ c^{n} exp\{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − \frac{c^{n}}{3T} (|x| − M)^{2}\}

as we have claimed. Part b). Let us introduce

R_{n}^{ε}f(x) = ∫_{\{y, ||y|−M|≥εc^{−0.4n}\}} exp(\frac{c^{n}}{T} xy)f(\frac{x + y}{2} )p_{n}(y) dy

and

Q_{n}^{ε}f(x) = ∫_{{y, ||y|−M|<εc^{−0.4n}, xy≤|x||y|−εc^{−0.4n}}} … dy .

Clearly, S_{n}^{ε}f(x) = R_{n}^{ε}f(x) + Q_{n}^{ε}f(x), and by (4.10)

0 ≤ R_{n}^{ε}f(x) ≤ 2π[∫_{0}^{M−εc^{−0.4n}} + ∫_{M+εc^{−0.4n}}^{∞}]

exp\{\frac{c^{n}}{T} |x|r + \frac{g_{n+1}}{2M} ((\frac{|x| + r}{2} )^{2} − M^{2})\}rp̄_{n}(r) dr .

Moreover, similarly to part a), we get by using (4.13) and the observation

(\frac{c^{n}}{T} + \frac{g_{n+1}}{4M} )|x|(r − M) + \frac{g_{n+1}}{4} (r − M) + \frac{g_{n+1}}{8M} (r − M)^{2}

< \frac{g_{n+1}}{4} (r − M)(1 + \frac{r − M}{2M} ) < −2c^{n/2} if − M < r − M < −c^{−0.4n}

that (4.16) ∫_{0}^{M−εc^{−0.4n}} … dr

≤ C_{1} exp \{−2c^{n/2} + \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − (\frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} )(|x| − M)^{2}\}

∫_{0}^{M−εc^{−0.4n}} rp̄_{n}(r) dr

≤ c^{−n/2} exp\{−2c^{n/2} + \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2})

− (\frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} )(|x| − M)^{2}\}

≤ c^{−n/2} exp\{−2c^{n/2} + \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2})\} ,

and by (4.13) and (4.11′′), similarly to (4.14), (4.14′)

∫_{M+εc^{−0.4n}}^{∞} … dr ≤ exp\{\frac{c^{n}}{T} M^{2}+ \frac{g_{n}}{2M} (|x|^{2}−M^{2})−(\frac{c^{n}}{2T} + \frac{g_{n+1}}{8M} )(|x|−M)^{2}\}J_{n}^{ε}(|x|)

with

J_{n}^{ε}(|x|) = C_{5} ∫_{εc^{0.6n}}^{∞} c^{n} exp\{(\frac{1}{T} + \frac{c}{4M} ḡ_{n+1})(|x|−M)t+(\frac{M}{T} + \frac{c}{2}ḡ_{n+1})t− \frac{t^{2}}{2} \} dt .

Since ||x| − M| ≤ c^{−0.4n}, and ḡ_{n+1} < D

sup_{t≥0}\{(\frac{1}{T} + \frac{c}{4M} ḡ_{n+1})(|x| − M)t + (\frac{M}{T} + \frac{c}{2}ḡ_{n+1})t − \frac{t^{2}}{6} \} ≤ C′ ,

and therefore

J_{n}^{ε}(|x|) ≤ C_{6} ∫_{εc^{0.6n}}^{∞} c^{n} exp(−\frac{t^{2}}{3} ) dt ≤ exp(−ε′c^{1.2n}) ≤ c^{−n} exp(−2c^{n/2})

with some ε′ = ε′(ε) > 0. Hence

∫_{M+εc^{−0.4n}}^{∞} … dr ≤ c^{−n} exp\{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − 2c^{n/2}\} .

The last inequality together with (4.16) imply that

(4.17)  R_{n}^{ε}f(x) ≤ 4πc^{−n/2} exp\{−2c^{n/2} + \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2})\} .

Now we estimate Q_{n}^{ε}f(x). We have

0 ≤ Q_{n}^{ε}f(x) ≤ ∫ \begin{matrix}\{y,||y|−M|≤c^{−0.4n}, \\ xy≤|x||y|−εc^{−0.4n} \} \end{matrix} exp\{\frac{c^{n}}{T} xy + \frac{g_{n+1}}{2M} [(\frac{x + y}{2} )^{2} −M^{2}]\}p_{n}(y) dy .

Since for |y| = r xy ≤ r|x| − εc^{−0.4n} and (\frac{x+y}{2} )^{2} ≤ (\frac{|x|+r}{2} )^{2} in the last integral, hence

Q_{n}^{ε}f(x) ≤ 2π ∫_{0}^{∞} exp\{\frac{c^{n}}{T} (|x|r − εc^{−0.4n}) + \frac{g_{n+1}}{2M} [(\frac{|x| + r}{2} )^{2} − M^{2}]\}rp̄_{n}(r) dr

≤ 2π exp(−\frac{εc^{0.6n}}{T} ) ∫_{0}^{∞} exp\{\frac{c^{n}}{T} |x|r + \frac{g_{n+1}}{2M} [(\frac{|x| + r}{2} )^{2} − M^{2}]\}

rp̄_{n}(r) dr .

The last integral has already appeared in (4.10). We have estimated it in Part a), and bounded it by the right hand side of (4.15). Hence

Q_{n}^{ε}f(x) ≤ 2πc^{n} exp\{−\frac{ε}{T} c^{0.6n} + \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2})\}

≤ \frac{1}{2} exp\{−2c^{n/2} + \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2})\} .

This inequality together with (4.17) imply that

S_{n}^{ε}f(x) ≤ exp\{−2c^{n/2} + \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2})\} .

Since \frac{1}{2M} (|x|^{2} − M^{2}) = (|x| − M) + \frac{1}{2M} (|x| − M)^{2} and \frac{g_{n}}{2M} (|x| − M)^{2} ≤ Dc^{0.2n} if ||x| − M| ≤ c^{−0.4n}, hence the last inequality implies that under the conditions of Part b)

S_{n}^{ε}f(x) ≤ exp\{−2c^{n/2} + \frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x| − M) + Kc^{0.2n}\}

≤ exp\{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M) − c^{n/2}\} ,

as we have claimed.

Proof of Lemma 3.

0 ≤ S_{n}f(x) ≤ ∫_{\{y, || \frac{x+y}{2} |−M|≤c^{−0.4(n+1)}\}} exp \{\frac{c^{n}}{T} xy + g_{n+1} (|\frac{x + y}{2} | − M)\}
p_{n}(y) dy .

Since

max_{|y|=r} \{\frac{c^{n}}{T} xy + g_{n+1} (|\frac{x + y}{2} | − M)\} ≤ \frac{c^{n}}{T} |x|r + g_{n+1} [(\frac{|x| + r}{2} ) − M] ,

hence

S_{n}f(x) ≤ 2π ∫_{0}^{∞} exp \{\frac{c^{n}}{T} |x|r + g_{n+1} (\frac{|x| + r}{2} − M)\} rp̄_{n}(r) dr .

Writing \frac{c^{n}}{T} |x|r = \frac{c^{n}}{T} M^{2} + \frac{c^{n}}{T} M(|x| − M + r − M) + \frac{c^{n}}{T} (|x| − M)(r − M) we get that

S_{n}f(x) ≤ 2π exp \{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M)\}

∫_{0}^{∞} exp \{g_{n}(r − M) + \frac{c^{n}}{T} (|x| − M)(r − M)\} rp̄_{n}(r) dr .

The change of variables r = M + c^{−n}t and the introduction of ḡ_{n} = c^{−n}g_{n} yields that

S_{n}f(x) ≤ 2π exp \{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M)\}

∫_{−c^{n}M}^{∞} exp \{ḡ_{n}t + \frac{(|x| − M)}{T}t\} (M + c^{−n}t)f_{n}(t) dt .

Since 0 < ḡ_{n} < D, and ||x| − M| < c^{−0.4n} relations (4.11′) and (4.11′′) imply that for large n

∫_{−c^{n}M}^{∞} exp \{ḡ_{n}t + \frac{1}{T} (|x| − M)t\} (M + c^{−n}t)f_{n}(t) dt < K̄

with some K̄ > 0 independent of n. Hence

S_{n}f(x) ≤ 2πK̄ exp \{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M)\}

≤ K exp \{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M)\} ,

as we have claimed.

5. The proof of Proposition 1′. In this Section we prove the estimates (4.1)– (4.7) which imply Proposition 1′.

a) The estimation of S_{n}^{1}f(x) for x ∈ Ω_{n}^{1}.

It follows from (2.15) and (2.15′) that

S_{n}^{1}f(x) = ∫_{\{y, \frac{x+y}{2} ∈Ω_{n+1}^{1}\}} exp\{\frac{c^{n}}{T} xy + g_{n+1}(\frac{x^{(1)} + y^{(1)}}{2} − M) + A_{n+1}

(\frac{x^{(2)} + y^{(2)}}{2} )^{2} + ε_{n+1}(\frac{x + y}{2} )\}p_{n}(y) dy .

Hence

S_{n}^{1}f(x) = exp(ε̄_{n}(x)) ∫_{\{y, \frac{x+y}{2} ∈Ω_{n+1}^{1}\}} exp\{\frac{c^{n}}{T} xy + g_{n+1} (\frac{x^{(1)} + y^{(1)}}{2} − M)

+ A_{n+1}(\frac{x^{(2)} + y^{(2)}}{2} )^{2}\}p_{n}(y) dy .

with some

(5.1)  sup_{x∈Ω_{n}^{1}} |ε̄_{n}(x)| ≤ q^{n+1} .

Let us rewrite the last expression in polar coordinate system. We get that

(5.2)  S_{n}^{1}f(x) = exp(ε̄_{n}(x)) ∫_{0}^{∞} I_{n}(r)p̄_{n}(r) dr

with

I_{n}(r) = ∫_{\{φ∈Γ_{n}(r,x)\}} r exp\{\frac{c^{n}}{T} (x^{(1)}y^{(1)} + x^{(2)}y^{(2)}) + g_{n+1} (\frac{x^{(1)} + y^{(1)}}{2} − M)

+ A_{n+1} (\frac{x^{(2)} + y^{(2)}}{2} )^{2}\} dφ

where y^{(1)} = r cos φ, y^{(2)} = r sin φ, −π < φ < π, y = (y^{(1)}, y^{(2)}) and Γ_{n}(r, x) = \{φ, \frac{x+y}{2} ∈ Ω_{n+1}^{1}\}. We shall express I_{n}(r) as an asymptotically Gaussian integral with respect to φ. For this aim we give some bounds on x^{(1)}, x^{(2)}, y^{(1)} and y^{(2)} if x ∈ Ω_{n}^{1} and \frac{x+y}{2} ∈ Ω_{n+1}^{1}. We have

(5.3)  \begin{matrix}|x^{(2)}| < c^{−0.4n} \\ |x^{(1)} − M| < \frac{3}{2}c^{−0.4n} \end{matrix}  if x ∈ Ω_{n}^{1} .

The second relation in (5.3) holds, since

|x^{(1)} − M| = |(|x|^{2} − x^{(2)2})^{1/2} − M| = | |x| − \frac{x^{(2)2}}{2|x|} + O(x^{(2)4}) − M|

≤ ||x| − M| + |\frac{x^{(2)2}}{2|x|} + O(x^{(2)4})| ≤ c^{−0.4n} + Kc^{−0.8n} ≤ \frac{3}{2}c^{−0.4n} .

Similarly, if \frac{x+y}{2} ∈ Ω_{n+1}^{1} then |\frac{x^{(2)}+y^{(2)}}{2} | < c^{−0.4n}, and |\frac{x^{(1)}+y^{(1)}}{2} − M| < 2c^{−0.4n}. Hence

(5.4)  |y^{(2)}| ≤ | − x^{(2)}| + |x^{(2)} + y^{(2)}| ≤ 3c^{−0.4n},
(5.4′)  |y^{(1)} − M| = |M − x^{(1)}| + |x^{(1)} + y^{(1)} − 2M| ≤ 6c^{−0.4n}

and

(5.4′′) |r − M| = ||y| − M| = |(y^{(1)2} + y^{(2)2})^{1/2} − M|

= |y^{(1)} + \frac{y^{(2)2}}{2y^{(1)}} + O(y^{(2)4}) − M| ≤ 10c^{−0.4n}

if x ∈ Ω_{n}^{1} and \frac{x+y}{2} ∈ Ω_{n+1}^{1}. In particular, (5.4′′) implies that

(5.5)  I_{n}(r) = 0  if |r − M| > 10c^{−0.4n}, and x ∈ Ω_{n}^{1} .

Furthermore, |φ| ≤ 2| sin φ| = \frac{2}{r}|y^{(2)}| = O(c^{−0.4n}) and y^{(1)} = r(1 − cos φ) = r(1 − φ^{2}/2) + O(c^{−1.6n}), y^{(2)} = r sin φ = rφ + O(c^{−1.2n}) and r = M + O(c^{−0.4n}) if φ ∈ Γ_{n}(r, x). Hence

I_{n}(r) = (1 + O(c^{−0.2n})) ∫_{\{φ∈Γ_{n}(r,x)\}} M exp\{\frac{c^{n}}{T} (x^{(1)}r(1 − \frac{φ^{2}}{2} ) + x^{(2)}rφ)

+ g_{n+1} (\frac{x^{(1)} + r(1 − \frac{φ^{2}}{2} )}{2} − M) + A_{n+1} (\frac{x^{(2)} + rφ}{2})^{2}\} dφ

= (1 + O(c^{−0.2n})) M exp \{\frac{c^{n}}{T} rx^{(1)} + \frac{g_{n+1}}{2} (\frac{x^{(1)} + r}{2} − M) + \frac{A_{n+1}}{4} x^{(2)2}\}

∫_{\{φ∈Γ_{n}(r,x)\}} exp\{−(\frac{c^{n}}{2T} x^{(1)} + \frac{g_{n+1}}{4} − r \frac{A_{n+1}}{4} )rφ^{2}

+ (\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )x^{(2)}rφ\} dφ .

Moreover, since c^{n}x^{(1)}rφ^{2} = c^{n}M^{2}φ^{2}+O(c^{−0.2n}), g_{n+1}rφ^{2} = g_{n+1}Mφ^{2}+O(c^{−0.2n}), A_{n+1}r^{2}φ^{2} = A_{n+1}M^{2}φ^{2} + O(c^{−0.2n}) and (\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )x^{(2)}rφ = (\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )x^{(2)}Mφ + O(c^{−0.2n}) in our case (observe that g_{n+1} ≤ Dc^{n+1} and A_{n+1} ≤ Āc^{n+1}), hence we make an error of order O(c^{−0.2n}) by substituting x^{(1)} and r by M in the last integral, i.e.

I_{n}(r) = (1 + O(c^{−0.2n})) M exp \{\frac{c^{n}}{T} rx^{(1)} + \frac{g_{n+1}}{2} (\frac{x^{(1)} + r}{2} − M) + \frac{A_{n+1}}{4} x^{(2)2}\}

∫_{\{φ∈Γ_{n}(r,x)\}} exp \{(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )M^{2}φ^{2} + (\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )Mx^{(2)}φ\} dφ ,

or equivalently

I_{n}(r) = (1 + O(c^{−0.2n})) exp\{\frac{c^{n}}{T} rx^{(1)} + \frac{g_{n+1}}{2} (\frac{x^{(1)} + r}{2} − M)

+ \frac{A_{n+1}}{4} x^{(2)2} + \frac{(\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )^{2}x^{(2)2}}{4(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )}\}

∫_{\{φ∈Γ_{n}(r,x)\}} M exp \{−M^{2}(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )(φ − γ_{n}x^{(2)})^{2}\} dφ ,

with

γ_{n} = \frac{(\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )}{(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )} .

By (2.5′) we get from this relation that

(5.6)

I_{n}(r) = (1 + O(c^{−0.2n})) exp \{\frac{c^{n}}{T} rx^{(1)} + \frac{g_{n+1}}{2} (\frac{x^{(1)} + r}{2} − M) + A_{n}x^{(2)2}\}

∫_{\{φ∈Γ_{n}(r,x)\}} M exp \{−M^{2}(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} ) (φ − γ_{n}x^{(2)})^{2}\} dφ .

Since

(5.7)  (\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} ) ≥ c^{n} (\frac{1}{2T} + c\frac{ḡ}{4M} − c\frac{Ā}{4}) = Kc^{n}

with K = \frac{1}{2T} + c\frac{ḡ}{4M} − c\frac{Ā}{4} = \frac{c(4−c)}{4(2−c)T} > 0 relation (5.6) implies that

(5.8)  I_{n}(r) ≤ \sqrt{π} (1 + O(c^{−0.2n})) (\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )^{−1/2}
exp \{\frac{c^{n}}{T} rx^{(1)} + \frac{g_{n+1}}{2} (\frac{x^{(1)} + r}{2} − M) + A_{n}x^{(2)2}\}

Moreover, we claim that (5.9)

\begin{aligned}I_{n}(r) & = (1 + O(c^{−0.2n})) \sqrt{π} (\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )^{−1/2} \\ & exp \{\frac{c^{n}}{T} rx^{(1)} + \frac{g_{n+1}}{2} (\frac{x^{(1)} + r}{2} − M) + A_{n}x^{(2)2}\} if |r − M| < ε̄c^{−0.4n} \end{aligned}

with some approppriate ε̄ > 0. Because of (5.6) and (5.7) to prove (5.9) it is enough to show that there is some ε = ε_{0}(ε̄) > 0 such that

(5.10)  {φ: |φ − γ_{n}x^{(2)}| < ε_{0}c^{−0.4n}} ⊂ Γ_{n}(x, r)  if |r − M| < ε̄c^{−0.4n} .

Since A_{n+1} ≤ Āc^{n+1} and g_{n+1} ≥ ḡc^{n+1}

γ_{n} = \frac{(\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )}{2(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )} ≤ \frac{1 + \frac{2−c}{2}}{(1 + \frac{c}{2−c} − \frac{2−c}{2})M} = \frac{2 − c}{cM} ,

and we prove (5.10) by showing that

(5.10′)  \{\frac{x^{(1)} + r cos φ}{2}, \frac{x^{(2)} + r sin φ}{2}\} ∈ Ω_{n+1}^{1}

if |φ − γ_{n}x^{(2)}| < ε_{0}c^{−0.4n}, |r − M| ≤ ε̄c^{−0.4n} and x ∈ Ω_{n}^{1}. But in this case

|\frac{x^{(2)} + r sin φ}{2}| ≤ \frac{1}{2} (|x^{(2)}| + r|φ|) ≤ \frac{1}{2} (|x^{(2)}| + r(γ_{n}|x^{(2)}| + |φ − γ_{n}x^{(2)}|))

≤ \frac{1}{2}[c^{−0.4n} + (M + ε̄c^{−0.4n})(\frac{2 − c}{cM}c^{−0.4n} + ε_{0}c^{−0.4n})]

≤ \frac{1}{\sqrt{c}}c^{−0.4n} ≤ c^{−0.4(n+1)}

if ε_{0} > 0 and ε̄ > 0 are sufficiently small. We also get with the help of (5.3′) that

|\frac{x^{(1)} + r cos φ}{2} − M| ≤ |\frac{x^{(1)} − M}{2} | + \frac{r}{2} |cos φ − 1| + |\frac{r − M}{2} |

≤ |\frac{x^{(1)} − M}{2} | + \frac{rφ^{2}}{2} + |\frac{r − M}{2} |

≤ |\frac{x^{(1)} − M}{2} | + \frac{r}{2} (γ_{n}|x^{(2)}| + ε_{0}c^{−0.4n})^{2} + |\frac{r − M}{2} |

≤ (\frac{3}{4} + ε̄) c^{−0.4n} + Kc^{−0.8n} ≤ c^{−0.1}c^{−0.4(n+1)}

if ε_{0} > 0 and ε̄ > 0 are sufficiently small. The above estimates imply (5.10′) hence also (5.10). Now we can estimate the term ∫_{0}^{∞} I_{n}(r)p̄_{n}(r) dr. Relation (5.9) yields that

∫_{|r−M|<ε̄c^{−0.4n}} I_{n}(r)p̄_{n}(r) dr = (1 + O(c^{−0.2n})) \sqrt{π}(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )^{−1/2}J_{n,ε̄}
exp \{\frac{c^{n}}{T} x^{(1)}M + \frac{g_{n+1}}{4} (x^{(1)} − M) + A_{n}x^{(2)2}\}

with

J_{n,ε̄} = J_{n,ε̄}(x^{(1)}) = ∫_{|r−M|<ε̄c^{−0.4n}} exp \{\frac{c^{n}}{T} x^{(1)}(r − M) + \frac{g_{n+1}}{4} (r − M)\} p̄_{n}(r) dr

= ∫_{|t|<ε̄c^{0.6n}} exp \{\frac{tx^{(1)}}{T} + c\frac{ḡ_{n+1}}{4} t\} f_{n}(t) dt

with the function f defined in (4.11). On the other hand, by (5.8) and (5.5)

∫_{|r−M|>ε̄c^{−0.4n}} I_{n}(r)p̄_{n}(r) dr

≤ (1 + O(c^{−0.2n})) \sqrt{π}(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )^{−1/2}

exp \{\frac{c^{n}}{T} x^{(1)}M + \frac{g_{n+1}}{4} (x^{(1)} − M) + A_{n}x^{(2)2}\} J̄_{n,ε̄}

with

J̄_{n,ε̄} = ∫_{10c^{0.6n}>|t|>ε̄c^{0.6n}} exp \{\frac{tx^{(1)}}{T} + c\frac{ḡ_{n+1}}{4} t\} f_{n}(t) dt .

Let us remark that J_{n,ε̄} = J_{n,ε̄}(x^{(1)}) depends on x^{(1)}. We show that this dependence is very weak. Namely, since

|\frac{d}{dx^{(1)}} J_{n,ε̄}(x^{(1)})| = ∫_{|t|<ε̄c^{0.6n}} exp \{\frac{tx^{(1)}}{T} + c\frac{ḡ_{n+1}}{4} t\} \frac{|t|}{T}f_{n}(t) dt ≤ C < ∞

by (4.11′) and (4.11′′), and for x^{(1)} = M the expression

J_{n,ε̄}(M) = ∫_{|t|<ε̄c^{0.6n}} exp \{\frac{tM}{T} + c\frac{ḡ_{n+1}}{4} t\} f_{n}(t) dt

satisfies the relation

(5.11)  0 < K_{1} < J_{n,ε̄}(M) < K_{2} < ∞

because of (4.11′), (4.11′′), the inequality 0 ≤ ḡ_{n+1} ≤ D and the relation f_{n}(t) ≥ const. > 0 for |t| < 1 that follows from Theorem A. Hence

J_{n,ε̄}(x^{(1)}) = (1 + O(c^{−0.4n})) J_{n,ε̄}(M) if x ∈ Ω_{n}^{1} .

Similarly,

0 ≤ J̄_{n,ε̄} = O (exp(−Kc^{0.6n})) .

The above relations imply that

∫ I_{n}(r)p̄_{n}(r) dr = (1 + O(c^{−0.2n})) \sqrt{π}(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )^{−1/2}J_{n,ε̄}(M)

exp \{\frac{c^{n}}{T} M^{2} + g_{n}(x^{(1)} − M) + A_{n}x^{(2)2}\} .

The last formula together with (5.1) and (5.2) imply (4.1) with

L_{n} = \sqrt{π}(\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} )^{−1/2}J_{n,ε̄}(M) exp (\frac{c^{n}}{T} M^{2}) .

Since (\frac{c^{n}}{2T} + \frac{g_{n+1}}{4M} − \frac{A_{n+1}}{4} ) < const.c^{n} relation (5.11) and the last formula imply (4.1′′). b) The estimation of S_{n}^{1}f(x) for x ∈ Ω_{n}^{2}. We divide Ω_{n}^{2} to two subsets Ω̄_{n}^{2} and Ω̂_{n}^{2}, where we apply different arguments.

Put

Ω_{n,ε}^{1} = {x, ||x| − M| ≤ c^{−0.4n}, |x^{(2)}| ≤ (1 + ε)c^{−0.4n}, x^{(1)} > 0} , ε > 0,
Ω̄_{n}^{2} = Ω_{n,ε}^{1} − Ω_{n}^{1} ,

and

Ω̂_{n}^{2} = {x, ||x| − M| ≤ c^{−0.4n}} − Ω_{n,ε}^{1} .

Clearly, Ω_{n}^{2} = Ω̄_{n}^{2} ∪ Ω̂_{n}^{2}. The domain Ω_{n,ε}^{1} is a slight enlargement of Ω_{n}^{1}. It is not difficult to see by analizing the proof of relation (4.1) that for sufficiently small ε > 0

S_{n}^{1}f(x) = L_{n} exp {g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} + ε̄(x) + ε̂(x)}

if x ∈ Ω_{n,ε}^{1} with some |ε̄_{n}(x)| < q^{n+1} and ε̂_{n}(x) < Kc^{−0.2n}. Since x^{(1)} − M = |x| − M − \frac{x^{(2)2}}{2M} + O(c^{−1.2n}) and |x^{(2)}| ≥ c^{−0.4n} for x ∈ Ω̄_{n}^{2}, the above relation implies that

S_{n}^{1}f(x) = L_{n} exp\{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n})x^{(2)2}
+ ε̄_{n}(x) + ε̂_{n}(x) + O (c^{−0.2n}) biggr}

≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) c^{−0.8n} + q^{n+1} + Kc^{−0.2n}\}

in this case, what we had to show. For x ∈ Ω̂_{n}^{2} we define

S_{n}^{ε}f(x) = ∫_{R^{2}−V_{n}^{ε}(x)} exp(\frac{c^{n}}{T} xy)f(\frac{x + y}{2} )p_{n}(y) dy

and

T_{n}^{ε}f(x) = ∫_{\{y, \frac{x+y}{2} ∈Ω_{n+1}^{1}, y∈V_{n}^{ε}(x)\}} exp (\frac{c^{n}}{T} xy) f (\frac{x + y}{2} ) p_{n}(y) dy ,

where V_{n}^{ε}(x) is defined in (4.8′), and ε > 0 is appropriately chosen. The function S_{n}^{ε}f(x) will be bounded with the help of Part b) of Lemma 2, and T_{n}^{ε}f(x) similarly to S_{n}^{1}f(x) in the case x ∈ Ω_{n}^{1}. To apply Lemma 2 first we show that under the conditions of Proposition 1′

(5.12)  0 ≤ f(x) ≤ 2 exp \{\frac{g_{n+1}}{2M} (|x|^{2} − M^{2})\}  for all x ∈ R^{2} .

For x ∈ Ω_{n+1}^{1}

x^{(1)} − M = |x| − M − \frac{x^{(2)2}}{2|x|} + O (x^{(2)4}) = |x| − \frac{x^{(2)2}}{2M} − M + O (c^{−1.2n}) ,

The large-scale limit of Dyson’s hierarchical model  25

hence

0 ≤ f(x) ≤ exp \{g_{n+1}(|x| − M) − (\frac{g_{n+1}}{2M} − A_{n+1})x^{(2)2} + ε_{n+1}(x) + O(c^{−0.2n})\} ,

and since \frac{g_{n+1}}{2M} − A_{n+1} > 0 hence

0 ≤ f(x) ≤ \frac{3}{2} exp\{g_{n+1}(|x| − M)\} .

This inequality together with the relation

|x| − M = \frac{1}{2M} [(|x|^{2} − M^{2}) − (|x| − M)^{2}] ≤ \frac{1}{2M} (|x|^{2} − M^{2})

imply (5.12) for x ∈ Ω_{n+1}^{1}. Similarly, for x ∈ Ω_{n+1}^{2} the relation

0 ≤ f(x) ≤ exp \{g_{n+1}(|x| − M) − (\frac{g_{n+1}}{2M} − A_{n+1})c^{−0.8(n+1)} + q^{n+1}\}

≤ \frac{3}{2} exp\{g_{n}(|x| − M)\}

implies (5.12), and this relation also holds for x ∈ Ω_{n+1}^{3} by relation (2.17). By (5.12) part b) of Lemma 2 can be applied for \frac{1}{2}f(x), and it yields that

S_{n}^{ε}f(x) ≤ 2 exp \{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M) − c^{n/2}\} if x ∈ Ω_{n}^{2} (or x ∈ Ω_{n}^{1}) .

Since L_{n} ≥ c^{−n} exp\{\frac{c^{n}}{T} M^{2}\}, and (\frac{g_{n}}{2M} − A_{n})c^{−0.8n} = O(c^{0.2n}) ≤ \frac{1}{3}c^{n/2}, the last inequality implies that

(5.13) S_{n}^{ε}f(x) ≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n})c^{−0.8n} − \frac{1}{2}c^{n/2}\}
if x ∈ Ω_{n}^{2} (or x ∈ Ω_{n}^{1}) .

To estimate T_{n}^{ε}f(x) first we show that if x ∈ Ω_{n}^{2}, y ∈ V_{n}^{ε}(x) and \frac{x+y}{2} ∈ Ω_{n+1}^{1} then

(5.14) |x^{(2)}| < 2\sqrt{ε}c^{−0.2n}, |y^{(2)}| < 2\sqrt{ε}c^{−0.2n}, |x^{(1)} − M| ≤ 2\sqrt{ε}c^{−0.2n},
|y^{(1)} − M| ≤ 2\sqrt{ε}c^{−0.2n}, ||y| − M| ≤ εc^{−0.4n} .

Indeed, in this case (x, y) ≥ |x||y| − εc^{−0.4n}, and

0 ≤ |x − y|^{2} = |x|^{2} + |y|^{2} − 2(x, y) ≤ |x|^{2} + |y|^{2} − 2|x||y| + εc^{−0.4n}
= (|x| − |y|)^{2} + εc^{−0.4n} ≤ 2(|x| − M)^{2} + 2(|y| − M)^{2} + εc^{−0.4n}
≤ 2(1 + ε)c^{−0.8n} + εc^{−0.4n} ≤ 2εc^{−0.4n} .

Hence |\frac{x^{(2)}−y^{(2)}}{2} | ≤ \sqrt{ε}c^{−0.2n} and \frac{1}{2}|(x^{(1)} − M) − (y^{(1)} − M)| ≤ \sqrt{ε}c^{−0.2n}. Since

\frac{x+y}{2} ∈ Ω_{n+1}^{1}, |\frac{x^{(2)}+y^{(2)}}{2} | ≤ c^{−0.4(n+1)} ≤ \sqrt{ε}c^{−0.2n} and \frac{1}{2}|(x^{(1)} − M) + (y^{(1)} − M)| ≤ \frac{3}{2}c^{−0.4(n+1)} ≤ \sqrt{ε}c^{−0.2n} by (5.3). These relations together with the definition of V_{n}^{ε}(x) imply (5.14) that enables us to estimate T_{n}^{ε}f(x) similarly to S_{n}^{1}f(x) for x ∈ Ω_{n}^{1}.

We get that

(5.15)  T_{n}^{ε}f(x) ≤ (1 + q^{n+1}) ∫_{0}^{∞} I_{n}(r)p̄_{n}(r) dr

with

(5.16)  \begin{aligned}I_{n}(r) & = ∫_{\{φ∈Γ_{n}(r,x)\}} r exp\{\frac{c^{n}}{T} (x^{(1)}y^{(1)} + x^{(2)}y^{(2)}) \\ & + g_{n+1}(\frac{x^{(1)} + y^{(1)}}{2} − M)A_{n+1}(\frac{x^{(2)} + y^{(2)}}{2} )^{2}\} dφ , \end{aligned}

where y^{(1)} = r cos φ, y^{(2)} = r sin φ, y = (y^{(1)}, y^{(2)}) and

Γ_{n}(r, x) = \{φ: \frac{x + y}{2} ∈ Ω_{n+1}^{1}, y ∈ V_{n}^{ε}(x)\}.

Observe that by (5.14) |r − M| < εc^{−0.4n}, |φ| ≤ 2| sin φ| = \frac{2}{r}|y^{(2)}| ≤ \frac{2}{r} \sqrt{ε}c^{−0.2n}. Let us make the change of variables z = sin φ in the integral I_{n}(r). We have y^{(2)} = rz, y^{(1)} = r(1 − \frac{y^{(2)2}}{r^{2}} )^{1/2} = r(1 − \frac{y^{(2)2}}{2r^{2}} ) + O(y^{(2)4}), z ≤ 2\frac{\sqrt{ε}}{r}c^{−0.2n}, hence |y^{(1)} − r(1 − \frac{z^{2}}{2} )| ≤ Kε^{2}c^{−0.8n} with some K > 0 independent of ε. These relations imply that if φ ∈ Γ_{n}(r, x) then

\frac{c^{n}}{T} (x^{(1)}y^{(1)} + x^{(2)}y^{(2)}) + g_{n+1}(\frac{x^{(1)} + y^{(1)}}{2} − M) + A_{n+1}(\frac{x^{(2)} + y^{(2)}}{2} )^{2}

≤ \frac{c^{n}}{T} r (x^{(1)}(1 − \frac{z^{2}}{2} ) + x^{(2)}z) + g_{n+1} (\frac{x^{(1)} + r(1 − \frac{z^{2}}{2} )}{2} − M)

+ A_{n+1}(\frac{x^{(2)} + rz}{2} )^{2} + Kε^{2}c^{0.2n}

with some K > 0. Since \frac{dz}{dφ} > \frac{1}{2} and |r − M| < εc^{−0.4n} if Γ_{n}(r, x) is not empty, the above inequality together with (5.16) imply that

(5.17)  I_{n}(r) = 0  if |r − M| ≥ εc^{−0.4n}

and

(5.17′)  I_{n}(r) ≤ 3 exp(Kε^{2}c^{0.2n})Î_{n}(r)  for |r − M| < εc^{−0.4n}

with

Î_{n}(r) = ∫_{−∞}^{∞} M exp\{\frac{c^{n}}{T} r (x^{(1)}(1 − \frac{z^{2}}{2} ) + x^{(2)}z)

+ g_{n+1} (\frac{x^{(1)} + r(1 − \frac{z^{2}}{2} )}{2} − M) + A_{n+1}(\frac{x^{(1)} + rz}{2} )^{2}\} dz .

The expression Î_{n}(r) can be calculated explicitly, and we get that for |r − M| < εc^{−0.4n}

Î_{n} = \sqrt{π}M(\frac{c^{n}}{2T} rx^{(1)}+\frac{g_{n+1}}{4} r − \frac{A_{n+1}}{4} r^{2})^{−1/2}
exp \{\frac{c^{n}x^{(1)}r}{T} + g_{n+1} (\frac{x^{(1)} + r}{2} − M) + A_{n}(x^{(1)}, r)x^{(2)2}\}

with

A_{n}(x^{(1)}, r) = \frac{A_{n+1}}{4} + \frac{(\frac{c^{n}}{T} + \frac{A_{n+1}}{2} )^{2}}{2(\frac{c^{n}x^{(1)}}{T r} + \frac{g_{n+1}}{2r} − \frac{A_{n+1}}{2} )} .

Hence

Î_{n}(r) ≤ Kc^{−n/2} exp \{\frac{c^{n}x^{(1)}r}{T} + g_{n+1} (\frac{x^{(1)} + r}{2} − M) + A_{n}(x^{(1)}, r)x^{(2)2}\}

= Kc^{−n/2} exp\{\frac{c^{n}}{T} M^{2} + g_{n}(x^{(1)} − M) + A_{n}(x^{(1)}, r)x^{(2)2}

+ (\frac{c^{n}x^{(1)}}{T} + \frac{g_{n+1}}{2} )(r − M)\} .

Observe that A_{n} = A_{n}(M, M), and if x ∈ Ω̂_{n}^{2}, y ∈ V_{n}^{ε}(x) and \frac{x+y}{2} ∈ Ω_{n}^{1} then

|A_{n}(x^{(1)}, r) − A_{n}| ≤ K_{0}c^{n}(|x^{(1)} − M| + |r − M|)

|x^{(1)} − M − (|x| − M − \frac{x^{(2)2}}{2M} )| ≤ K_{1}x^{(2)2}(x^{(2)2} + |x^{(1)} − M|) ,

hence (5.17′) implies that (5.18)

I_{n}(r) ≤ K̄c^{−n/2} exp\{Kε^{2}c^{0.2n} + \frac{c^{n}}{T} M^{2} + g_{n}(x^{(1)} − M) + A_{n}x^{(2)2}

+ K_{0}c^{n}(|x^{(1)} − M| + |r − M|)x^{(2)2} + (\frac{c^{n}x^{(1)}}{T} + \frac{g_{n+1}}{2} ) (r − M)\}

≤ K̄c^{−n/2} exp\{Kε^{2}c^{0.2n} + \frac{c^{n}}{T} M^{2} + g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n}) x^{(2)2}

+ [K_{0}c^{n}(|x^{(1)} − M| + |r − M|) + K_{1}g_{n}(x^{(2)2} + |x^{(1)} − M|)] x^{(2)2}

+ (\frac{c^{n}x^{(1)}}{T} + \frac{g_{n+1}}{2} ) (r − M)\}

if |r−M| ≤ εc^{−0.4n}. Since \frac{g_{n}}{2M} −A_{n} > αc^{n} with some α > 0 and x^{(2)2} > (1+ε)c^{−0.4n} if x ∈ Ω̂_{n}^{2}

Kε^{2}c^{0.2n} + [K_{0}c^{n}(|x^{(1)} − M| + |r − M|) + K_{1}g_{n}(x^{(2)2} + |x^{(1)} − M|)] x^{(2)2}

− (\frac{g_{n}}{2M} − A_{n}) x^{(2)2}

= Kε^{2}c^{0.2n} − (\frac{g_{n}}{2M} − A_{n} + O (c^{0.8n})) x^{(2)2}

≤ Kε^{2}c^{0.2n} − (\frac{g_{n}}{2M} − A_{n} + O (c^{0.8n})) (1 + ε)^{2}c^{−0.8n}

≤ − (\frac{g_{n}}{2M} − A_{n}) (1 + ε)c^{−0.8n}

if ε > 0 is chosen sufficiently small. This relation together with (5.18) imply that

I_{n}(r) ≤ Kc^{−n/2} exp\{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M) − (1 + ε)(\frac{g_{n}}{2M} − A_{n})c^{−0.8n}

+ (\frac{c^{n}x^{(1)}}{T} + \frac{g_{n+1}}{2} )(r − M)\} if |r − M| < εc^{−0.4n} .

With the help of this inequality, the estimate we have on the function p̄_{n}(r) and (5.17) we can bound the integral in (5.15) from above. We get that

T_{n}^{ε}f(x) ≤ L_{n} exp \{g_{n}(|x| − M) − (1 + \frac{ε}{2})(\frac{g_{n}}{2M} − A_{n})c^{−0.8n}\} .

Since S_{n}^{1}f(x) ≤ S_{n}^{ε}f(x) + T_{n}^{ε}f(x) the last inequality together with (5.13) imply (4.4) for x ∈ Ω̂_{n}^{2}.

c) The estimation of S_{n}^{2}f(x) for x ∈ Ω_{n}^{1} and x ∈ Ω_{n}^{2}.

We have

0 ≤ S_{n}^{2}f(x) = ∫_{\{y, \frac{x+y}{2} ∈Ω_{n+1}^{2}\}} exp(\frac{c^{n}}{T} xy)f(\frac{x + y}{2} )p_{n}(y) dy

≤ exp \{−(\frac{g_{n+1}}{2M} − A_{n+1})c^{−0.8(n+1)} + q^{n+1}\}

∫_{\{y, \frac{x+y}{2} ∈Ω_{n+1}^{2}\}} exp \{\frac{c^{n}}{T} xy + g_{n+1}(\frac{|x + y|}{2} − M)\} p_{n}(y) dy .

It follows from Lemma 3 that for x ∈ Ω_{n}^{1} ∪ Ω_{n}^{2}

0 ≤ S_{n}^{2}f(x) ≤ K exp\{−(\frac{g_{n+1}}{2M} − A_{n+1})c^{−0.8(n+1)}

+ q^{n+1} + \frac{c^{n}}{T} M^{2} + g_{n}(|x| − M)\} .

By Lemma 1 g_{n+1}c^{−(n+1)} ≥ g_{n}c^{n}, A_{n}c^{−n} ≥ A_{n+1}c^{−(n+1)} i.e. g_{n+1} ≥ cg_{n} and cA_{n} ≥ A_{n+1}. Hence

(\frac{g_{n+1}}{2M} − A_{n+1})c^{−0.8(n+1)} ≥ (\frac{g_{n}}{2M} − A_{n})c^{−0.8n}c^{0.2} ≥ (1 + ε)(\frac{g_{n}}{2M} − A_{n})c^{−0.8n}

if 0 < ε < c^{0.2} − 1. Since ε(\frac{g_{n}}{2M} − A_{n})c^{−0.8n} > ε′c^{0.2n} with some appropriate ε′(ε) > 0 we get that

(\frac{g_{n+1}}{2M} − A_{n+1})c^{−0.8(n+1)} ≥ (\frac{g_{n}}{2M} − A_{n})c^{−0.8n} + ε′c^{0.2n}

Therefore

(5.19)

S_{n}^{2}f(x) ≤ K exp \{\frac{c^{n}}{T} M^{2} + g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n})c^{−0.8n} + q^{n+1} − ε′c^{0.2n}\}

≤ L_{n} exp \{g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n})c^{−0.8n} − \frac{ε′}{2} c^{0.2n}\} .

This is estimate (4.5) (with \frac{ε′}{2} instead of ε′). For x ∈ Ω_{n}^{1}

g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} = g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n})x^{(2)2} + O(c^{−0.2n}) ,

hence (5.19) implies that for x ∈ Ω_{n}^{1}

S_{n}^{2}f(x) ≤ L_{n} exp \{g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} − \frac{ε′}{4} c^{0.2n}\} ,

and this is relation (4.2) (with \frac{ε′}{4} ).

d) The estimation of S_{n}^{3}f(x) for x ∈ Ω_{n}^{1} and x ∈ Ω_{n}^{2}.

Clearly

S_{n}^{3}f(x) = S_{n}^{ε}f(x) + ∫_{\{y, \frac{x+y}{2} ∈Ω_{n+1}^{3}\}∩V_{n}^{ε}(x)} exp(\frac{c^{n}}{T} xy)f(\frac{x + y}{2} )p_{n}(y) dy ,

where S_{n}^{ε}f(x) and V_{n}^{ε} are defined in (4.8) and (4.8′). The term S_{n}^{ε}f(x) is bounded in (5.13). On the other hand, we claim that there is some ε_{0} = ε_{0}(c) such that if 0 < ε < ε_{0} and x ∈ Ω_{n}^{1} ∪ Ω_{n}^{2} then the set \{y, \frac{x+y}{2} ∈ Ω_{n+1}^{3}\} ∩ V_{n}^{ε}(x) is empty, hence the last integral is zero. We have to show that if ||x| − M| < c^{−0.4n}, y ∈ V_{n}^{ε}, i.e. ||y| − M| < εc^{−0.4n} and xy > |x||y| − εc^{−0.4n} then \frac{x+y}{2} ∉ Ω_{n+1}^{3}, i.e. −c^{−0.4(n+1)} ≤ |\frac{x+y}{2} | − M ≤ c^{−0.4(n+1)}.

Estimation from above:

|\frac{x + y}{2} | − M ≤ \frac{|x| − M}{2} + \frac{|y| − M}{2} ≤ \frac{1 + ε}{2}c^{−0.4n} ≤ c^{−0.4(n+1)}

if \frac{1+ε}{2} ≤ c^{−0.4}, what holds for sufficiently small ε.

Estimation from below:

|\frac{x + y}{2} |^{2} − M^{2} = \frac{|x|^{2} + |y|^{2} + 2xy}{4} − M^{2}

≥ \frac{|x|^{2} + |y|^{2} + 2|x||y| − 2εc^{−0.4n}}{4} − M^{2} = (\frac{|x| + |y|}{2})^{2} − M^{2} − \frac{ε}{2}c^{−0.4n}

≥ (\frac{M − c^{−0.4n} + M − εc^{−0.4n}}{2} )^{2} − M^{2} − \frac{ε}{2}c^{−0.4n}

≥ −(1 + ε)Mc^{−0.4n} − \frac{ε}{2}c^{−0.4n} = −M (1 + ε + \frac{ε}{2M} ) c^{−0.4n} .

Hence

|\frac{x + y}{2} | ≥ (M^{2} − M(1 + ε + \frac{ε}{2M} )c^{−0.4n})^{1/2}

≥ M − \frac{(1 + ε + \frac{ε}{2M})}{2}c^{−0.4n} − Kc^{−0.8n}

= M − [(\frac{1}{2} + \frac{ε}{2}(1 + \frac{1}{2M} )) + Kc^{−0.4n}] c^{−0.4n} ≥ M − c^{−0.4(n+1)}

if \frac{1}{2} + \frac{ε}{2}(1 + \frac{1}{2M}) + Kc^{−0.4n} ≤ c^{−0.4}, what we had to show.

Hence S_{n}^{3}f(x) = S_{n}^{ε}f(x), and (5.13) implies (4.6). To prove (4.3) we still have to remark that for x ∈ Ω_{n}^{1}

g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n})c^{−0.8n} − \frac{1}{2}c^{n/2}

≤ g_{n}(|x| − M) − (\frac{g_{n}}{2M} − A_{n})x^{(2)2} − \frac{1}{2}c^{n/2} ≤ g_{n}(x^{(1)} − M) + A_{n}x^{(2)2} − \frac{1}{6}c^{n/2} .

e) The estimation of S_{n}f(x) for x ∈ Ω_{n}^{3}.

We get from (5.12) and Part a) of Lemma 2 that

0 ≤ S_{n}f(x) ≤ 2c^{n} exp \{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − \frac{c^{n}}{3T} (|x|^{2} − M^{2})\} .

For x ∈ Ω_{n}^{3} \frac{c^{n}}{3T} (|x|^{2} − M^{2}) ≥ \frac{c^{0.2n}}{3T}, hence (4.1′′) implies that

0 ≤ S_{n}f(x) ≤ 2c^{n} exp \{\frac{c^{n}}{T} M^{2} + \frac{g_{n}}{2M} (|x|^{2} − M^{2}) − \frac{c^{0.2n}}{3T} \}

≤ L_{n} exp \{\frac{g_{n}}{2M} (|x|^{2} − M^{2})\} ,

i.e. relation (4.7) holds, as we have claimed. Proposition 1′ is proved.

6. The proof of Theorem 1 and Proposition 2. Existence of the thermodinamical limit. First we show with the help of Proposition 1, Lemma 1 and Theorem A that for all q, c^{−0.2} < q < 1, there exist some thresholds n_{0} and N_{0}(n, q) for n ≥ n_{0} such that if n ≥ n_{0} and N ≥ N_{0}(n, q) then

(6.1)  \frac{dμ_{n,N}^{h_{N}}}{dμ_{n}} (x_{1}, …, x_{2^{n}}) = f_{n,N}^{h_{N}}(2^{−n} ∑_{j=1}^{2^{n}} x_{j})

with

(6.2) f_{n,N}^{h_{N}}(x) = L_{n} exp {ḡc^{n}(x^{(1)} − M) + Āc^{n}x^{(2)2} + ε_{n}(x)}  for x ∈ Ω_{n}^{1} ,

where

(6.2′)  sup_{x∈Ω_{n}^{1}} |ε_{n}(x)| ≤ q^{n} ,

(6.3) f_{n,N}^{h_{N}}(x) ≤ L_{n} exp \{ḡc^{n}(|x| − M) − (\frac{ḡ}{2M} − Ā)c^{0.2n} + q^{n}\}  for x ∈ Ω_{n}^{2}

(6.4)  f_{n,N}^{h_{N}}(x) ≤ L_{n} exp \{\frac{ḡc^{n}}{M} (|x|^{2} − M^{2})\}  if x > M + c^{−0.4n}

(6.4′)  f_{n,N}^{h_{N}}(x) ≤ L_{n} exp \{\frac{ḡc^{n}}{2M} (|x|^{2} − M^{2})\}  if 0 < x < M − c^{−0.4n}

with some appropriate norming constant L_{n} which satisfies the relation

(6.5)  C_{1} < c^{−n/2}L_{n} < C_{2}  with some 0 < C_{1} < C_{2} < ∞ .

Indeed, Proposition 1 and Lemma 1 imply (6.1)–(6.4′) with some norming constant L_{n} = L_{n}(N, h_{N}). (In the domain Ω_{n}^{3} we have divided the cases |x|^{2} − M^{2} > 0 and |x|^{2} − M^{2} < 0, since here we apply that ḡc^{n} < g_{n} < 2ḡc^{n}. ) It remains to prove (6.5) and to show that L_{n} can be chosen independently of N and h_{N}. For this aim we observe that

(6.6)  1 = μ_{n,N}^{h_{N}}((R^{2})^{2^{n}}) = ∫ f_{n,N}^{h_{N}}(x)p_{n}(x) dx = ∫_{Ω_{n}^{1}} + ∫_{Ω_{n}^{2}} + ∫_{Ω_{n}^{3}} .

By applying the change of variables r = M + c^{−n}t and by using the function f_{n}(t) defined in (4.11) we get that

∫_{Ω_{n}^{3}} f_{n,N}^{h_{N}}(x)p_{n}(x) dx ≤ ∫_{0}^{M−c^{−0.4n}} L_{n}(N, h_{N}) exp \{\frac{ḡc^{n}}{2M} (r^{2} − M^{2})\} rp̄_{n}(r) dr

+ ∫_{M+c^{−0.4n}}^{∞} L_{n}(N, h_{N}) exp \{\frac{ḡc^{n}}{M} (r^{2} − M^{2})\} rp̄_{n}(r) dr

= L_{n}(N, h_{N})[∫_{−c^{n}M}^{−c^{0.6n}} exp \{\frac{ḡ}{2M} (2Mt + \frac{t^{2}}{c^{n}} )\} (M + c^{−n}t) f_{n}(t) dt

+ ∫_{c^{0.6n}}^{∞} exp \{\frac{ḡ}{M} (2Mt + \frac{t^{2}}{c^{n}} )\} (M + c^{−n}t) f_{n}(t) dt] .

Relations (4.11′) and (4.11′′) imply that

(6.7)  ∫_{Ω_{n}^{3}} = L_{n}(N, h_{N})O (exp(−Kc^{0.6n}))  with some K > 0 .

Similarly,

(6.7′)  ∫_{Ω_{n}^{2}} ≤ L_{n}(N, h_{N}) exp \{−\frac{1}{2}(\frac{ḡ}{2M} − Ā)c^{0.2n}\} .

(Observe that \frac{ḡ}{2M} − Ā > 0.)

Define the number T_{n},

(6.8)  T_{n} = ∫_{Ω_{n}^{1}} exp {ḡc^{n}(x^{(1)} − M) + Āc^{n}x^{(2)2}} p_{n}(x) dx .

It follows from Theorem A that

(6.9)  C_{1}c^{−n/2} < T_{n} < C_{2}c^{−n/2}  with some 0 < C_{1} < C_{2} < ∞ .

Indeed, since the expression in the exponent of (6.8) can be written in the form

ḡc^{n}(x^{(1)} − M) + Āc^{n}x^{(2)2} = ḡc^{n}(|x| − M) − (\frac{ḡ}{2M} − Ā)c^{n}x^{(2)2} + O (c^{−0.2n}) ,

we get (6.9) by integrating (6.8) first by the variable x^{(2)}. Some calculation with the help of (6.2) and (6.2′) shows that

(6.10)  |∫_{Ω_{n}^{1}} f_{n,N}^{h_{N}}(x)p_{n}(x) dx − L_{n}(N, h_{N})T_{n}| ≤ L_{n}(N, h_{N})T_{n}q^{n} .

Relations (6.6), (6.7), (6.7′) and (6.10) imply that

1 = L_{n}(N, h_{N})T_{n} (1 + ε_{n} + O(exp(−c^{0.1n}))) , and ε_{n} ≤ q^{n} .

The last relation implies that relations (6.1)–(6.4′) remain valid if we choose L_{n} = T_{n}^{−1}, and this L_{n} satisfies (6.5) by (6.9).

We prove Theorem 1 with the help of (6.1)–(6.5). Fix some integer k ≥ 0, and define for all n ≥ k and measurable sets A ⊂ (R^{2})^{2^{k}} the cylindrical set A(n) = A × (R^{2})^{2^{n}−2^{k}} ⊂ (R^{2})^{2^{n}}. Put

μ̃_{n}(A) = L_{n} ∫_{Ã(n)} exp {ḡc^{n}2^{−n} ∑_{j=1}^{2^{n}}(x_{j}^{(1)} − M) + Āc^{n}4^{−n}(∑_{j=1}^{2^{n}} x_{j}^{(2)})^{2}}
μ_{n}( dx_{1} …, dx_{2^{n}})

with Ã(n) = A(n) ∩ {(x_{1}, …, x_{2^{n}}), 2^{−n} ∑_{j=1}^{2^{n}} x_{j} ∈ Ω_{n}^{1}}. We claim that if n > n_{0} and N > N_{0}(n, q) then

(6.11)  |μ̃_{n}(A(n)) − μ_{n,N}^{h_{N}}(A(n))| ≤ Kq^{n}

with some K > 0 independent of the set A. Indeed,

|μ̃_{n}(A(n)) − μ_{n,N}^{h_{N}}(A(n))| ≤ ∫_{Ω_{n}^{2}∪Ω_{n}^{3}} f_{n,N}^{h_{N}}(x)p_{n}(x) dx

+ ∫_{Ã(n)} μ_{n}( dx_{1} …, dx_{2^{n}})|f_{n,N}^{h_{N}}(2^{−n} ∑_{j=1}^{2^{n}} x_{j})

− L_{n} exp {ḡc^{n}2^{−n} ∑_{j=1}^{2^{n}}(x_{j}^{(1)} − M) + Āc^{n}4^{−n}(∑_{j=1}^{2^{n}} x_{j}^{(2)})^{2}}| = I_{1} + I_{2} .

It follows from (6.5), (6.7) and (6.7′) that

(6.12)  I_{1} ≤ exp (−c^{0.1n}) .

On the other hand we increase the term I_{2} by enlarging the domain of integration to the set {(x_{1}, …, x_{2^{n}}), 2^{−n} ∑_{j=1}^{2^{n}} x_{j} ∈ Ω_{n}^{1}}. Hence

I_{2} ≤ ∫_{Ω_{n}^{1}} |f_{n,N}^{h_{N}}(x) − L_{n} exp {ḡc^{n}(x^{(1)} − M) + Āc^{n}x^{(2)2}}| p_{n}(x) dx .

The last inequality together with (6.2) and (6.2′) imply that

I_{2} ≤ 2q^{n}μ_{n,N}^{h_{N}} ((R^{2})^{2^{n}}) = 2q^{n} .

The last inequality together with (6.12) imply (6.11). Since for all k ≥ 0 and measurable sets A ∈ (R^{2})^{2^{k}}, k ≤ n ≤ N we have μ_{n,N}^{h_{N}}(A(n)) = μ_{k,N}^{h_{N}}(A), relation (6.11) implies that for all ε > 0 there is some N_{0}(ε) such that for N > N_{0}(ε) and N′ > N_{0}(ε) the relation |μ_{n,N}^{h_{N}}(A)−μ_{k,N′}^{h_{N′}}(A)| < ε holds true. Let us emphasize that the threshold N(ε) does not depend on the set A. Hence the last relation means that the limit μ̄_{k}(A) = lim_{N→∞} μ_{k,N}^{h_{N}}(A) exists, and the convergence is uniform in A. This implies that μ_{k,N}^{h_{N}} → μ̄_{k} in variational metric. To complete the proof of Theorem 1 we have to show that the measure μ̄_{k} does not depend on the sequence h_{N}. But it is not difficult to see with the help of (6.11) that this statement holds, since μ̄_{k}(A) = lim_{n→∞} μ̃(A(n)), and the right hand side of the last expression does not depend on h_{N}.

Proof of Proposition 2. Let n > n_{0} and N > N_{0}(n, q). Relations (6.1)–(6.5) hold for such pairs n and N. By Theorem 1 the measures μ_{n,N}^{h_{N}} converge in variational metric to the projection μ̄_{n} of the measure μ̄ to (R^{2})^{2^{n}} as N → ∞. Since all measures are absolute continuous with respect to the measure μ_{n} the above convergence is

equivalent to the convergence of the Radon–Nikodym derivatives \frac{dμ_{n,N}^{h_{N}}}{dμ_{n}} = f_{n,N}^{h_{N}} to \frac{dμ̄_{n}}{dμ_{n}} = f̄ in L_{1} norm in the space ((R^{2})^{2^{n}}, μ_{n}) as N → ∞. Since all functions f_{n,N}^{h_{N}}(x) satisfy (6.1)–(6.5) for N > N_{0}(n, q), their limit f̄ also has this property. Hence Proposition 2 holds true.

7. The proof of Theorem 2. Existence of the large-scale limit. First we need some results about the transformation Q_{n} = Q_{n}(k) of probability measures on (R^{2})^{2^{n+k}} to probability measures on (R^{2})^{2^{k}} to be defined below. First we define a transformation Q_{n} = Q_{n}(k), Q_{n}: (R^{2})^{2^{n+k}} → (R^{2})^{2^{k}} in the following way: For all (x_{1}, …, x_{2^{n+k}}), x_{j} ∈ R^{2}, j = 1, …, 2^{n+k}

Q_{n}(x_{1}, …, x_{2^{n+k}}) = (y_{1}, …, y_{2^{k}}),  y_{j} = 2^{−n} ∑_{l=(j−1)2^{n}+1}^{j2^{n}}x_{l},  j = 1, …, 2^{k}.

This transformation induces the transformation Q_{n} of probability measures on (R^{2})^{2^{n+k}} to probability measures on (R^{2})^{2^{k}} in a natural way. Namely, if ν is a probability measure on (R^{2})^{2^{n+k}} and (η(1), …, η(2^{n+k})) is a ν distributed vector then Q_{n}ν is the distribution of the random vector Q_{n}(η(1), …, η(2^{n+k})). In Theorem 2 we have to study an appropriately rescaled version of the measure Q_{n}μ̄_{n+k}. It is not difficult to see that relation (2.10) implies that

(7.1)  \frac{dQ_{n}μ̄_{n+k}}{dQ_{n}μ_{n+k}} (x_{1}, …, x_{2^{k}}) = f̄_{n+k}(2^{−k} ∑_{j=1}^{2^{k}} x_{j}).

We formulate below Theorem C which follows from the relatively simple Theorem 1 in [4]. For the sake of completeness we present its proof in Appendix B.

Theorem C. The above defined measure Q_{n}μ_{n+k} = Q_{n}μ_{n+k}(T, t) has a density function h_{k}(x_{1}, …, x_{2^{k}}) of the form

h_{k}(x_{1}, …, x_{2^{k}}) = L(T, t, n, k) exp \{−\frac{1}{T} H_{k} (c^{n/2}x_{1}, …, c^{n/2}x_{2^{k}})\} ∏_{j=1}^{2^{k}} p_{n}(x_{j})
x_{j} ∈ R^{2}, j = 1, …, 2^{k}.

Here H_{k} is the Hamiltonian function defined in (1.2′) of Part I, p_{n}(x) is the function appearing in Theorem A, and L is an appropriate norming constant.

Formula (7.1) and Theorem C enable us to express the density function of the random vector {R_{n}σ^{(1)}(j), R_{n}σ^{(2)}(j), 1 ≤ j ≤ 2^{k}} with the help of the functions p_{n}(x) and f_{n}(x), where the sequence {σ(j), j ∈ Z} is μ̄ distributed, and R_{n}σ^{(1)}, R_{n}σ^{(2)} are defined in (1.2) and (1.3) of Part II. This density equals to

(7.2)

h_{n,k}(x_{1}, …, x_{2^{k}})

= L_{n,k}f̄_{n+k}(2^{−k} ∑_{j=1}^{2^{k}} x̃_{j}) exp \{−\frac{1}{T} H_{k}(c^{n/2}x̃_{1}, …, c^{n/2}x̃_{2^{k}})\} ∏_{j=1}^{2^{k}} p_{n}(x̃_{j})

with

(7.2′)  x̃ = x̃(x) = (M + c^{−n}x^{(1)}, c^{−n/2}x^{(2)})  for x = (x^{(1)}, x^{(2)}) .

Let us define the sets W_{n} ⊂ R^{2} and W̄_{n} ⊂ R^{2} by the formulas

W̄_{n} = \{(x^{(1)}, x^{(2)}), M − \frac{nη}{c^{n}} < |x| < M + \frac{ηn^{1/α}}{c^{n}} , |x^{(2)}| < c^{−0.45n}, x^{(1)} > 0\}

W_{n} = {(x^{(1)}, x^{(2)}), x̃(x) ∈ W̄ } ,

where η and α are the same constants as in Theorem A, and x̃(x) is defined in (7.2′). We shall show that there is some n_{0} > 0 and 0 < q < 1 such that

(7.3)  P ( (R_{n}σ^{(1)}(j), R_{n}σ^{(2)}(j) ) ∉ W_{n}) ≤ q^{n}  if n ≥ n_{0}

for a μ̄ distributed random field σ(j), j ∈ Z, and give a good asymptotic formula for the expression h_{n,k}(x_{1}, …, x_{2^{k}}) defined in (7.2) if x_{j} ∈ W_{n} for all 1 ≤ j ≤ 2^{k}. First we prove (7.3).

P ((R_{n}σ^{(1)}(j), R_{n}σ^{(2)}(j) ) ∉ W_{n} ) = P (2^{−n} ∑_{l=1}^{2^{k}} σ(k) ∉ W̄_{n})

= ∫_{R^{2}−W̄_{n}} f̄_{n}(x)p_{n}(x) dx = ∫_{Ω_{n}^{1}−W̄_{n}} + ∫_{Ω_{n}^{2}} + ∫_{Ω_{n}^{3}} = I_{1} + I_{2} + I_{3} .

We get similarly to the estimates (6.7) and (6.7′) that

I_{3} ≤ exp (−Cc^{0.6n}) ,

I_{2} ≤ const. exp \{−(\frac{ḡ}{2M} − Ā)c^{0.2n}\} ≤ exp (−Cc^{0.2n})

The term I_{1} has to be estimated a little more carefully.

Define

Ω̃_{n}^{1} = {(x^{(1)}, x^{(2)}), ||x| − M| < c^{−0.4n}, |x^{(2)}| < c^{−0.45n}, x^{(1)} > 0} ,

and write

I_{1} = I_{1,1} + I_{1,2} = ∫_{Ω_{n}^{1}−Ω̃_{n}^{1}} + ∫_{Ω̃_{n}^{1}−W̄_{n}} .

We get, similarly to the estimation of I_{3} and I_{2} that I_{1,1} ≤ exp(−Kc^{0.1n}), and we can write by (2.11) and (2.11′) that

I_{1,2} ≤ 2L_{n} ∫_{Ω̃_{n}^{1}−W̄_{n}} exp {ḡc^{n}(x^{(1)} − M) + Āc^{n}x^{(2)2}} p_{n}(x) dx

≤ 3L_{n} ∫_{Ω̃_{n}^{1}−W̄_{n}} exp \{ḡc^{n}(|x| − M) − (\frac{ḡ_{n}}{2M} − A)c^{n}x^{(2)2}\} p_{n}(x) dx .

Then integrating first by x^{(2)} we get that

I_{1,2} ≤ KL_{n}c^{−n/2}[∫_{M−c^{−0.4n}}^{M−ηnc^{−n}} + ∫_{M+ηn^{1/α}c^{−n}}^{M+c^{−0.4n}}] exp (ḡc^{n}(r − M)) p̄_{n}(r) dr ≤ q^{n} with the help of relations (1.8) and (1.10) in Theorem A. Let us emphasize that it was the multiplying term q^{n} in (1.8) that enabled us to give an exponentially small bound for the second term in the last integral. The above estimates imply (7.3).

To estimate the expression (7.2) in the case x_{j} ∈ W_{n}, j = 1, 2, …, 2^{k}, we make some preparatory remarks. Put

ℓ_{n}(x) = c^{n}(|x̃(x)| − M) = c^{n} {[(M + c^{−n}x^{(1)})^{2} + c^{−n}x^{(2)2}]^{1/2} − M}

We have

(7.4) ℓ_{n}(x) = x^{(1)} + \frac{x^{(2)2}}{2M} + O (c^{−n}(x^{(2)4} + |x^{(1)}|x^{(2)2}))

= x^{(1)} + \frac{x^{(2)2}}{2M} + O(c^{−0.8n}) if x ∈ W_{n} ,

because, as it is not difficult to see, c^{−n/2}|x^{(2)}| < c^{−0.45n}, and M − 2c^{−0.9n} < M + c^{−n}x^{(1)} < M + 2c^{−0.9n} if x ∈ W_{n}. We show with the help of Theorem A and (7.4) that (7.5)

p_{n}(x̃) = exp \{−\frac{a_{0}}{T} (Mx^{(1)} + \frac{x^{(2)2}}{2} )\} g (\frac{a_{1}}{T} (Mx^{(1)} + \frac{x^{(2)2}}{2} )) (1 + O(q^{n}))

with some 0 < q < 1 for x ∈ W_{n} if η > 0 is chosen sufficiently small in Theorem A. Indeed, by Theorem A

p_{n}(x̃) = exp \{−\frac{a_{0}M}{T} ℓ_{n}(x)\} g (\frac{a_{1}M}{T} ℓ_{n}(x)) (1 + O(q^{n})) ,

exp \{−\frac{a_{0}M}{T} ℓ_{n}(x)\} = exp \{−\frac{a_{0}}{T} (Mx^{(1)} + \frac{x^{(2)2}}{2} ) + O (c^{−0.8n})\} ,

and

(7.6)  |g(\frac{a_{1}}{T} Mℓ_{n}(x)) − g(\frac{a_{1}}{T} (Mx^{(1)} + \frac{x^{(2)2}}{2} ))| = O(c^{−0.8n})

by (7.4) and the boundedness of the function \frac{d}{dx}g(x). (See Lemma 13 in Part I.) On the other hand, by Lemma 17 of Part I the relation −2ηn < x^{(1)} + \frac{x^{(2)2}}{2M} < 2ηn^{1/α} if x ∈ W_{n}, which holds because of the definition of W_{n}, and the inequality

|c^{n}(|x̃(x)| − M) − (x^{(1)} + \frac{x^{(2)2}}{2M} )| ≤ Kc^{−0.8n}

we have

g (\frac{a_{1}}{T} (Mx^{(1)} + \frac{x^{(2)2}}{2} )) > c^{−0.3n}

if x ∈ W_{n}, and η is chosen in Theorem A sufficiently small. Hence (7.6) can be rewritten as

g (\frac{a_{1}}{T} Mℓ_{n}(x)) = g(\frac{a_{1}}{T} (Mx^{(1)} + \frac{x^{(2)2}}{2} )) (1 + O(c^{−n/2})) .

The above relations imply (7.5).

We also claim that

(7.7) H_{k} (c^{n/2}x̃_{1}, …, c^{n/2}x̃_{2^{k}})

= − ∑_{i=1}^{2^{k}−1} ∑_{j=i+1}^{2^{k}} U(i, j) [M(x_{i}^{(1)} + x_{j}^{(1)}) + x_{i}^{(2)}x_{j}^{(2)} + c^{n}M^{2}] + O(q^{n})

and

(7.8) f̄_{n+k}(2^{−k} ∑_{j=1}^{2^{k}} x̃_{j}) = L_{n+k} exp \{\frac{c^{k}}{2^{k}} ḡ ∑_{j=1}^{2^{k}} x_{j}^{(1)} + \frac{c^{k}}{4^{k}} Ā(∑_{j=1}^{2^{k}} x_{j}^{(2)})^{2} + O(q^{n})\}

if x_{j} ∈ W_{n}, j = 1, …, 2^{k}.

Indeed,

H_{k} (c^{n/2}x̃_{1}, …, c^{n/2}x̃_{2^{k}})

= − ∑_{i=1}^{2^{k}−1} ∑_{j=i+1}^{2^{k}} U(i, j) [c^{n}(M + c^{−n}x_{i}^{(1)})(M + c^{−n}x_{j}^{(1)}) + x_{i}^{(2)}x_{j}^{(2)}] ,

hence to prove (7.7) it is enough to remark that in the last expression the terms c^{−n}x_{i}^{(1)}x_{j}^{(1)} are negligibly small, since c^{−n}x_{i}^{(1)}x_{j}^{(1)} = O(c^{−0.8n}) if x_{i} ∈ W_{n} and x_{j} ∈ W_{n}.

To prove (7.8) we have to show that 2^{−k} ∑_{j=1}^{2^{k}} x̃_{j} ∈ Ω_{n+k}^{1} if x̃_{j} ∈ W̄_{n} for all j = 1, …, 2^{k} and then apply Proposition 2. We can write with the notation x̃_{j} = (x̃_{j}^{(1)}, x̃_{j}^{(2)}) that |2^{−k} ∑_{j=1}^{2^{k}} x̃_{j}^{(2)}| < c^{−0.45n} ≤ c^{−0.4n}4^{−k}, and |2^{−k} ∑_{j=1}^{2^{k}} x̃_{j}^{(1)} − M| ≤ 2c^{−0.45n} ≤ 4^{−k}c^{−0.4n} if n is sufficiently large (k is fixed, n → ∞) and x̃_{j} ∈ W_{n} for j = 1, …, 2^{k}. These relations imply that 2^{−k} ∑_{j=1}^{2^{k}} x̃_{j} ∈ Ω_{n+k}^{1}. We get, by putting (7.5), (7.7) and (7.8) into (7.2) that

h_{n,k}(x_{1}, …, x_{2^{k}}) = L̄_{n,k} exp\{\frac{c^{k}}{2^{k}} ḡ ∑_{j=1}^{2^{k}} x_{1}^{(j)} + \frac{c^{k}}{4^{k}} Ā(∑_{j=1}^{2^{k}} x_{j}^{(2)})^{2}

− \frac{1}{T} ∑_{i=1}^{2^{k}−1} ∑_{j=i+1}^{2^{k}} U(i, j) (Mx_{i}^{(1)} + Mx_{j}^{(1)} + x_{i}^{(2)}x_{j}^{(2)}) − ∑_{j=1}^{2^{k}} \frac{a_{0}}{T} (Mx_{j}^{(1)} + \frac{x_{j}^{(2)2}}{2} )\}

∏_{j=1}^{2^{k}} g(\frac{a_{1}}{T} (Mx_{j}^{(1)} + \frac{x_{j}^{(2)2}}{2} )) (1 + O(q^{n}))

with some 0 < q < 1 if x_{j} ∈ W_{n}, j = 1, 2, …, 2^{k}.

Simple calculation shows that − ∑_{i=1}^{2^{k}} U(i, j) = \frac{1−(c/2)^{k}}{1−c/2} , hence the coefficient of

x_{j}^{(1)}, (c/2)^{k}ḡ + \frac{M}{T} ∑_{j=1}^{2^{k}} U(i, j) − \frac{a_{0}M}{T} equals zero, and

(7.9)  h_{n,k}(x_{1}, …, x_{2^{k}}) = h_{k}(x_{1}, …, x_{2^{k}})(1 + O(q^{n}))

if x_{j} ∈ W_{n}, j = 1, 2, …, 2^{k}, where h_{k} is defined in (1.12) (with p = 2). It is not difficult to see that (7.3) also holds with a random vector with density function (1.12). Hence (7.3) and (7.9) imply Theorem 2.

8. Some open problems and conjectures. Dyson [12] has defined a more general class of models than that considered in this work. He defined, with the help of a real function φ: Z → R^{1}, models with the Hamiltonian function

(8.1)  H(σ) = − ∑_{i∈Z} ∑_{\begin{matrix}j∈Z \\ j>i \end{matrix}} φ(d(i, j))S_{n}^{(}f(x)i)σ(j),  σ = \{σ(i), i ∈ Z\} ,

where d(·, ·) denotes the hierarchical distance on Z given in formula (1.1) of Part I. In this work we have considered models in the special case φ(x) = |x|^{−a} with a = 2 − \frac{log c}{log 2}. One question we are going to discuss here is that which are the functions φ for which Dyson’s model with the Hamiltonian (8.1) has a phase transition at low temperatures. In the boundary case some more delicate phenomena appear which we also want to discuss. The behaviour of vector and scalar-valued models is different. First we discuss the vector-valued case.

The quantities M_{n} = M_{n}(T) considered in Part I can be defined in a natural way in the general case. The arguments of Part I suggest that the relation

(8.2)  M_{n+1} = M_{n} − \frac{1}{4^{n}M_{n}φ(2^{n})}

holds true. The existence or non-existence of phase transition depends on whether M = lim_{n→∞} M_{n} equals to zero or not if T is small, i.e. if M_{0} is large. Hence formula (8.2) suggests that a phase transition at low temperatures occurs if and only if ∑ \frac{1}{4^{n}φ(2^{n})} is convergent. Dyson has formulated the same conjecture in [13] and proved its convergent part in the special case when σ(i) ∈ R^{3}. He has also solved the problem for scalar-valued models. He proved that there is a phase transition at low temperatures if φ(n) ≥ C \frac{log log n}{n^{2}} with some C > 0, and there is none if φ(n)\frac{log log n}{n^{2}} → 0. Moreover, in the boundary case φ(n) = C \frac{log log n}{n^{2}} the following Thouless effect occurs: There is some critical parameter T_{cr.} such that M(T) = lim_{n→∞} M_{n}(T) > 0 for T ≤ T_{cr.} and M(T) = 0 for T > T_{cr.}. The quantity M(T) has a physical content, it is called the spontaneous magnetization. The interesting feature of the above result is that it states that the function M(T) has a discontinuity at T = T_{cr.}. This particular behaviour of the spontaneous magnetization appears only in the boundary case φ(n) = C \frac{log log n}{n^{2}}. On the other hand, the Thouless effect occurs in some other models too, like in the one-dimensional Ising model with \frac{1}{|x−y|^{2}} interaction, in one-dimensional percolation models if the probability of the event that the points i and j are connected has the order C(T)|i−j|^{−2}, e.t.c.. In recent time several interesting papers appeared on this subject, (see e.g. [1], [2], [17]). On the other hand, there are some other interesting phenomena connected with the Thouless effect, like the irregular behaviour of the correlation function, whose investigation requires essentially new ideas.

The appearance of phase transitions and the Thouless effect in scalar-valued models are connected with the behaviour of the sequence M_{n}. The quantity M_{n+1} can be expressed asymptotically in a simple way with the help of M_{n} and the function φ in scalar-valued models too. But this formula is essentially different from his vector-valued counterpart, namely

(8.3)  M_{n+1} ∼ M_{n} [1 − exp\{−\frac{1}{T} M_{n}^{2}4^{n}φ(2^{n})\}] .

In the particular case φ(n) = \frac{log log n}{n^{2}} we have

(8.3′)  _{n+1} ∼ M_{n} [1 − exp\{−\frac{1}{T} M_{n}^{2} log n\}] .

Formula (8.3′) may help us to understand the cause of the Thouless effect, at least at a heuristic level. If M_{n}(T) < \sqrt{T} for some n then relation (8.3) implies that M(T) = lim_{n→∞} M_{n}(T) = 0, hence either M(T) ≥ \sqrt{T} or M(T) = 0. Since M(T) ≠ 0 for small T, this relation implies the discontinuity of the function M(T). In vector-valued models relation (8.2) does not suggest such a behaviour. We expect however that some delicate effects appear in this case too, and we are going to study them in the future.

Let us remark that the study of existence or non-existence of phase transitions at low temperatures seems to be an essentially simpler problem than the study of the Thouless effect and related questions. In the first problem it is enough to consider sufficiently low temperatures, and in the case of vector-valued models with Hamiltonian function of the form (8.1) for instance the method of the present paper works without any essential changes. In the second problem however, one has to study the behaviour of the model near the critical temperature, and this requires more work and new ideas.

Another problem we are going to discuss here is the description of the large-scale limit of vector-valued equilibrium states with translation invariant Hamiltonian function. We have discussed its scalar-valued counterpart in Section 8 of our paper [6], and formulated our conjectures about it.

Let us consider vector-valued models on the d-dimensional integer lattice with Hamiltonian function

H(σ) = − ∑_{\begin{matrix}|i−j|=1 \\ i,j∈Z^{d} \end{matrix}} σ(i)σ(j) − ∑_{i∈Z^{d}} p(σ(i))

with

p(x) = −\frac{t}{4}|x|^{4} − \frac{|x|^{2}}{2} , t > 0 ,

and the Lebesgue measure on R^{p} with some p ≥ 2 as the free measure of the model. The expression σ(i)σ(j) in the above formulas denotes scalar product.

Let {X(k) = (X_{k}^{(1)}, …, X_{k}^{(p)}), k ∈ Z^{d}} be a random field with the distribution of a (pure) equlibrium state with the above Hamiltonian function at a certain temperature T. If d ≥ 3 then there exists a spontaneous magnetization at sufficiently low temperatures. This is proved with the help of the infrared bounds (see e.g. [14]). In case of phase transition we consider that pure state for which the direction of the spontaneous magnetization is e_{1} = (1, 0, …, 0), i.e. E X_{k}^{(1)} = M > 0, and E X_{k}^{(s)} = 0 for s = 2, …, p.

Define the “renormalized” random fields {Y_{k}(N) = ( Y_{k}(N)^{1}, …, Y_{k}(N)^{p})}, N = 1, 2, … by the formulas

(8.4)  Y_{k}(N)^{1} = A(N)^{−1} ∑_{j∈D_{k}(N)}(X_{j}^{(1)} − EX_{j}^{(1)})

(8.4′)  Y_{k}(N)^{s} = B(N)^{−1} ∑_{j∈D_{k}(N)} X_{j}^{(s)},  s = 2, …, p,

with

D_{k}(N) = {j = (j^{(1)}, …, j^{(d)}) ∈ Z^{d}, k^{(r)}N + 1 ≤ j^{(r)} ≤ (k^{(r)} + 1)N,
r = 1 …, d} .

We are interested in the question that for which choice of A(N) and B(N) the fields Y_{k}(N) have a non-trivial limit, i.e. a limit which is not concentrated on a single configuration. We also want to describe the distribution of the limit field.

Dyson’s hierarchical model with parameter c, 1 < c < 2 can be considered as an approximation of translation invariant models with nearest neighbour interaction on the d-dimensional lattice Z^{d} with d = \frac{2}{1−log_{2} c}. (See paper [21] for a discussion of this approximation.) It must be admitted that the above approximation is made only at a heuristic level, but it helps us to get a better understanding about the behaviour of the large-scale limit. On the basis of the present work and [5] we can formulate the following conjectures about the large-scale limit of translation invariant models at low temperatures.

The behaviour of the large-scale limit is different in the cases d > 4, d = 4 and d < 4, and they correspond to the cases c > \sqrt{2}, c = \sqrt{2} and c < \sqrt{2} in Dyson’s hierarchical model. In accordance with [5] we expect that for d > 4 the large-scale limit exists at low temperatures with A(N) = N^{d/2} and B(N) = N^{(d+2)/2}. The limit field {Y_{k} = (Y_{k}^{(1)}, …, Y_{k}^{(p)}), k ∈ Z^{d}} is such that the fields {Y_{k}^{(j)}, k ∈ Z^{d}}, j = 1, 2, …, p are independent, {Y_{k}^{(1)}, k ∈ Z^{d}} consists of independent identically distributed Gaussian random variables with zero mean, {Y_{k}^{(s)}, k ∈ Z^{d}}, s = 2, …, p, are massless free Gaussian fields, i.e. they have the same distribution as the field

Y_{k} = C ∫ \frac{exp(ikx)}{|x|} ∏_{j=1}^{d}\{\frac{exp(ix^{(j)}) − 1}{ix^{(j)}}\}W ( dx), k ∈ Z^{d},

with some C > 0, where W ( dx) is a complex-valued white noise field on R^{d} with the conjugation property W (x) = \overline{W (−x)}. (For the definition of W (x) see e.g. [16].) The situation is similar in the case d = 4, i.e. the investigation of Dyson’s model suggests a similar behaviour with the only difference that a logarithmic factor appears in the normalizing term A(n). More precisely, for d = 4 the fields Y_{k}^{(N)} defined in (8.4) and (8.4′ ) with A(N) = N^{2}\sqrt{log N} and B(N) = N^{(d+2)/2} = N^{3} have a Gaussian limit as N → ∞ which consists of independent components s = 1, …, p, similarly to the case d > 4. The limit of the fields Y_{k}^{(s)}, s = 2, …, p, is a massless free field.

The result of the present work motivates the following conjecture for d = 3.

Conjecture. For d = 3 the large-scale limit exists at low temperatures with the normalizations A(N) = N^{2} and B(N) = N^{5/2}. The large-scale limit has the same distribution as the random field {Y_{k} = (Y_{k}^{(1)}, …, Y_{k}^{(p)}), k ∈ Z^{3}} defined by the formulas

(8.5) Y_{k}^{(s)} = C ∫ \frac{exp(ikx)}{|x|} ∏_{j=1}^{3} \frac{exp(ix^{(j)}) − 1}{ix^{(j)}}W_{s}( dx), k ∈ Z^{3} , s = 2, …, p

and

(8.6)

Y_{1}^{(s)} = −\frac{C^{2}}{2M} ∑_{s=2}^{p} ∫∫ \frac{exp(ik(x + y))}{|x||y|} ∏_{j=1}^{3}\{\frac{exp[i(x^{(j)} + y^{(j)})] − 1}{i(x^{(j)} + y^{(j)})}\}W_{s}( dx)W_{s}( dy),
k ∈ Z^{3} ,

where C > 0 is an appropriate positive constant, and W_{s}( dx), s = 1, …, p are independent complex valued white noise fields on R^{3} with the conjugation property W (x) = \overline{W (−x)}. (For the definition of two-fold stochastic integrals with respect to a Gaussian field see e.g. [16]. Such an integral appears in the definition of Y_{1}^{(s)}.)

The fields Y_{s}^{(k)}, s = 2, …, p defined in (8.5) are massless free fields, the field defined in (8.6) belongs to the class of self-similar fields constructed in Dobrushin’s paper [11]. It is a quadratic functional of a Gaussian field, just as the corresponding field in Dyson’s model for 1 < c < \sqrt{2}.

The large-scale limit of the equilibrium state in Dyson’s model described in Theorem 2 of Part II has the following independence property: The random variables Y_{k}^{(1)} + \frac{1}{2M} ∑_{s=2}^{p} Y_{k}^{(s)2} are independent for different k. This independence property does not hold for their translation invariant counterpart defined in (8.5) and (8.6). It cannot be preserved, because translation invariant models have less symmetry. Nevertheless, the following non-rigorous argument shows some analogy between the behaviour of the fields defined in (8.5) and (8.6) and the above mentioned independence property. In this non-rigorous argument we consider the limit field appearing in the Conjecture as the discretization of a generalized field.

Let δ(t) denote the Dirac-delta function in the point t, and consider the generalized field which at δ(t) takes the value Y (δ(t)) = ( Y^{(1)}(δ(t)), …, Y^{(s)}(δ(t)) ),

Y^{(s)}(δ(t)) = C ∫ \frac{exp(itx)}{|x|}W^{(s)}( dx) , s = 2, …, p ,

Y^{(1)}(δ(t)) = −\frac{C^{2}}{2M} ∑_{s=2}^{p} ∫∫ \frac{exp(it(x + y))}{|x||y|}W^{(s)}( dx) W^{(s)}( dy) .

Actually this definition is not correct, since the above stochactic integrals are meaningless because of the divergence of the integrals ∫_{R^{3}} \frac{dx}{|x|^{2}} and ∫_{R^{3}} ∫_{R^{3}} \frac{dx dy}{|x|^{2}|y|^{2}} . But the integral

Y^{(1)}(φ) = ∫ Y^{(1)}(δ(t))φ(t) dt = −\frac{C^{2}}{2M} ∑_{s=2}^{p} ∫∫ \frac{φ̃(x + y)}{|x||y|}W^{(s)}( dx) W^{(s)}( dy) .

and

Y^{(s)}(φ) = ∫ Y^{(s)}(δ(t))φ(t) dt = C ∫ \frac{φ̃(x)}{|x|}W^{(s)}( dx), 2 ≤ s ≤ p ,

are meaningful for nice funtions φ. In particular, they are meaningful for the

indicator functions of the unit cubes ∏_{i=1}^{3}[k_{i}, k_{i} + 1) which we denote by φ_{k} if k =

(k_{1}, k_{2}, k_{3}). The random field appearing in the Conjecture can be considered as the discretization of the above defined generalized field if we identify Y_{k} with Y (φ(k)).

A formal application of Itô’s formula (see e.g. [16]) would supply the relation Y^{(1)}(δ(t)) − \frac{1}{2M} ∑_{s=2}^{p} Y^{(s)}(δ(t))^{2} = const., and this can be considered as the analogue of the independence property of the large-scale limit of Dyson’s model for the above defined generalized field Y (δ(·)). On the other hand by our Conjecture the discretization of this generalized field is the large-scale limit of the three-dimensional translation invariant vector-valued model at low temperatures.

Let us finally discuss the cases d = 1 and d = 2. The case d = 1 is rather simple. In this case there is no phase transition, and if {X_{k}, k ∈ Z^{1}} is a random field with the distribution function of an equilibrium state at any temperature then it satisfies the central limit theorem with the usual normalization. The case d = 2 is more delicate. In this case the dimension p, (σ(i) ∈ R^{p}), also plays an important role. In this case there is no symmetry breaking, but for d = 2, p = 2 a more delicate phenomenon, the so-called Kosterlitz–Thouless effect occurs. (See [15]). This means that at low temperatures the correlation function decreases rather slowly, only power-like. Hence a non-trivial large-scale limit should appear in this case. For d = 2, p ≥ 3 it is expected that the Kosterlitz–Thouless effect does not occur, but for the time being it is proved only at a physical level (see[18]). Hence, it is expected that for d = 2, p > 2 the large-scale limit has the same (trivial) behaviour as for d = 1.

When typing the final version of this work the authors learned about some recent results about the Kosterlitz–Thouless effect (see [22], [23]). The arguments of these works, also supported by computer simulation, suggest that the situation in twodimensional translation invariant models is essentially different from what we had expected. In particular, the difference between the cases p = 2 and p > 2 in the models we have discussed at the end of this Section does nevertheless not occur.

Appendix

Appendix A. The proof of the basic recursive relations (2.1) and (2.1′) in Part I. Formula (2.1′) immediately follows from (1.4) in Part I with n = 0. To prove (2.1) let us first observe that the recursive relation

(A1) H_{n+1} (x_{1}, …, x_{2^{n+1}}) = H_{n} (x_{1}, …, x_{2^{n}}) + H_{n} (x_{2^{n}+1}, …, x_{2^{n+1}})

− c^{n}(2^{−n} ∑_{i=1}^{2^{n}} x_{i})(2^{−n} ∑_{j=2^{n}+1}^{2^{n+1}} x_{j})

holds for n ≥ 0, where

H_{n}(x_{1}, …, x_{2^{n}}) = − ∑_{i=1}^{2^{n}} ∑_{j=i+1}^{2^{n}} U(i, j)x_{i}x_{j} ,

and U(i, j) is defined by (1.1) and (1.2′) in Part I. By relation (1.4) in Part I

p_{n+1}(x, T) = \frac{1}{Z_{n+1}(T, t)} ∫ exp\{−\frac{1}{T} H_{n}(x_{1}, …, x_{2^{n+1}})\} (A2)

δ(2^{−(n+1)} ∑_{i=1}^{2^{n+1}} x_{i} − x) ∏_{i=1}^{2^{n+1}} p(x_{i}) dx_{i} ,

where Z_{n+1}(T, t) is an appropriate norming constant, and δ(2^{−(n+1)} ∑_{i=1}^{2^{n+1}} x_{i} − x)

means that integration in (A2) is taken on the hyperplane 2^{−(n+1)} ∑_{i=1}^{2^{n+1}} x_{i} = x with respect to the Lebesgue measure. Let us fix some number u, and calculate the integral on the right-hand side of (A2) by integrating first on the hyperplane defined by the relations 2^{−n} ∑_{i=1}^{2^{n}} x_{i} = x + u and 2^{−n} ∑_{i=2^{n}+1}^{2^{n+1}} x_{i} = x − u and then by integrating by u. We get with the help of relations (A1) and (A2) that

p_{n}(x, T) = \frac{1}{Z_{n+1}(T, t)} ∫ exp \{\frac{c^{n}}{T} (x + u)(x − u)\}

[∫ exp \{−\frac{1}{T} H_{n}(x_{1}, …, x_{2^{n}})\} δ (2^{−n} ∑_{i=1}^{2^{n}} x_{i} − (x + u)) ∏_{i=1}^{2^{n}} p(x_{i}) dx_{i}]

[∫ exp \{−\frac{1}{T} H_{n}(x_{2^{n}+1}, …, x_{2^{n+1}})\} δ(2^{−n} ∑_{i=2^{n}+1}^{2^{n+1}} x_{i} − (x − u))

∏_{i=2^{n}+1}^{2^{n+1}}p(x_{i}) dx_{i}] du

= C_{n} ∫ exp \{\frac{c^{n}}{T} (x^{2} − u^{2})\} p_{n}(x + u)p_{n}(x − u) du ,

as we have claimed.

Appendix B. The proof of Theorem C. Since the measure μ_{n+k} has the density function

\frac{1}{Z}_{n+k} exp \{−\frac{1}{T} H_{n+k}(z_{1}, …, z_{2^{n+k}})\} ∏_{i=1}^{2^{n+k}} p(z_{i}) ,

the density function of the measure Q_{n}μ_{n+k}, the function h_{k}(x_{1}, …, x_{2^{k}}) equals to

h_{k}(x_{1}, …, x_{2^{k}}) = \frac{1}{Z}_{n+k} ∫ exp \{−\frac{1}{T} H_{n+k}(z_{1}, …, z_{2^{n+k}})\} (B1)

∏_{l=1}^{2^{k}} δ(2^{−n} ∑_{j=(l−1)2^{n}+1}^{l2^{n}}z_{j} − x_{l}) ∏_{i=1}^{2^{n+k}} p(z_{i}) dz_{i} ,

where ∏_{l=1}^{2^{k}} δ(2^{−n} ∑_{j=(l−1)2^{n}+1}^{l2^{n}} z_{j} − x_{l}) in the integral (B1) means that integra-

tion is taken on the hyperplane defined by the relations 2^{−n} ∑_{j=(l−1)2^{n}+1}^{l2^{n}} z_{j} = x_{l}, l = 1, …, 2^{k}, with respect to the Lebesgue measure. The special structure of the hierarchical distance implies that

H_{n+k}(z_{1}, …, z_{2^{n+k}}) = ∑_{l=1}^{2^{k}} H_{n}(z_{(l−1)2^{n}+1}, …, z_{l2^{n}}) − ∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{k}} c^{n}U(i, j)z̄_{i}z̄_{j} ,

with z̄_{i} = 2^{−n} ∑_{p=(i−1)2^{n}+1}^{i2^{n}} z_{p}, i = 1 …, 2^{k}.

Hence relation (B1) can be rewritten as

h_{k}(x_{1}, …, x_{2^{k}}) = \frac{1}{Z}_{n+k} exp \{\frac{1}{T} ∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{k}} c^{n}U(i, j)x_{i}x_{j}\}

· ∏_{l=1}^{2^{k}} ∫ exp \{−\frac{1}{T} H_{n}(z_{(l−1)2^{n}+1}, …, z_{l2^{n}})\}

δ(2^{−n} ∑_{j=(l−1)2^{n}+1}^{l2^{n}} z_{j} − x_{l}) ∏_{j=(l−1)2^{n}+1}^{l2^{n}} p(z_{j}) dz_{j}

= C_{k,n} exp \{−\frac{1}{T} H_{k}(c^{n/2}x_{1}, …, c^{n/2}x_{2^{k}})\} ∏_{l=1}^{2^{k}} p_{n}(x_{l}, T) ,

as we have claimed.

Appendix C. The calculation of the Radon–Nikodym derivatives. The proof of formulas (2.1)—(2.3′) in Part II. For n = N relations (2.1) and (2.2) of Part II immediately follow from formula (1.4) in Part II. Hence it is enough to prove our relations by induction from n + 1 to n. Clearly,

P_{n+1}(x_{1}, …, x_{2^{n+1}}) = C_{n}P_{n}(x_{1}, …, x_{2^{n}})P_{n}(x_{2^{n}+1}, …, x_{2^{n+1}})
exp \{\frac{c^{n}}{T} (2^{−n} ∑_{j=1}^{2^{n}} x_{j})(2^{−n} ∑_{j=2^{n}+1}^{2^{n+1}}x_{j})\}

with some norming constant C_{n}. Given some measurable set A ⊂ (R^{p})^{2^{n}} define the cylindrical set Ã ⊂ (R^{p})^{2^{n+1}} as Ã = A × (R^{p})^{2^{n}}. By our inductive hypothesis for n + 1

μ_{n,N}^{h_{N}}(A) = ∫Ã f_{n+1,N}^{h_{N}}(2^{−(n+1)} ∑_{j=1}^{2^{n+1}} x_{j})P_{n+1}(x_{1}, …, x_{2^{n+1}}) dx_{1} … dx_{2^{n+1}}

= C_{n} ∫Ã f_{n+1,N}^{h_{N}}(2^{−n}(∑_{j=1}^{2^{n}} \frac{x_{j}}{2} + ∑_{j=2^{n}+1}^{2^{n+1}} \frac{x_{j}}{2} ))

P_{n}(x_{1}, …, x_{2^{n}})P_{n}(x_{2^{n}+1}, …, x_{2^{n+1}})

exp \{\frac{c^{n}}{T} (2^{−n} ∑_{j=1}^{2^{n}} x_{j})(2^{−n} ∑_{j=2^{n}+1}^{2^{n+1}} x_{j})\} dx_{1} … dx_{2^{n+1}} .

Let us calculate the last integral by first integrating on the hyperplanes where x_{1}, …, x_{2^{n}} and y = \frac{1}{2^{n}} ∑_{j=2^{n}+1}^{2^{n+1}} x_{j} are fixed. Since P_{n}(x_{2^{n}+1}, …, x_{2^{n+1}}) is the only term in the integrand which is not constant on such a hyperplane, and its integral equals p_{n}(y) on it, we get that

μ_{n,N}^{h_{N}}(A) = C_{n} ∫_{A×R^{p}} f_{n+1,N}^{h_{N}}(2^{−n}(∑_{j=1}^{2^{n}} \frac{x_{j}}{2} + \frac{y}{2} )P_{n}(x_{1}, …, x_{2^{n}}))p_{n}(y)

exp\{\frac{c^{n}}{T} 2^{−n}(∑_{j=1}^{2^{n}} x_{j})y\} dx_{1} … dx_{2^{n}}dy .

Hence we get, by integrating first by the variable y that

μ_{n,N}^{h_{N}}(A) = C′_{n} ∫_{A} S_{n}f_{n+1,N}^{h_{N}}(2^{−n} ∑_{j=1}^{2^{n}} x_{j})P_{n}(x_{1}, …, x_{2^{n}}) dx_{1} … dx_{2^{n}} .

Since this relation holds for all measurable sets A ⊂ (R^{p})^{2^{n}}, it implies our inductive hypothesis for n.

Appendix D. On limit Gibbs states. Here we briefly describe the definition of limit Gibbs states (also called equilibrium states in the literature,) and discuss some important questions related to this definition. Limit Gibbs states are defined with the help of a Hamiltonian (often called energy) function, a free measure and a physical parameter, the temperature T. The Hamiltonian function is a formal series. Let us have a subset Z ⊂ Z^{d} of the d-dimensional integer lattice and a closed set K ⊂ R^{p} in the p-dimensional Euclidedan space. We consider a Hamiltonian function H(σ) of the form

H(σ) = − ∑_{i,j∈Z} U(i, j)σ(i)σ(j), σ = {σ(j), σ(j) ∈ K, j ∈ Z},

where U(·, ·) is a given function, U : Z × Z → R^{1}, and σ(i)σ(j) denotes scalar product. (There is a more general definition of Hamiltonian functions, but this special class is sufficiently large for our purposes.) Given some finite set V ⊂ Z, we define the energy function H_{V} (σ) as

(D1)  H_{V} (σ) = − ∑_{i,j∈V} U(i, j)σ(i)σ(j) ,  σ = {σ(i), i ∈ V },

and the conditional energy in V with respect to a configuration σ̄ in Z − V as

(D2)  H_{V} (σ|σ̄) = H_{V} (σ) − ∑_{i∈V} ∑_{j∈Z−V} U(i, j)σ(i)σ̄(j),
σ = {σ(i), i ∈ V }, σ̄ = {σ̄(i), i ∈ Z − V },

provided that the last sum is convergent. Given some h ∈ R^{p}, we also introduce the energy of a configuration σ in the volume V with respect to the external field h as

(D3)  H_{V}^{h} (σ) = H_{V} (σ) − h ∑_{i∈V} σ(i).

Given some finite set V ⊂ Z, a configuration σ̄ = {σ̄(j), j ∈ Z − V } outside V, a Hamiltonian function H(σ)) and a free measure P ( dx) on K, we define the Gibbs measure in volume V with respect to the external field σ̄ at temperature T as the probability measure μ_{V,T} ( · |σ̄) on K^{V} given by the formula

(D4)  μ_{V,T} (σ ∈ A|σ̄) = \frac{1}{Z(V, T, σ̄)} ∫_{A} exp\{−\frac{1}{T} H_{V} (σ|σ̄)\} ∏_{j∈V} P ( dσ(j)),

where A ∈ K^{V} is an arbitrary measurable set, σ = {σ(j), j ∈ V } and Z(V, T, σ̄) is an appropriate norming constant, provided that the above expression is meaningful. Now we formulate the following

Definition of Gibbs states. A probability measure μ̄ is a Gibbs state with Hamiltonian function H and free measure P at temperature T if a μ̄ distributed random field σ(j), j ∈ Z satisfies the following relation: For any finite set V and measurable set A ⊂ K^{V} the conditional probability of the event σ ∈ A, σ = {σ(j), j ∈ V } with respect to the condition {σ(j) = σ̄, j ∈ Z − V } with a configuration σ̄ = {σ̄(j), j ∈ Z − V } equals to

μ̄(σ ∈ A | {σ(j), j ∈ Z − V } = σ̄) = μ_{V,T} (A | σ̄)

with μ̄ probability one, where μ_{V,T} is defined in (D4).

The question arises whether Gibbs states on K^{Z} exist, and whether they are unique. A natural way to construct Gibbs states is to carry out the following procedure. Choose an increasing family of sets V_{n} ⊂ Z, ∪V_{n} = Z, fix a configuration σ̄ = σ̄^{(n)} = {σ̄(j), j ∈ Z − V_{n}} for each V_{n}, and consider the measures μ_{V_{n},T} ( · |σ̄) defined in (D4). Prove that under some mild restrictions there is a convergent subsequence of this sequence, and the limit of this subsequence is a Gibbs state. The problem about the uniqueness of Gibbs states is closely related to the question whether, in dependence of the choice of the external configuration σ̄^{(n)}, different limits can appear in the above construction. A slightly different, and often useful approach is to choose a sequence h_{n} ∈ R^{n}, h_{n} → 0, and try to construct Gibbs states as the limit of a sequence of measures of the form μ_{V_{n},T}^{h_{n}}, where we define the probability measure μ_{V,T}^{h} as (D5)

μ_{V,T}^{h} (A) = \frac{1}{Z(V, T, h)} ∫_{A} exp\{−\frac{1}{T} (∑_{i∈V} ∑_{j∈V} U(i, j)x_{i}x_{j} − h ∑_{i∈V} x_{i})\} ∏_{i∈V} P (dx_{i}) .

If K is a compact subset of R^{p} then standard results in probability theory imply the compactness of the measures μ_{V_{n},T} ( · |σ̄) or of μ_{V_{n},T}^{h_{n}} in weak topology, i.e. the existence of a convergent subsequence in this topology. (See e.g. [3].) Nevertheless, there are many interesting models, where the set K is non-compact (e.g. K = R^{p},) and in such cases a hard analysis is needed to prove the existence of such a convergent subsequence. (See e.g. [10] or [20] as an example.) In order to prove that the limit of the sequence of measures μ_{V_{n},T} ( · |σ̄) (or μ_{V_{n},T}^{h_{n}}) is really a Gibbs state it is worth while to rewrite the definition of Gibbs states in an equivalent integral form. Let f = f(x_{j_{1}}, …, x_{j_{k}}) and g = g(x_{l_{1}}, …, x_{l_{k}}), x_{j_{i}} ∈ R^{p}, x_{l_{i}} ∈ R^{p}, be two bounded and continuous functions with finitely many arguments, V = {j_{1}, …, j_{k}}, W = {l_{1}, …, l_{k}}, V ⊂ Z, W ⊂ Z such that V ∩ W = ∅.

The measure μ̄ on K^{Z} is a Gibbs state if and only if

(D6)  ∫_{K^{Z}} fg dμ̄ = ∫_{K^{Z−V}} μ(f)g dμ̄

for all functions f and g with the above properties, where

(D7) μ(f) = μ(f)(σ̄) = ∫_{K^{V}} f(σ(j_{1}), …, σ(j_{k})) μ_{V,T} ( dσ|σ̄)
σ = {σ(j), j ∈ V }, σ̄ = {σ(j), j ∈ Z − V },

and μ_{V,T} is defined in (D4).

Let us consider an arbitrary sequence of sets V_{n} ⊂ Z, ∪V_{n} = Z, and numbers h_{n} ∈ R^{p}, h_{n} → 0. Some calculation shows that for sufficiently large n (if V ⊂ V_{n}, W ⊂ V_{n})

(D8)  ∫_{K^{Vn}} fg dμ_{V_{n},T}^{h_{n}} = ∫_{K^{Vn−V}} μ^{h_{n}}(f)g dμ_{V_{n},T}^{h_{n}}

with

μ^{h_{n}}(f) = μ_{V,V_{n},T}^{h_{n}} (f; σ(j), j ∈ V_{n} − V)

= \frac{∫ f(σ(j_{1}), …, σ(j_{k})) exp\{−\frac{1}{T} (H_{V,V_{n}}(σ) − h_{n} ∑_{i∈V} σ(i))\} ∏_{i∈V} P ( dσ(i))}{∫ exp\{−\frac{1}{T} (H_{V,V_{n}}(σ) − h_{n} ∑_{i∈V} σ(i))\} ∏_{i∈V} P ( dσ(i))} ,

where

H_{V,V_{n}}(σ) = ∑_{i∈V} ∑_{j∈V_{n}} σ(i)σ(j) .

If the sequence μ_{V_{n},T}^{h_{n}} tends weakly to the measure μ̄ then the left-hand side of (D8) converges to that of (D6). Hence to prove that the limit measure μ̄ is a Gibbs state it suffices to establish the convergence of the right-hand side of (D8) to that of (D6). If the Hamiltonian function has a finite range interaction, i.e. there is some number r > 0 such that U(i, j) = 0 if |i − j| ≥ r then it is not difficult to see that μ^{h_{n}}(f)(σ̄) → μ(f)(σ̄), and the required convergence can be proved with the help of this relation. In case of infinite range interaction one must be more careful, especially if the state space K is non-compact. Dyson’s model which we are investigating is such a model. In Theorem 1 of Part II we have proved the weak convergence of the measures μ_{N}^{h_{n}} to μ̄. (Actually, we have proved a stronger form of convergence.) In Appendix E we prove Theorem B, i.e. we show that the limit measure is a Gibbs state. In the proof we approximate Dyson’s model with a model with finite range interaction, and this enables us to carry out the required limiting procedure. In Appendix E we restrict ourselves to Dyson’s model, although the argument also works in more general cases.

Appendix E. The proof of Theorem B. We apply the argument of Appendix D. The proof of Theorem B can be completed by showing that also in the case of Dyson’s model the right-hand side of (D8) tends to that of (D6). We formulate this statement in more detail.

It suffices to consider the case when V = {1, 2, …, 2^{k}}, W = {2^{k} + 1, 2, …, 2^{m}} with some 0 ≤ k < m, i.e. f = f(x_{1}, …, x_{2^{k}}), g = g(x_{2^{k}+1}, …, x_{2^{m}}) and V_{N} = {1, 2, …, 2^{N}}. (We apply the notation of Appendix D.) Introduce the functions

p_{k,N}^{h}(x_{1}, …, x_{2^{k}}|x_{2^{k}+1}, …, x_{2^{N}}) (E1)

= \frac{exp\{−\frac{1}{T} (∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{N}} U(i, j)x_{i}x_{j} − h ∑_{i=1}^{2^{k}} x_{i})\}}{∫ exp\{−\frac{1}{T} (∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{N}} U(i, j)x_{i}x_{j} − h ∑_{i=1}^{2^{k}} x_{i})\} ∏_{i=1}^{2^{k}} p(x_{i}) dx_{i}} ,

x_{j} ∈ R^{p}, j = 1, …, 2^{N}

and

(E1′)  p_{k}(x_{1}, …, x_{2^{k}}|x_{2^{k}+1}, …)

= \frac{exp\{−\frac{1}{T} ∑_{i=1}^{2^{k}} ∑_{j=i+1}^{∞} U(i, j)x_{i}x_{j}\}}{∫ exp\{−\frac{1}{T} ∑_{i=1}^{2^{k}} ∑_{j=i+1}^{∞} U(i, j)x_{i}x_{j}\} ∏_{i=1}^{2^{k}} p(x_{i}) dx_{i}} ,

x_{j} ∈ R^{p}, j = 1, …,

where the function U(·, ·) is defined in (1.2) and p(x) in (1.3) of Part I. Put

(E2)μ_{k,N}^{h}(f)(x_{2^{k}+1}, …, x_{2^{N}}) = ∫ f(x_{1}, …, x_{2^{k}})

p_{k,N}^{h}(x_{1}, …, x_{2^{k}}|x_{2^{k}+1}, …, x_{2^{N}}) ∏_{i=1}^{2^{k}} p(x_{i}) dx_{i}

and

(E2′) μ_{k}(f)(x_{2^{k}+1}, …) = ∫ f(x_{1}, …, x_{2^{k}})p_{k}(x_{1}, …, x_{2^{k}}|x_{2^{k}+1}, …) ∏_{i=1}^{2^{k}} p(x_{i}) dx_{i}.

The convergence of the right hand side of (D8) to that of (D6) is equivalent to the relation

(E3)  lim_{N→∞} ∫ gμ_{k,n}^{h_{N}}(f) dμ_{N}^{h_{N}} = ∫ gμ_{k}(f) dμ̄

in our case, where μ_{N}^{h_{N}} and μ̄ are the same probability measures on (R^{p})^{2^{N}} and (R^{p})^{Z} as in Theorem 1 of Part I.

To prove this relation let us introduce the sets A(K, k, n, N) and A(K, k, n),where K ∈ R^{1}, K > 0, k, n, N ∈ Z and k < n < N, defined by the formulas

A(K, k, n, N) = {(x_{2^{k}+1}, …, x_{2^{N}}), x_{j} ∈ R^{p}, j = 2^{k} + 1, … 2^{N}, |x_{j}| < K
if 2^{k} < j ≤ 2^{n}, and |x_{j}| < 2^{lα} if 2^{l} < j ≤ 2^{l+1}, l = n, …, N − 1}

and

A(K, k, n) = {(x_{2^{k}+1}, …), x_{j} ∈ R^{p}, j = 2^{k} + 1, …, |x_{j}| < K
if 2^{k} < j ≤ 2^{n}, and |x_{j}| < 2^{lα} if 2^{l} < j ≤ 2^{l+1}, l = n, n + 1, … },

where α = \frac{3}{4} − \frac{1}{2} \frac{log c}{log 2}.

We claim that for all ε > 0 some n = n(ε) and K = K(ε, n) can be chosen in such a way that

(E4)  μ_{N}^{h_{N}} ((x_{2^{k}+1}, …, x_{2^{N}}) ∉ A(K, k, n, N)) < ε

and

(E4′)  μ̄((x_{2^{k}+1}, …,) ∉ A(K, k, n,)) < ε .

To prove relations (E4) and (E4′) let us first observe that there is a universal constant C, C > 0, such that if S_{n}^{(}f(x)j) is the j-th coordinate of a μ_{N}^{h_{N}} or μ̄ distributed random variable then the inequality Eσ^{2}(j) < C holds for all j ∈ Z and measures μ_{N}^{h_{N}} and μ̄. This can be seen by observing that the argument of Section 6 in Part II actually implies that all moments of the j-th coordinate σ(j) of a μ_{N}^{h_{N}} distributed random vector σ converge to the corresponding moment of the j-th coordinate of a μ̄ distributed random vector as N → ∞. Then we get,by exploiting that 1 − 2α > 0 that

μ_{N}^{h_{N}} \{(x_{2^{k}+1}, …, x_{2^{N}}) ∉ A(K, k, n, N)\} ≤ C(\frac{2^{n} − 2^{k}}{K^{2}} + ∑_{j=n}^{N} 2^{j(1−2α)}) < ε

if first n and then K is chosen sufficiently large. The proof of (E4′) is the same. (We remark that relations (E4) and (E4′) hold with arbitrary α > 0 in the definition of the sets A(·, ·, ·). To prove it we have to apply the stronger statement Eσ^{2k} < C_{k} for all k ≥ 1. This observation is needed if we want to prove Theorem B in the case \sqrt{2} < c < 2 too.)

We claim that for all ε > 0 there is some N_{0} = N_{0}(ε, K, n) such that for N > N_{0}

|p_{k,N}^{h_{n}}(x_{1}, …, x_{2^{k}}|x_{2^{k}+1} …, x_{2^{N}}) − p_{k,N_{0}}^{h_{n}}(x_{1}, …, x_{2^{k}}|x_{2^{k}+1} …, x_{2^{N_{0}}} )| (E5)

< ε(∑_{i=1}^{2^{k}} |x_{i}|) exp{(2Kn + 1) ∑_{i=1}^{2^{k}} |x_{i}|}

and

(E5′) |p_{k}(x_{1}, …, x_{2^{k}}|x_{2^{k}+1} …) − p_{k,N_{0}}^{0}(x_{1}, …, x_{2^{k}}|x_{2^{k}+1} …, x_{2^{N_{0}}} )|

< ε(∑_{i=1}^{2^{k}} |x_{i}|) exp{(2Kn + 1) ∑_{i=1}^{2^{k}} |x_{i}|}

if (x_{2^{k}+1}, …, x_{2^{N}}) ∈ A(K, k, n, N) and (x_{2^{k}+1}, …) ∈ A(K, k, n). (The constants K and n in formulas (E5) and (E5′) are the same as in the definition of the sets A(K, k, n, N).) First we show that (E5) and (E5′) together with (E4) and (E4′) imply (E3), hence also Theorem B. Indeed, since p_{n}(x) decreases at infinity faster than exp(−x^{2}/2) hence (E5) and (E5′) imply that

|μ_{k,N}^{h_{N}}(f)(x_{2^{k}+1}, …, x_{2^{N}}) − μ_{k,N_{0}}^{0}(f)(x_{2^{k}+1}, …, x_{2^{N_{0}}} )| < const.ε

and

|μ_{k}(f)(x_{2^{k}+1}, …) − μ_{k,N_{0}}^{0}(f)(x_{2^{k}+1}, …, x_{2^{N_{0}}} )| < const.ε .

if (x_{2^{k}+1}, …) ∈ A(K, k, n). This implies that (x_{2^{k}+1}, …, x_{2^{N}}) ∈ A(K, k, n, N). Since this relation holds on a set of 1 − ε μ_{N}^{h_{N}} resp. μ̄ probability by (E4) and (E4′), the functions f, g, μ_{k,N}^{h_{N}}(f) and μ(f) are bounded, hence an error less than const.ε is committed if μ_{k,N}^{h_{N}} and μ_{k}(f) is replaced by μ_{k,N_{0}}^{0} in formula (E3). After this replacement relation (E3) holds, because the projections of the measures μ_{N}^{h_{N}} to (R^{p})^{2^{N}} converge to the projection of μ̄ to the same subspace. Since ε > 0 can be chosen arbitrary small, relation (E3) holds in its original form.

We prove only (E5) the proof of (E5′) being the same. Let us first observe that for any η > 0 there is some N_{0} = N_{0}(K, n, η) such that for N ≥ N_{0} and (x_{2^{k}+1}, …, x_{2^{N}}) ∈ A(K, k, n, N)

(E6)

|exp\{−\frac{1}{T} (∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{N}} U(i, j)x_{i}x_{j} − h_{N} ∑_{i=1}^{2^{k}} x_{i})\} − exp\{−\frac{1}{T} ∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{N_{0}}} U(i, j)x_{i}x_{j}\}|

= exp\{−\frac{1}{T} ∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{N}} U(i, j)x_{i}x_{j}\}

|exp \{−\frac{1}{T} (∑_{i=1}^{2^{k}} ∑_{j=2^{N_{0}} +1}^{2^{N}} U(i, j)x_{i}x_{j} − h_{N} ∑_{i=1}^{2^{k}} x_{i})\} − 1|

≤ η \frac{∑_{i=1}^{2^{k}} |x_{i}|}{T} exp \{\frac{2Kn + η}{T} ∑_{i=1}^{2^{k}} |x_{i}|\} .

In the last relation we have applied the inequality |e^{x} − 1| ≤ |x|e^{|x|} together with the relations h_{N} < η/2, | ∑_{j=2^{N_{0}} +1}^{2^{N}} U(i, j)x_{j}| ≤ | ∑_{j=N_{0}}^{∞} 2^{j(α−1)}(\frac{c}{4})^{j}| < η/2 and

| ∑_{j=i+1}^{2^{N}} U(i, j)x_{j}| ≤ 2Kn if N > N_{0}, j > 2^{k}, (x_{2^{k}+1}, …, x_{2^{N}}) ∈ A(K, k, n, N) and N_{0} is sufficently large. Integrating inequality (E6) with respect to the measure ∏_{i=1}^{2^{k}} p(x_{i}) dx_{i} we get that

(E7) |∫ exp\{−\frac{1}{T} (∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{N}} U(i, j)x_{i}x_{j} − h_{N} ∑_{i=1}^{2^{k}} x_{i})\} ∏_{i=1}^{2^{k}} p(x_{i}) dx_{i}

− ∫ exp\{−\frac{1}{T} ∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{N_{0}}} U(i, j)x_{i}x_{j}\} ∏_{i=1}^{2^{k}} p(x_{i}) dx_{i}| < const.η ,

where the const. may depend on K and n. In formulas (E6) and (E7) we have shown that both the numerators and the denominators of the functions p_{k,N}^{h_{N}} and p_{k,N_{0}}^{0} defined in (E1) are close to each other. The number η can be chosen arbitrary small in these estimates by fixing first n then K = K(n) and finally N_{0} = N_{0}(K, n) in an appropriate way. Moreover, given some appropriately chosen n and K the number η > 0 can be taken arbitrary small if N_{0} = N_{0}(K, n, η) is sufficiently large. Hence we prove (E5) by showing that

(E8)  ∫ exp\{−\frac{1}{T} ∑_{i=1}^{2^{k}} ∑_{j=i+1}^{2^{N_{0}}} U(i, j)x_{i}x_{j}\} ∏_{i=1}^{2^{k}} p(x_{i}) dx_{i} > D

with some D > 0 on the set A(K, k, n, N_{0}), i.e. the integral in (E8) is separated from zero. Here the constant D may depend on K and n but not on N_{0}. Relation (E8) holds, since if |x_{j}| < 1, j = 1, 2, …, 2^{k} and (x_{2^{k}+1}, …, x_{2^{N}}) ∈ A(K, k, n, N_{0}) then the integrand in (E8) is separated from zero.

Table of Contents

Preface.

Part I. Limit theorems for the average spin.

1. Introduction

2. On the content of Theorem 1. Convergence to the solution of the fixed point equation.

3. On the strategy of the proof. The inductive procedure.

4. The proof of Proposition 1 and its Corollary. The first step of the inductive procedure.

5. The proof of Proposition 2. The second step of the inductive procedure.

6. The proof of Proposition 3 and some of its consequences.

7. On the fixed point equation T_{M}g = g.

8. The proof of Theorems 1 and 1′.

9. The proof of Theorem 2. The behaviour of the density of the average spin at infinity.

10. The proof of Lemmas 17 and 18.

Part II. Description of the large-scale limit.

1. Introduction.

2. On the basic estimates needed during the proof. Reduction to integral equations.

3. The proof of Lemma 1.

4. Some preparatory remarks to the proof of Proposition 1′.

5. The proof of Proposition 1′.

6. The proof of Theorem 1 and Proposition 2. Existence of the thermodynamical limit.

7. The proof of Theorem 2. Existence of the large-scale limit.

8. Some open problems and conjectures.

Appendix. A. The proof of the basic recursive relations (2.1) and (2.1′) in Part I. B. The proof of Theorem C. C. The calculation of the Radon–Nikodym derivatives. The proof of formulas

(2.1)—(2.3′) in Part II. D. On limit Gibbs states. E. The proof of Theorem B.