THE LARGE-SCALE LIMIT OF DYSON’S HIERARCHICAL

VECTOR-VALUED MODEL AT LOW TEMPERATURES.

THE MARGINAL CASE c = \sqrt{2}.

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

1 Keldysh Institute for Applied Mathematics of the Academy of Sciences of the U.S.S.R., Moscow A–47 Miusskaya Square 4, U.S.S.R.

2 Mathematical Institute of the Hungarian Academy of Sciences,

Budapest H–1364 P.O.B. 127, Hungary

Dedicated to Professor R.L. Dobrushin for his sixtieth birthday

Summary: In this paper we construct the equilibrium states of Dyson’s vector-valued hierarchical model with parameter c = \sqrt{2} at low temperatures and describe their largescale limit. The analogous problems for \sqrt{2} < c < 2 and 1 < c < \sqrt{2} were solved in our papers [1] and [2]. In the present case the large-scale limit is similar to the case \sqrt{2} < c < 2, i.e. it is a Gaussian self-similar field with long-range dependence in the direction orthogonal to and a field consisting of independent Gaussian random variables in the direction parallel with the magnetization. The main difference between the two cases is that now the normalizing factor in the direction of the magnetization contains, beside the square-root of the volume, a logarithmic term too.

1. Introduction

First we briefly describe the model we are investigating. Dyson’s hierarchical model is a one-dimensional classical spin model on the lattice Z = {1, 2, … }. Its Hamiltonian function depends on a parameter a, 1 < a < 2, and is defined as

H(σ) = − ∑_{i∈Z} ∑_{\begin{matrix}j∈Z \\ j>i \end{matrix}} d(i, j)^{−a}σ(i)σ(j) ,  (1.1)

where d(i, j) = 2^{n(i,j)−1}, and

n(i, j) = min {n, there exists some k such that (k − 1)2^{n} < i, j ≤ k2^{n}} .

We are dealing with vector-valued models, where σ(j) ∈ R^{p} with some p ≥ 2. If x ∈ R^{p} and y ∈ R^{p} then xy denotes scalar product. We consider models with the free measure ν,

\frac{dν}{dx}(x) = p_{0}(x) = p_{0}(x, t) = C(t) exp \{−\frac{x^{2}}{2} − \frac{t}{4}|x|^{4}\} ,  x ∈ R^{p},  (1.2)

where t > 0 is a sufficiently small number, and C(t) is an appropriate norming constant which turns p_{0}(x) into a density function. For the sake of convenience we shall work in the sequel with the number c = 2^{2−a} instead of the parameter a.

Typeset by AMS-TEX

We investigate the following problem: First we construct an equilibrium state μ = μ(T) at low temperatures with magnetization in the direction of the first coordinate x^{(1)} and then we want to describe its large-scale limit. In more detail, let

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

be a random field with the distribution of the equilibrium state μ = μ(T), and define for all n = 1, 2, … the random field

R_{n}σ = {(R_{n}σ^{(1)}(j), …, R_{n}σ^{(p)}(j)) ∈ R^{p}, j ∈ Z } ,  (1.3)
R_{n}σ^{(1)}(j) = \frac{1}{A_{n}} ∑_{k=(j−1)2^{n}+1}^{j2^{n}} [σ^{(1)}(k) − Eσ^{(1)}(k)] , j ∈ Z ,  (1.4)

R_{n}σ^{(s)}(j) = \frac{1}{B_{n}} ∑_{k=(j−1)2^{n}+1}^{j2^{n}} σ^{(s)}(k) , j ∈ Z, s = 2, …, p,  (1.5)

where A_{n} and B_{n} are appropriate norming constants. We want to choose them in such a way that the finite dimensional distributions of the fields R_{n}σ converge as n → ∞, and also want to describe the limit field. Here A_{n} is the norming constant in the direction of the magnetization and B_{n} in the direction orthogonal to it.

We have solved this problem for \sqrt{2} < c < 2 in our paper [1] and for 1 < c < \sqrt{2} in [2]. In both cases we have to choose a “critical” normalization B_{n} = 2^{n}c^{−n/2} in the direction orthogonal to the magnetization, and the limit is a self-similar Gaussian field with long-range correlation. On the other hand, in the direction of the magnetization we have a different situation in the two cases. For \sqrt{2} < c < 2 we have to choose A_{n} = 2^{n/2} and get a field of independent Gaussian variables for the limit. For 1 < c < \sqrt{2} the right choice in (1.4) is A_{n} = 2^{n}c^{−n}, and the limit is a non-Gaussian field which we have described explicitly in [2]. Our aim in this paper is to solve this problem for c = \sqrt{2}. The answer is very similar to the case \sqrt{2} < c < 2. Namely, we have to choose B_{n} = 2^{n}c^{−n/2} = 2^{3n/4} and get a dependent Gaussian field in the direction orthogonal to the direction of the magnetization. In the direction of the magnetization we have to choose A_{n} = 2^{n/2}\sqrt{n}, and the limit is a field consisting of independent Gaussian random variables. The main difference between the cases \sqrt{2} < c < 2 and c = \sqrt{2} is the appearance of a multiplying term \sqrt{n} in the normalizing factor A_{n} in the latter case. It is expected that translation invariant models with short-range interaction in the cases d < 4, d = 4 and d > 4 show a behaviour similar to Dyson’s model in the cases 1 < c < \sqrt{2}, c = \sqrt{2} and \sqrt{2} < c < 2. Thus Dyson’s model with c = \sqrt{2} corresponds to four-dimensional translation invariant models.

Let us formulate our results in more detail. In Theorem 1 formulated below we construct the equilibrium state whose large-scale limit will be investigated.

Given some h ∈ R^{1}, h ≥ 0, and a positive integer n let us define the Gibbs measure μ_{n}^{h} = μ_{n}^{h}(T, t) on (R^{p})^{2^{n}} with the 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} , given by the formula

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}} d(i, j)^{−3/2}x_{i}x_{j} − h ∑_{j=1}^{2^{n}} x_{j}^{(1)})\} ∏_{j=1}^{2^{n}} p_{0}(x_{j}, t) ,

(1.6)

where

Z_{n} = ∫ exp\{−\frac{1}{T} (− ∑_{i=1}^{2^{n}−1} ∑_{j=i+1}^{2^{n}} d(i, j)^{−3/2}x_{i}x_{j} − h ∑_{j=1}^{2^{n}} x_{j}^{(1)})\} ∏_{j=1}^{2^{n}} p_{0}(x_{j}, t) dx_{j}

is the grand partition function, and p_{0}(x, t) is defined in (1.2). Let p_{n}^{h}(x) = p_{n}^{h}(x, T) 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_{n}(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.

Let us introduce the functions

q_{n}(x) = q_{n}(x, T) = K_{n} exp\{\frac{a_{0}}{2a_{1}} 2^{n/2}x^{2}\}p_{n}(\sqrt{\frac{T}{a_{1}}} x)  (1.7)

with a_{0} = \frac{2}{2−\sqrt{2}}, a_{1} = a_{0} +1 and the above defined functions p_{n}(x), where the norming constant K_{n} will be appropriately chosen. The function q_{n}(x, T) is rotation invariant, i.e. the function q̄_{n}(z, T), z ∈ R^{1}, defined by the formula q̄_{n}(z, T) = q_{n}((z, 0), T), z ∈ R^{1}, 0 = (0, …, 0) ∈ R^{p−1} satisfies the relation q_{n}(x, T) = q̄_{n}(|x|, T). Choose the constant K_{n} in (1.7) in such a way that

∫_{0}^{∞} q̄_{n}(x, T) dx = 1 ,

and define the numbers

M_{n} = ∫_{0}^{∞} xq̄_{n}(x, T) dx .  (1.8)

Now we formulate the following

Theorem 1. There are some thresholds T_{0} > 0 and t_{0} > 0 such that if 0 < T < T_{0} and 0 < t < t_{0} then the limit M = lim_{n→∞} M_{n} > 0 exists, and M^{2} = \frac{a_{1}(a_{0}−T)}{tT^{2}} + O(1) with a_{0} = \frac{2}{2−\sqrt{2}} and a_{1} = a_{0} + 1. Moreover, the following relation holds: Put

M̄ = \sqrt{\frac{T}{a_{1}}} M,  (1.9)

and consider an arbitrary sequence of real numbers h_{n}, n = 0, 1, 2, … such that

\frac{2M̄}{2 − \sqrt{2}} (\frac{1}{\sqrt{2}})^{n} ≤ h_{n} ≤ D (\frac{1}{\sqrt{2}})^{n}  (1.10)

with some ∞ > D > \frac{2M̄}{2−\sqrt{2}}. Then the measures μ_{n}^{h_{n}} tend to a probability measure

μ̄ = μ̄(t, T) on (R^{p})^{Z}. More precisely, for all k ≥ 0 the measures μ_{k,n}^{h_{n}}, the projections of the measures μ_{n}^{h} to (R^{p})^{2^{k}}, converge to the projection of μ̄ to the first 2^{k} coordinates in variational metric as n → ∞. The measure μ̄ does not depend on the choice of sequences h_{n}.

The main result of this paper is 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 (1.3), (1.4), (1.5) tend, with the choice A_{n} = 2^{n/2}\sqrt{n} and B_{n} = 2^{3n/4}, to those of a Gaussian random field Y = (Y (n) = (Y^{(1)}(n), …, Y^{(p)}(n)) ∈ R^{p}, n ∈ Z). For all numbers k ≥ 0 the density func-

tion 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

\begin{matrix}h_{k} (x_{1}, …, x_{2^{k}}) \\ = C(k) exp\{−\frac{1}{T} [∑_{s=2}^{p}(\frac{2 + \sqrt{2}}{2} ∑_{j=1}^{2^{k}} x_{j}^{(s)2} − (\sqrt{2} − 1) (\frac{\sqrt{2}}{4})^{k} (∑_{j=1}^{2^{k}} x_{j}^{(s)})^{2} \\ − ∑_{i=1}^{2^{k}−1} ∑_{j=i+1}^{2^{k}} d(i, j)^{−3/2}x_{i}^{(s)}x_{j}^{(s)}) + (6 + 2\sqrt{2})M^{2} ∑_{j=1}^{2^{k}} x_{j}^{(1)2}]\}. \end{matrix} (1.11)

It follows from the result in Appendix E of [2] that the measure constructed in Theorem 1 is an equilibrium state. We restricted ourselves to the construction of equilibrium states for low temperatures where we are interested in their large-scale limit. The proofs of Theorems 1 and 2 are based, similarly to the papers [1] and [2], on two analytic problems, where the action of an integral operator must be investigated. We formulate these problems in the next Section.

2. The basic steps of the proof

In this Section we discuss two analytical problems which play a central role in the proof of Theorems 1 and 2. The first one is connected with the asymptotical behaviour of the density function p_{n}(x) of the average of a μ_{n} distributed vector defined after formula (1.6). It is proved (see e.g. Appendix A in [2]) that p_{n}(x) satisfies the recursive relation

p_{n+1}(x) = C_{n}(T) ∫ exp \{\frac{2^{n/2}}{T} (x^{2} − u^{2})\} p_{n}(x − u)p_{n}(x + u) du  (2.1)

with the starting function p_{0}(x) defined in formula (1.2). For us it is more convenient to work with the functions q_{n}(x) defined in (1.7) instead of the functions p_{n}(x). Simple calculation shows that relations (2.1) and (1.7) imply the recursive relations

q_{n+1}(x) = K_{n} ∫ exp{−2^{n/2}u^{2}}q_{n}(x − u)q_{n}(x + u) du  (2.2)

with the starting function

q_{0}(x) = q_{0}(x, T, t) = K_{0} exp \{\frac{a_{0} − T}{2a_{1}} x^{2} − \frac{tT^{2}}{4a_{1}^{2}} |x|^{4}\} ,  (2.2′)

where a_{0} = \frac{2}{2−\sqrt{2}}, a_{1} = a_{0} + 1, and K_{n} is an appropriate norming constant. (The numbers a_{0} and a_{1} will denote these numbers in the whole paper.)

In Theorem A formulated below we describe the asymptotic behaviour of the function q_{n}(x). We recall that we have introduced the functions q̄_{n}(z) = q̄_{n}(z, T), z ∈ R^{1} in Section 1, and they satisfy the relation q_{n}(x, T) = q̄_{n}(|x|, T).

Theorem A. There are some thresholds t_{0} and T_{0} such that for 0 < t < t_{0} and 0 < T < T_{0} the functions q_{n}(x) defined by formulas (2.2) and (2.2′) satisfy the following relations:

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

2^{−n/2}\sqrt{n}q_{n}(x, T) = 2^{−n/2}\sqrt{n}q̄_{n}(|x|, T)
= \frac{\sqrt{2}M}{\sqrt{π}} exp \{−\frac{2^{n+1}M^{2}}{n} (|x| − M)^{2}\} + r_{n}(x)  (2.3)

with

|r_{n}(x)| ≤ \frac{K}{\sqrt{n}}  (2.3′)

and

|M^{2} − \frac{a_{1}(a_{0} − T)}{tT^{2}} | ≤ K  (2.4)

with some K > 0. Also the estimate

2^{−n/2}\sqrt{n}q̄_{n}(x, T) ≤ K exp \{−\frac{2^{n/2}μ}{\sqrt{n}}|x − M|\} for all x > 0  (2.5)

holds with some K > 0 and μ > 0 depending on T and t.

For x − M ≫ 0 or x − M ≪ 0 we need a better bound on q̄_{n}(x, T) than that given in (2.5). This is given in the following Proposition A. Under the conditions of Theorem A

2^{−n/2}\sqrt{n}q̄_{n}(x, T) ≤ K exp \{−β \frac{2^{n}}{n} (x − M)^{2}\} for x > M  (2.6)

with some β > 0 and K > 0 depending on t and T.

If 0 < x < M then for all ε > 0 there are some thresholds t_{0} = t_{0}(ε), T_{0} = T_{0}(ε) and a real number r_{n}, C_{1}n2^{−n/2} < M − r_{n} < C_{2}n2^{−n/2} with C_{2} > C_{1} > 0 such that if 0 < t < t_{0} and < T < T_{0} then

2^{−n/2}\sqrt{n}q̄_{n}(x, T) ≤ K exp \{−β \frac{2^{n}}{n} (x − M)^{2}\} for r_{n} < x < M  (2.7)

and

2^{−n/2}\sqrt{n}q̄_{n}(x, T) ≤ K exp \{−(1 − ε)2^{n/2} (r_{n}^{2} − x^{2}) − β \frac{2^{n}}{n} (r_{n} − M)^{2}\}
for 0 < x < r_{n} .  (2.7′)

(With some extra-work it can be shown that the number r_{n} can be chosen in the form r_{n} = M − Cn2^{−n/2} with some C > 0. We do not prove it, because the slightly weaker statement formulated above is sufficient for our purposes. Also the dependence of the thresholds t_{0} and T_{0} on ε can be dropped with the help of some additional investigation. We do not do it, because we do not need Proposition A with very small ε. What we need is that it holds with some ε > 0 such that 1 − ε > \frac{a_{0}}{2a_{1}} .)

Now we formulate the second problem we are interested in. Given some integers 0 ≤ n ≤ N and positive real number h > 0 consider the measure μ_{N}^{h} with density function p_{N}^{h}(x_{1}, …, x_{2^{N}}) defined in (1.6) (we replace the number n by N in it), and define its projection μ_{n,N}^{h} to the first 2^{n} coordinates x_{1}, …, x_{2^{n}}. We want to give a

good asymptotic formula for the Radon–Nikodym derivative \frac{dμ_{n,N}^{h}}{dμ_{n}} (x_{1}, …, x_{2^{n}}), where μ_{n} is μ_{n}^{h} with h = 0. It can be expressed explicitly by the following recursive integral formula: (See e.g. Appendix C in [2].)

Formula for the Radon–Nikodym derivative.

\frac{dμ_{n,N}^{h}}{dμ_{n}} (x_{1}, …, x_{2^{n}}) = f_{n,N}^{h}(2^{−n} ∑_{j=1}^{2^{n}} x_{j}), n ≤ N  (2.8)

f_{N,N}^{h}(x) = K(N, h) exp (\frac{2^{N}hx^{(1)}}{T} )  (2.9)
f_{n,N}^{h}(x) = K(n, N, h)S_{n}f_{n+1,N}^{h}(x)  (2.10)

with

S_{n}f(x) = ∫_{R^{p}} exp(\frac{2^{n/2}}{T} xy)f(\frac{x + y}{2} )p_{n}(y) dy  (2.10′)

where K(n, N, h) are appropriate norming factors and p_{n}(x) is the density function of a μ_{n} distributed random vector.

In Theorem B formulated below we give an asymptotic formula for the function f_{n,N}^{h}(x) if h = h_{N} satisfies the relation

\frac{2M̄}{2 − \sqrt{2}} (\frac{1}{\sqrt{2}})^{N} ≤ h_{N} ≤ D (\frac{1}{\sqrt{2}})^{N}  (2.11)

with some ∞ > D > \frac{2M̄}{2−\sqrt{2}}. To formulate it we introduce the sequences g_{n}, A_{n}, n = 1, 2, …, N defined by the recursive relations

g_{N} = g_{N}(N, h_{N}) = \frac{2^{N}h_{N}}{T}  (2.12)
g_{n} = g_{n}(N, h_{N}) = \frac{g_{n+1}}{2} + \frac{2^{n/2}}{T} M̄  for n < N  (2.12′)
A_{N} = A_{N}(N, h_{N}) = 0  (2.13)

A_{n} = A_{n}(N, h_{N}) = \frac{A_{n+1}}{4} + \frac{(\frac{2^{n/2}}{T} + \frac{A_{n+1}}{2} )^{2}}{\frac{2^{(n+2)/2}}{T} + \frac{g_{n+1}}{M̄} − A_{n+1}}  for n < N , (2.13′)

where M̄ is defined in (1.9), and M and T are the same as in Theorem A.

For the sake of simpler notations let us restrict ourselves to the case R^{p} = R^{2}. Let us define the domains

Ω_{n}^{1} = {x ∈ R^{2}, ||x| − M̄| < 2^{−0.2n} , |x^{(2)}| < 2^{−0.2n} , x^{(1)} > 0}  (2.14)
Ω_{n}^{2} = {x ∈ R^{2}, ||x| − M̄| < 2^{−0.2n}} \ Ω_{n}^{1}  (2.14′)
Ω_{n}^{3} = {x ∈ R^{2}, ||x| − M̄| ≥ 2^{−0.2n}} .  (2.14′′)

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

Theorem B. For all q, 2^{−0.1} < q < 1, there is some n_{0} = n_{0}(T, M̄, D, q) such that if (2.11) holds, and N ≥ n ≥ n_{0} then the Radon–Nikodym derivative f_{n}(x) = f_{n,N}^{h_{N}}(x) defined by formulas (2.9)–(2.10′) satisfies the following relations:

a) In the domain Ω_{n}^{1}

f_{n}(x) = L_{n} exp {g_{n} (x^{(1)} − M̄) + A_{n}x^{(2)2} + ε_{n}(x)}  (2.15)

with

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

b) In the domain Ω_{n}^{2}

0 ≤ f_{n}(x) ≤ L_{n} exp \{g_{n}(|x| − M̄) − (\frac{g_{n}}{2M̄} − A_{n})2^{−0.4n} + q^{n}\} .  (2.16)

c) In the domain Ω_{n}^{3}

0 ≤ f_{n}(x) ≤ L_{n} exp \{\frac{g_{n}}{2M̄} (|x|^{2} − M̄^{2})\} ,  (2.17)

where the numbers A_{n} and g_{n} are defined in (2.12)–(2.13′), and L_{n} = L_{n}(N, h_{N}) is an appropriate norming constant.

We also need the following result which describes the asymptotic behaviour of the sequences g_{n} and A_{n} defined by (2.12)–(2.13′).

Proposition B. Let us choose some integer N and real number h_{N} > 0. Define the sequences g_{n} and A_{n}, 0 ≤ n ≤ N, by formulas (2.12)–(2.13′) and put ḡ_{n} = 2^{−n/2}g_{n}, Ā_{n} = 2^{−n/2}A_{n}. If h_{N} satisfies relation (2.11) then ḡ_{N} ≥ ḡ_{N−1} ≥ ⋯ ≥ ḡ_{0} ≥ ḡ and 0 = Ā_{N} ≤ Ā_{N−1} ≤ ⋯ ≤ Ā_{0} ≤ Ā with ḡ = \frac{2}{2−\sqrt{2}} \frac{M̄}{T}, and Ā = \frac{\sqrt{2}−1}{T}. If the relations N > N_{0} and N > n^{B} also hold with some appropriate N_{0} = N_{0}(M̄, T, D) and B = B(M̄, T, D) then |ḡ_{n} − ḡ| < 4^{−n} and |Ā_{n} − Ā| < 4^{−n}.

The above results enable us to carry out a limiting procedure analogous to that in Sections 6 and 7 in Part II of [2], which leads to the proof of Theorems 1 and 2. The main step of this limiting procedure is to give a good estimate for the expression

p_{n}(2^{−n} ∑_{j=1}^{2^{n}} x_{j})f_{n,N}^{h_{N}}(2^{−n} ∑_{j=1}^{2^{n}} x_{j}) .  (2.18)

Since we can express the function p_{n}(x) through q_{n}(x) Theorems A and B together with Proposition B enable us to give a good asymptotic formula for this expression in a typical domain around the point (M̄, 0) ∈ R^{2}. Then Theorem B together with Proposition A guarantee that the region outside this typical domain has a negligible effect.

3. On Theorem A. The method of the proof

The proof both of Theorem A and B is based on the ideas of [2]. Most proofs can be carried out in almost the same way, only the number c must be replaced by \sqrt{2}. The proofs of such parts will be omitted, we only refer to the corresponding result in [2]. From now on the letters C, C_{1}, K etc. will denote absolute constants. The same letter in different formulas may denote different constants if it is not stated otherwise.

Let us introduce, similarly to Part I in [2], the numbers M_{n} defined in (1.8) and the functions

f_{n}(x) = f_{n}(x, T) = 2^{−n/2}q̄_{n} (M_{n} + 2^{−n/2}x, T ) ,  (3.1)

where the function q̄_{n}(x) was defined after formula (1.7). We shall deduce Theorem A from the following

Theorem A′. Under the conditions of Theorem A the limit lim_{n→∞} M_{n} = M > 0 exists, and

M^{2} = \frac{a_{1}(a_{0} − T)}{tT^{2}} + R(T, t)  (3.2)

with some |R(T, t)| < const. Moreover, there is some n_{0} = n_{0}(t, T) such that for n > n_{0}

M_{n} = M + \frac{2 + \sqrt{2}}{4M}2^{−n/2} + δ(n)2^{−n/2},  |δ(n)| < K2^{−n/2}  (3.3)

with some K > 0. The function f_{n} satisfies the relations

|f_{n}(x, T) − \frac{\sqrt{2}M}{\sqrt{nπ}} exp \{−\frac{2M^{2}}{n} x^{2}\}| < \frac{K}{n}  for x > −2^{n/2}M_{n}  (3.4)

and

f_{n}(x, T) ≤ \frac{KM}{\sqrt{n}} exp\{−μ|x|\}  for x > −2^{n/2}M_{n}  (3.5)

for n > n_{0} with some μ > 0 and K > 0.

To prove Theorem A′ let us introduce, similarly to [2], the operator Q̄_{n,M};

Q̄_{n,M}f(x) = ∫ exp {−2^{−n/2}u^{2} − v^{2}}
f (2^{n/2} (\sqrt{(M + 2^{−(n+1)/2}x + 2^{−n/2}u)^{2} + 2^{−n/2}v^{2}} − M))

f (2^{n/2} (\sqrt{(M + 2^{−(n+1)/2}x − 2^{−n/2}u)^{2} + 2^{−n/2}v^{2}} − M)) du dv ,

(3.6)

its standardization defined by the formula

Q_{n,M}f(x) = \frac{Q̄_{n,M}f(x + m_{n})}{∫_{−2^{(n+1)/2}M}^{∞} Q̄_{n,M}f(x) dx}  (3.7)

with

m_{n} = \frac{∫_{−2^{(n+1)/2}M}^{∞} xQ̄_{n,M}f(x) dx}{∫_{−2^{(n+1)/2}M}^{∞} Q̄_{n,M}f(x) dx}  (3.7′)

together with their approximations T_{M} and T̄_{M} given by the formulas

T̄_{M}f(x) = ∫ e^{−v^{2}}f (\frac{x}{\sqrt{2}} + u + \frac{v^{2}}{2M} ) f (\frac{x}{\sqrt{2}} − u + \frac{v^{2}}{2M} ) du dv  (3.8)

and

T_{M}f(x) = \sqrt{\frac{2}{π}} T̄_{M}f (x − \frac{\sqrt{2}}{4M} ) .  (3.8′)

The same calculation as that in (2.20) of [2] yields that the Fourier transforms of the operators T_{M} and T̄_{M} defined by the formulas ˜ T̄_{M} f̃ = (T̄_{M}f)˜ and T̃_{M} f̃ = (T_{M}f)˜ satisfy the relation

˜T̄_{M} f̃(ξ) = \sqrt{\frac{π}{2}} \sqrt{1 + \frac{iξ}{\sqrt{2}M}}^{f̃(\frac{ξ}{\sqrt{2}})2}  (3.9)

and

T̃_{M} f̃(ξ) = \begin{matrix}exp & (\frac{i\sqrt{2}ξ}{4M} ) \\ √ & \overline{1 + \frac{iξ}{\sqrt{2}M}} \end{matrix} f̃(\frac{ξ}{\sqrt{2}})^{2} .  (3.9′)

The relation

(f_{n+1}(x), M_{n+1}) = (Q_{n,M_{n}}f_{n}(x), M_{n} + 2^{−(n+1)/2}m_{n})  (3.10)

holds with the starting pair (f_{0}(x), M_{0}) defined by the relations

M_{0} = ∫_{0}^{∞} xq̄_{0}(x) dx  f_{0}(x) = q̄_{0}(x − M_{0}) ,  (3.10′)

where the function q̄_{0}(x) was defined after formula (1.7) (with n = 0).

We have

Q_{n,M}f(x) = T_{M}f(x) + ε_{n}(x) ,  (3.11)

where ε_{n}(x) is a small error term. We get a heuristic explanation of Theorem A′ by investigating the expression T_{M}^{n}f(x) for large n with a function f(x) satisfying the relations ∫ f(x) dx = 1 and ∫ xf(x) dx = 0. Put

φ_{k}(ξ) = log T̃_{M}^{k} f̃(ξ) = ∑_{j=2}^{∞} d_{j,k}ξ^{k} .

It follows from (3.9) that

d_{j,k+1} = 2^{(2−j)/2}d_{j,k} + \frac{(−i)^{j}}{2j(\sqrt{2}M)^{j}} , j ≥ 2 .

Hence

lim_{n→∞} d_{j,n} = \frac{(−i)^{j}}{2j(\sqrt{2}M)^{j}(1 − 2^{(2−j)/2})} for j ≥ 3 ,

and

d_{2,n} = −\frac{n}{8M^{2}} + d_{2,0} .

The above relations imply that

lim_{n→∞} φ_{n} (\frac{ξ}{\sqrt{n}}) = −\frac{1}{8M^{2}} ξ^{2} .

Since f_{n}(x) behaves similarly to T_{M}^{n}f_{0}(x), the above calculation suggests that the expression \sqrt{n}f_{n}(\sqrt{n}x) is asymptotically Gaussian with variance \frac{1}{8M^{2}} . We justify this heuristic argument similarly to the method of [2]. First we show that if t and T are sufficiently small then for all not too large n f_{n}(x) is asymptotically normal with variance σ = \frac{a_{1}}{2(a_{0}−T)}. More precisely, we prove the following

Proposition 1. For all positive integers N ≥ 1 there are some thresholds t_{0} and T_{0} such that if 0 < T < T_{0} and 0 < t < t_{0} then for all n ≤ N

|\frac{d^{j}}{dx^{j}} [f_{n}(x) − ϕ(x, σ)]| ≤ \frac{B(n)}{\sqrt{M_{n}}} exp \{−2^{(n+2)/2}|x|\}
if |x| < log M_{n}, j = 0, 1, 2,  (3.12)

|\frac{d^{j}}{dx^{j}} f_{n}(x)| ≤ B(n) exp \{−2^{n/2} |2x + \frac{2^{−n/2}x^{2}}{M_{n}} |\} if x > −2^{−n/2}M_{n}, j = 0, 1, 2,

(3.13)

and

|M_{n} − M̂_{0}| ≤ B(n)t^{1/2}T ,  (3.14)

where M̂_{0}^{2} = \frac{a_{1}(a_{0}−T)}{T t^{2}}, σ^{2} = \frac{a_{1}}{2(a_{0}−T)}, ϕ(x, σ) denotes the normal density function with expectation zero and variance σ, and B(n) is some appropriate multiplying factor depending on n, but not on t and T.

If t_{0} and T_{0} are sufficiently small then M̂_{0} is very large, therefore (3.14) states that for fixed n (depending on t and T) M_{n} is very close to M̂_{0}. Then (3.12) gives a good Gaussian approximation of f_{n}(x) and (3.13) a good bound on its tail behaviour.

The proof of Proposition 1 is based on the observation that M_{0} almost agrees with the positive maximum M̂_{0} of the function q̄_{0}(x), f_{0}(x) is almost Gaussian, and we commit a small error by substituting the operator Q̄_{n,M} for small n by the operator T̂_{n},

T̂_{n}f_{n}(x) = C ∫ exp {−v^{2} − 2^{−n/2}u^{2}} f_{n}(x + u)f_{n}(x − u) du dv
= C\sqrt{π} ∫ exp \{−2^{−n/2}u^{2}\} f_{n}(x + u)f_{n}(x − u) du .

Since the proof is almost the same as the proof of the corresponding result for 1 < c < \sqrt{2} given in Section 4 of Part I in [2] we omit it. By the same reason we omit the proof of its Corollary formulated below. To formulate this result first we have to introduce the following notion:

Definition of the regularization of a function. Let us choose some fixed function φ(x) ∈ C_{0}^{∞}(R^{1}) such that 1 ≥ φ(x) ≥ 0 for all x ∈ R^{1}, φ(x) = 1 for |x| < 1, and φ(x) = 0 for |x| ≥ 2. Put φ_{n}(x) = φ (\frac{1}{100}2^{−n/2}x). Given some function f(x), f(x) ≥ 0, ∫ f(x) dx < ∞ we define its n-th regularization φ_{n}(f) as φ_{n}(f)(x) = \frac{1}{A_{n}} φ_{n}(x+B_{n})f(x+ B_{n}) with A_{n} = ∫ φ_{n}(x)f(x) dx and B_{n} = \frac{1}{A_{n}} ∫ xφ_{n}(x)f(x) dx, provided that the above formula is meaningful, i.e. A_{n} > 0.

Now we formulate the following

Corollary of Proposition 1. Under the conditions of Proposition 1 we have for all n ≤ N

|φ̃_{n}(f_{n})(t + is)| ≤ \frac{exp s^{2}}{1 + \frac{t^{2}}{200}} for |s| < 2, t ∈ R^{1} ,

and

|\frac{d^{j}}{dx^{j}} f_{n}(x)| ≤ 10^{5} exp \{− |2x + \frac{2^{−n/2}x^{2}}{M_{n}} |\} for x > −2^{n/2}M_{n}, j = 0, 1, 2.

Let us fix some positive integer N, and define the sequences α_{n}, β_{n}, n = N, N+1, …, as

α_{N} = \frac{1}{200} ,  (3.15)
α_{n+1} = (1 − 2^{−n/4}) α_{n} + \frac{10^{−12}}{M_{n}^{2}}  for n ≥ N  (3.15′)

and

β_{N} = 1 ,  (3.16)
β_{n+1} = (1 + 2^{−n/4}) β_{n} + \frac{10}{M_{n}^{2}}  for n ≥ N ,  (3.16′)

where M_{n} is defined in (1.8).

Now we define the following Properties I(n) and J(n).

Property I(n). Let n ≥ N. The function f(x) satisfies Property I(n) (with the starting index N and parameter C) if

|\frac{d^{j}}{dx^{j}} f(x)| ≤ \frac{C}{β_{n}^{(j+1)/2}} exp \{−\frac{1}{\sqrt{β_{n}}} |2x + \frac{2^{−n/2}x^{2}}{M_{n}} |\} for x > −2^{n/2}M_{n}, j = 0, 1, 2

with the above defined sequence β_{n} and the number M_{n} defined in (1.8).

Property J(n). Let n ≥ N. The function f(x) satisfies Property J(n) (with the starting index N) if

|φ̃_{n}(f)(t + is)| ≤ \frac{exp\{β_{n}s^{2}\}}{1 + α_{n}t^{2}} for |s| < \frac{2}{\sqrt{β_{n}}} , t ∈ R^{1} ,

with the above defined sequences α_{n} and β_{n}.

Now we formulate Proposition 2. The multiplying factor C and the starting index N can be chosen in Properties I(n) and J(n) in such a way that under the additional conditions M_{n} > K with some universal constant K, |M_{n} − M_{n−1}| < 1, 100n > β_{n} > max(9M_{n}^{−2}, 4^{−n}) Properties I(n) and J(n) for the function f_{n}(x) imply Properties I(n + 1) and J(n + 1) for the function f_{n+1}(x) (with the same parameters N and C). Also the following relations hold true:

M_{n+1} = M_{n} + 2^{−(n+1)/2}m_{n},  m_{n} = −\frac{\sqrt{2}}{4M_{n}} + γ(n)
with γ(n) < C_{1}2^{−n/2}\sqrt{β_{n}}  (3.17)

|\frac{d^{j}}{dx^{j}} f_{n+1}(x) − T_{M_{n}}φ_{n}(f_{n})(x)| ≤ \frac{C_{1}C^{4}}{β_{n+1}^{(j+1)/2}} 2^{−n/2}

[exp \{−\frac{1}{\sqrt{β_{n+1}}} |2x + \frac{2^{−(n+1)/2}x^{2}}{M_{n+1}} |\} + exp \{−\frac{2|x|}{\sqrt{β_{n+1}}} \}]

for x > −2^{(n+1)/2}M_{n+1}, j = 0, 1, 2,  (3.18)

and

|\frac{d^{j}}{dx^{j}} T_{M_{n}}φ_{n}(f_{n})(x)| ≤ \frac{C_{1}C^{2}}{β_{n+1}^{(j+1)/2}} exp \{−\frac{2|x|}{\sqrt{β_{n+1}}} \} , x ∈ R^{1}, j = 0, 1, 2, 3, 4 (3.19)

with some absolute constant C_{1}. As a consequence, if 0 < T < T_{0} and 0 < t < t_{0} with some suficiently small t_{0} > 0 and T_{0} > 0 then Properties I(n) and J(n) hold for the functions f_{n}(x) with some appropriate parameters C and N, and these functions satisfy relations (3.17)–(3.19). Also the relation β_{n} < 100n holds.

Proposition 2 is proved similarly to the analogous result for 1 < c < \sqrt{2} in Sections 5 and 6 in Part I of [2], only the number c must be replaced by \sqrt{2} everywhere. The expressions Q_{n,M}f(x), T_{M}φ_{n}(f)(x) and Q_{n,M}f(x) − T_{M}φ_{n}(f)(x) can be bounded with the help of Property I(n), as it is formulated in Proposition 3 and proved in Section 5 in Part I of [2]. This enables us to reduce the problem to the investigation of T_{M_{n}}φ_{n}(f_{n})(x), which can be done with the help of Property J(n) and formula (3.9′).

The only difference between the cases 1 < c < \sqrt{2} and c = \sqrt{2} is that for c = \sqrt{2} the condition β < 100 must be replaced by the condition β < 100n when the operator Q_{n,M} is investigated. This is so, because we apply our estimates with β = β_{n}, and the sequence β_{n} defined in (3.16), (3.16′) is of order const. n. (In the case 1 < c < \sqrt{2} it was bounded by a constant.) Nevertheless, this difference causes no problem. The condition β < 100 was applied in [2] for such arguments as to show that the estimate (5.4) of that work implies Lemma 3 for large x. To make such a conclusion we need some upper bound on β, but the estimate β < 100n is sufficient for our purposes.

Proposition 2 enables us to bound the error term ε_{n}(x) in (3.11) when the operator Q_{n,M_{n}} is applied for f_{n}(x). With the help of this estimate we can turn the heuristic argument after formula (3.11) into a rigorous proof.

4. The proof of Theorem A

We prove Theorem A by estimating the Fourier transforms φ̃_{n}(f_{n})(t). Let us fix some constants N and C in such a way that Propositions 1 and 2 hold with this choice of the parameters. Let us introduce the functions ψ_{n}(t) = log φ̃_{n}(f_{n})(t) and the numbers β̄_{n} = − \frac{d^{2}}{dt^{2}} ψ_{n}(t)|_{t=0}, provided that these quantities are well-defined, i.e. we can take logarithm in these expressions. We shall prove the following

Lemma 1. If 0 < t < t_{0}, 0 < T < T_{0} with some sufficiently small t_{0} > 0 and T_{0} > 0 then a)

β̄_{N} = \frac{a_{1}}{2(a_{0} − T)} + δ(N)  |δ(N)| ≤ 4^{−N}  (4.1)
β̄_{n+1} = β̄_{n} + \frac{1}{4M_{n}^{2}} + δ(n)  |δ(n)| ≤ 2^{−n/4}  for n ≥ N  (4.2)

b) For |t| < (\frac{n}{β_{n}} )^{1/3} and n ≥ N ψ_{n}(t) is well-defined, and

|\frac{d^{3}}{dt^{3}} ψ_{n}(t)| ≤ \frac{2}{M_{n}^{3}} + 2^{−n/4}  for |t| ≤ (\frac{n}{β̄_{n}} )^{1/3} and n ≥ N  (4.3)

Proof of Lemma 1. Because of Proposition 1 φ̃_{N}(f_{N})(t) is very close to the Fourier transform of the normal density function ϕ(x, σ) with σ^{2} = \frac{a_{1}}{2(a_{0}−T)}, and the analogous result also holds for its derivatives. This implies (4.2) and (4.3) for n = N, since if φ̃_{N}(f_{N})(t) were exactly normal then we would have β̄_{N} = \frac{a_{1}}{2(a_{0}−T)} and \frac{d^{3}}{dt^{3}} ψ_{N}(t) = 0. We prove (4.2′) and (4.3) in the general case by induction from n to n + 1.

Let us introduce the operator T̂_{n} by the formula T̂_{n}ψ(t) = log T̃_{M_{n}} exp ψ(t). It follows from (3.9′) that

− \frac{d^{2}}{dt^{2}} T̂_{n}ψ_{n}(t)|_{t=0} = \frac{1}{4M_{n}^{2}} + β̄_{n} ,  (4.4)

and

\frac{d^{3}}{dt^{3}} T̂_{n}ψ_{n}(t) = \frac{1}{\sqrt{2}}ψ_{n} (\frac{t}{\sqrt{2}}) + \frac{\sqrt{2}i}{16M_{n}^{3} (1 + \frac{it}{\sqrt{2}M_{n}} )^{3}} .  (4.4′)

Since M_{n} is very large, (4.4′) together with our inductive hypothesis imply that

|\frac{d^{3}}{dt^{3}} T̂_{n}ψ_{n}(t)| ≤ \frac{2}{M_{n+1}^{3}} + \frac{1}{\sqrt{2}}2^{−n/4} if |t| < (\frac{n + 1}{β̄_{n+1}})^{1/3} .  (4.5)

Because of the identities T̂_{n}ψ_{n}(0) = \frac{d}{dt} T̂_{n}ψ_{N}(t)|_{t=0} = 0 it follows from (4.4) and (4.5) that

ℜT̂_{n}ψ_{n}(t) ≥ − (β̄_{n} + \frac{1}{4M_{n}^{2}} ) \frac{t^{2}}{2} − (\frac{2}{M_{n+1}^{3}} + \frac{2^{−n/4}}{\sqrt{2}} ) \frac{|t|^{3}}{6} ≥ −\frac{n}{10}

for |t| ≤ (\frac{n + 1}{β̄_{n+1}})^{1/3} ,

where ℜ denotes real part. (Observe that 1 < β̄_{n} < n/10.) This relation implies that

|T̃_{M_{n}} φ̃_{n}(f_{n})(t)| ≥ e^{−n/10}  for |t| ≤ (\frac{n + 1}{β̄_{n+1}})^{1/3} .  (4.6)

We get similarly, by expressing the derivatives of T̃_{M_{n}} φ̃_{n}(f_{n})(t) through ψ_{n}(t) and its derivatives, that

|\frac{d^{j}}{dt^{j}} T̃_{M_{n}} φ̃_{n}(f_{n})(t)| ≤ e^{n/10}  for |t| ≤ (\frac{n + 1}{β̄_{n+1}})^{1/3} j = 1, 2, 3.  (4.6′)

On the other hand, some calculation with the help of (3.18) yields that

|\frac{d^{j}}{dt^{j}} φ̃_{n+1}(f_{n+1})(t) − \frac{d^{j}}{dt^{j}} T̃_{M_{n}} φ̃_{n}(f_{n})(t)| ≤ K2^{−n/2} for |t| ∈ R^{1} and j = 0, 1, 2, 3 .

(4.7)

By expressing \frac{d^{3}}{dt^{3}} ψ_{n+1}(t) and \frac{d^{3}}{dt^{3}} T̂_{n}ψ_{n}(t) by the corresponding Fourier transforms we get that relations (4.6), (4.6′) and (4.7) imply that

|\frac{d^{3}}{dt^{3}} ψ_{n+1}(t) − \frac{d^{3}}{dt^{3}} T̂_{n}ψ_{n}(t)| ≤ \frac{1}{100}2^{−n/4}.

The last relation together with (4.5) imply (4.3) for n + 1.

It can be proved similarly that

|\frac{d^{2}}{dt^{2}} ψ_{n+1}(t)|_{t=0}−\frac{d^{2}}{dt^{2}} T̂_{n}ψ_{n}(t)|_{t=0}| ≤ 2^{−n/4}

which relation together with (4.4) imply (4.2′) for n + 1. Lemma 1 is proved.

The proof of Theorem A′. It follows from Lemma A that

φ̃_{n}(f_{n})(t) = exp \{−\frac{β̄_{n}}{2} t^{2} + R_{n}(t)t^{3}\} with |R_{n}(t)| < \frac{2}{M_{n}^{3}} + 2^{−n/4} if t < (\frac{n}{β̄_{n}} )^{1/3} .

Hence

|∫_{|t|<( \frac{n}{β̄n} )^{1/3}} e^{−itx} [e^{− \frac{β̄nt^{2}}{2}} − φ̃_{n}(f_{n})(t)] dt| ≤ 2 (\frac{2}{M_{n}^{3}} + 2^{−n/4}) \frac{1}{β̄_{n}^{2}}  (4.8)

On the other hand

|∫_{|t|>( \frac{n}{β̄n} )^{1/3}} e^{−itx}e^{− \frac{β̄nt^{2}}{2}} dt| ≤ exp \{−\frac{n^{2/3}}{2} β̄_{n}^{1/3}\} ,  (4.8′)

and by Property J(n) and the relation α_{n} > 10^{−14}β̄_{n}

|∫_{|t|>( \frac{n}{β̄n} )^{1/3}} e^{−itx}φ̃_{n}(f_{n})(t) dt| ≤ 10^{14}n^{−1/3}β̄_{n}^{−2/3}.  (4.8′′)

Relations (4.8), (4.8′) and (4.8′′) imply that

|φ_{n}(f_{n})(x) − \frac{1}{\sqrt{2πβ̄_{n}}} exp \{−\frac{x^{2}}{2β̄_{n}} \}|

= |∫ e^{−itx} [φ̃_{n}(f_{n})(t) − exp{−β̄_{n}t^{2}} dt]|

≤ \frac{2}{β̄_{n}^{2}} (\frac{2}{M_{n}^{3}} + 2^{−n/4}) + 10^{14}n^{−1/3}β̄_{n}^{−2/3} + exp \{−\frac{n^{2/3}}{2} β̄_{n}^{1/3}\}

≤ \frac{1}{β̄_{n}^{2}} (\frac{4}{M_{n}^{2}} + 2^{−n/4}) + 2 · 10^{14}n^{−1/3}β̄_{n}^{−2/3}.  (4.9)

In relation (4.9) φ_{n}(f)(x) can be replaced by f_{n}(x), since for |x| < 2^{n/2} they are very close to each other by (3.8), and for |x| > 2^{n/2} both terms at the left-hand side of (4.9) are negligible small. (The norming constants A_{n} and B_{n} appearing in the regularization are almost 0 and 1.)

Hence (4.9) implies that

|f_{n}(x) − \frac{1}{\sqrt{2πβ̄_{n}}} exp \{−\frac{x^{2}}{2β̄_{n}} \}| ≤ \frac{1}{β̄_{n}} (\frac{4}{M_{n}^{2}} + 2^{−n/5} + 10^{15} (\frac{β̄_{n}}{n} )^{1/3}) for n ≥ N .

(4.10)

Since |β̄_{n} − \frac{n}{4M^{2}} | < 10, hence

|\frac{1}{\sqrt{2πβ̄_{n}}} exp \{−\frac{1}{2β̄_{n}} x^{2}\} − \frac{\sqrt{2}M}{\sqrt{πn}} exp \{−\frac{2M^{2}}{n} x^{2}\}| ≤ \frac{const.}{n} .  (4.10′)

For large n the term \frac{1}{β̄_{n}} can be replaced by \frac{5M^{2}}{n} in (4.10), hence (4.10) and (4.10′) imply (3.4). Relation (3.5) holds because of Property I(n), and relations (3.2) and (3.3) can be deduced from Proposition 2 in the same way as the analogous result in [2] in Lemma 10 of Part I. Theorem A′ is proved.

The proof of Theorem A. By Theorem A′ and (3.1)

2^{−n/2}\sqrt{n}q̄_{n}(x, T) = \frac{\sqrt{2}M}{\sqrt{π}} exp \{−\frac{2^{n+1}M^{2}}{n} (x − M_{n})^{2}\} + r_{n}(x)  (4.11)

with

|r_{n}(x)| < \frac{K}{\sqrt{n}} .

We have to check that an error of order O (\frac{1}{\sqrt{n}}) is committed if M_{n} is replaced by M in (4.11). We have

|exp \{−\frac{2^{n+1}M^{2}}{n} (x − M)^{2}\} − exp \{−\frac{2^{n+1}M^{2}}{n} (x − M_{n})^{2}\}|

≤ exp \{−\frac{2^{n+1}M^{2}}{n} (x − M)^{2}\} \frac{2^{n+1}}{n} M^{2} |(x − M)^{2} − (x − M_{n})^{2}|

≤ \frac{2^{n+1}M^{2}}{n} |M − M_{n}| (2|x − M| + 2|M − M_{n}|) exp \{−\frac{2^{n+1}}{n} M^{2}(x − M)^{2}\}

≤ \frac{K}{\sqrt{n}} ,

since |M − M_{n}| < CM^{−1}2^{−n/2}. This estimate together with (4.11) imply (2.3). The remaining statements of Theorem A also follow from Theorem A′.

Theorem A gives a good Gaussian approximation only for large n. On the other hand, the error term in (4.10) is small for all n ≥ N. Beside this, Proposition 1 yields a good Gaussian approximation for all n ≤ N if M̂_{0} is very large. These observations imply the following

Corollary of Theorem A′. Define the sequence β̄_{n} by (4.1) and (4.2) for n ≥ N and β̄_{n} = β̄_{N} for n ≤ N. For all δ > 0 some positive integer N and thresholds t_{0} > 0 and T_{0} > 0 can be chosen in such a way that

|f_{n}(x) − \frac{1}{2\sqrt{πβ̄_{n}}} exp \{−\frac{1}{2β̄_{n}} x^{2}\}| < δ  for all n ≥ 0 and x ∈ R^{1}  (4.12)

if 0 < t < t_{0} and 0 < T < T_{0}. As a consequence, for arbitrary L > 0 the inequality

f_{n}(x) < \frac{10}{\sqrt{β̂_{n}}} exp \{−\frac{1}{2β̂_{n}} x^{2}\}  for |x| < L\sqrt{β̂_{n}}, n = 0, 1, 2, …  (4.13)

holds with the sequence

β̂_{n} = 10 + \frac{n}{M_{n}^{1/2}}  for 0 ≤ n ≤ N  (4.14)
β̂_{n+1} = β̂_{n} (1 + 2^{−n/4}) + \frac{1}{8M_{n}^{2}}  for n ≥ N  (4.14′)

if the conditions of (4.12) hold with a sufficiently small δ = δ(L).

5. The proof of Proposition A

First we prove formula (2.6). Choose an appropriately small ε > 0 and a large L = L(ε) > 0. We are going to show that if 0 < t < t_{0} and 0 < T < T_{0} with some t_{0} = t_{0}(ε, L), and T_{0} = T_{0}(ε, L) then

f_{n}(x) ≤ \frac{10}{\sqrt{β̂_{n}}} exp \{−\frac{1}{2β̂_{n}} x^{2}\} for |x| < L\sqrt{β̂_{n}}, n = 0, 1, 2, …  (5.1)

and

f_{n}(x) ≤ \frac{ε}{\sqrt{β̂_{n}}} exp \{−\frac{1}{4β̂_{n}} x^{2}\} for |x| > L\sqrt{β̂_{n}}, n = 0, 1, 2, ….  (5.1′)

Since lim_{n→∞} \frac{β̂_{n}}{n} = \frac{1}{8M^{2}} , relations (5.1) and (5.1′) imply (2.6). Because of the Corollary of Theorem A′ we may assume that relation (5.1) and relation (5.1′) for

L\sqrt{β̂_{n}} < |x| < 3L\sqrt{β̂_{n}} hold. It is enough to apply this Corollary for 3L, and to choose

L in such a way that exp\{−\frac{L^{2}}{4} \} < \frac{ε}{10}. Moreover, it can be seen from the form of f_{0}(x)

that for n = 0 (5.1′) holds for all x > L\sqrt{β̂_{0}}. Hence it is enough to prove (5.1′) for

x > 2L\sqrt{β̂_{n}} by induction from n to n + 1. We shall do it with the help of the following

Lemma 2. If ε > 0 and L > L(ε) > 0 are appropriately chosen (in dependence of the number C appearing in the conditions of this Lemma), n is some non-negative integer, M > K > 0 with an appropriate K > 0 and

f(x) ≤ \frac{10}{\sqrt{β}} exp \{−\frac{1}{β} x^{2}\} for |x| < L\sqrt{β}  (5.2)
f(x) ≤ \frac{ε}{\sqrt{β}} exp \{−\frac{1}{2β} x^{2}\} for |x| > L\sqrt{β}  (5.2′)
f(x) ≤ \frac{C}{\sqrt{β}} for all x ∈ R^{1}  (5.2′′)

then

Q̄_{n,M}f(x) ≤ \frac{ε^{3/2}}{\sqrt{β}} exp \{−\frac{1}{2β} x^{2}\} for x > 2L\sqrt{β}

Proof of Lemma 2. The proof applies the same ideas as that of Lemma 19 in Part I of [2]. Let us introduce the functions

ℓ_{n,M}^{±}(x, u, v) = 2^{n/2} (\sqrt{(M + 2^{−(n+1)/2}x ± 2^{−n/2}u)^{2} + 2^{−n/2}v^{2}} − M) ,

P (x, u) = ∫ exp {−v^{2}} f (ℓ_{n,M}^{+}(x, u, v)) f (ℓ_{n,M}^{−}(x, u, v)) dv  (5.3)

and

P (x) = P (x, 0) .  (5.3′)

Then

Q̄_{n,M}f(x) = 2 ∫_{0}^{∞} exp {−2^{−n/2}u^{2}} P (x, u) du ,  (5.4)

and by the Schwarz inequality

P (x, u) ≤ [P (x + \sqrt{2}u)P (x − \sqrt{2}u)]^{1/2} .  (5.5)

Let us estimate P (x). It follows from (5.2)–(5.2′′) and the inequality ℓ_{n,M}^{±}(x, 0, v) ≥ ℓ_{n,M}^{±}(x, 0, 0) that

P (x) ≤ \frac{ε^{2}}{β} \sqrt{π} exp \{−\frac{1}{2β} x^{2}\} for x > \sqrt{2β}L  (5.6)
P (x) ≤ \frac{100}{β}\sqrt{π} exp \{−\frac{1}{β} x^{2}\} for |x| < \sqrt{2β}L  (5.6′)
P (x) ≤ \frac{C^{2}\sqrt{π}}{β} for all x ∈ R^{1}.  (5.6′′)

These estimates together with (5.4) and (5.5) imply that for x ≥ 2L\sqrt{β}

Q̄_{n,M}f(x) ≤ 2 ∫_{0}^{\frac{x}{\sqrt{2}} −L\sqrt{β}} \frac{ε^{2}\sqrt{π}}{β} exp \{−\frac{x^{2}}{2β} − \frac{u^{2}}{β} \} du

+ 2 ∫_{\frac{x}{\sqrt{2}} −L\sqrt{β}}^{\frac{x}{\sqrt{2}} +L\sqrt{β}} \frac{10ε}{β}\sqrt{π} exp \{−\frac{(x − \sqrt{2}u)^{2}}{β} − \frac{(x + \sqrt{2}u)^{2}}{2β} \} du

+ 2 ∫_{\frac{x}{\sqrt{2}} +L\sqrt{β}}^{∞} \frac{Cε}{β}\sqrt{π} exp \{−\frac{(x + \sqrt{2}u)^{2}}{2β} \} du ≤ \frac{ε^{3/2}}{\sqrt{β}} exp \{−\frac{1}{2β} x^{2}\}

if L = L(ε) is sufficiently large. Lemma 2 is proved.

Let us apply Lemma 2 with f(x) = f_{n}(x), β = 2β̂_{n} and M = M_{n}. Since f_{n+1}(x) = Q_{n,M_{n}}f_{n}(x) ≤ C_{1}Q̄_{n,M_{n}}f_{n}(x + m_{n}) with some C_{1} > 0 hence in order to carry out our inductive procedure it is enough to show that

C_{1}\sqrt{ε} exp \{−\frac{1}{4β̂_{n}} (x + m_{n})^{2}\} ≤ exp \{−\frac{1}{4β̂_{n+1}} \} .

This can be deduced from the inequality

β̂_{n+1}(x + m_{n})^{2} + Kβ̂_{n}β̂_{n+1} ≥ β̂_{n}x^{2}  (5.7)

with sufficiently large K > 0 if ε > 0 is chosen sufficiently small. Since |m_{n}| < CM_{n}^{−1} for n ≤ N, |m_{n}| < \frac{1}{2M} + C_{1}2^{−n/2} for n > N and β̂_{n} > 10 one gets formula (5.7) with the help of simple calculation from (4.13) and (4.13′).

The proof of formulas (2.7) and (2.7′) is based on the following Lemma 3. Let the function f(x) satisfy the conditions of Lemma 2. Let some numbers r > 0, β > 0 and α > 0 be given in such a way that r > β > 10, β < \frac{9}{10}Mn and \frac{1}{100} < α < 1 − ε^{1/8}. Let us assume that the function f(x) satisfies, beside the conditions of Lemma 2, the estimates

f(x) ≤ \frac{ε}{\sqrt{β}} exp \{−\frac{1}{2β} x^{2}\}  for − r < x < −L\sqrt{β}  (5.8)
f(x) ≤ \frac{ε}{\sqrt{β}} exp \{−\frac{1}{2β} r^{2} − α2^{(n−1)/2} [(M − 2^{−n/2}r)^{2} − (M + 2^{−n/2}x)^{2}]\}
for − 2^{−n/2}M < x < −r  (5.8′)

Put ᾱ = min((1 + ε)α, 1 − ε^{1/8}), ¯ ᾱ = (1 + ε^{1/8})ᾱ and

r̄ = \frac{\sqrt{2}¯ᾱβM}{1 + ¯ ᾱβ2^{−n/2}} .  (5.8′′)

If r̄ < \sqrt{2}r then

Q̄_{n,M}f(x) ≤ \frac{ε^{3/2}}{\sqrt{β}} exp \{−\frac{1}{2β} x^{2}\}  for − r̄ < x < −2L\sqrt{β}  (5.9)
Q̄_{n,M}f(x) ≤ \frac{ε^{3/2}}{\sqrt{β}} exp \{−\frac{1}{2β} r̄^{2} − ᾱ2^{n/2} [(M − 2^{−(n+1)/2}r̄)^{2} − (M + 2^{−(n+1)/2}x)^{2}]\}
for − 2^{(n+1)/2}M < x < −r̄.  (5.9′)

The proof of Lemma 3 is similar to that of Lemma 2. The main difference is that in Lemma 2, i.e. when x > 0, the main contribution to the integral Q̄_{n,M}f(x) is given in a small neighbourhood of the point (u, v) = (0, 0). For x < 0 this statement remains valid only for x > −r̄. For x < −r̄ the main contribution to this integral is given in a small neighbourhood of the points (u, v) = (0, ±v^{∗}) with v^{∗2} = 2^{n/2} {(M − 2^{−(n+1)/2}r̄)^{2} − (M + 2^{−(n+1)}x)^{2}}.

Proof of Lemma 3. Define the function

K(x) = \{ \begin{matrix}\frac{10}{\sqrt{β}} exp \{−\frac{1}{β} x^{2}\}  for |x| < −\sqrt{2β}L \\ \frac{ε}{\sqrt{β}} exp \{−\frac{1}{2β} x^{2}\}  for x > L\sqrt{β} or − r < x < −\sqrt{2β}L \\ \frac{ε}{\sqrt{β}} & exp & \{−\frac{1}{2β} r^{2} − α2^{(n−1)/2} [(M − 2^{−n/2}r)^{2} − (M + 2^{−n/2}x)^{2}]\} \\ for − 2^{n/2}M < x < −r. \end{matrix}

Some calculation shows that for fixed x the function

K̄(x, v) = exp{−¯ᾱv^{2}}K^{2}(ℓ_{n,M}^{±}(x, 0, v))

takes its maximum in the point v = 0 for x > −\frac{r̄}{\sqrt{2}} and in the points ±v^{∗} satisfying the equation ℓ_{n,M}^{±}(x, 0, v^{∗}) = −\frac{r̄}{\sqrt{2}} for x < −\frac{r̄}{\sqrt{2}}. (At this point we need the condition r̄ < \sqrt{2}r which guarantees that the estimate (5.8) holds in the point \frac{r̄}{\sqrt{2}}.) The function P (x) defined in formula (5.3) can be estimated in the following way:

P (x) ≤ ∫ exp \{−ε^{1/8}v^{2}\} K̄(x, v) dv ≤ ε^{−1/4}\sqrt{π} sup_{v} K̄(x, v).

Hence we obtain that

P (x) ≤ \frac{ε^{7/4}\sqrt{π}}{β} exp \{−\frac{1}{2β} x^{2}\}  for − r̄ < x < −\sqrt{2β}L  (5.10)
P (x) ≤ \frac{ε^{7/4}π}{β} exp \{−\frac{1}{2β} r̄^{2} − ¯ ᾱ2^{n/2} [(M − 2^{−(n+1)/2}r̄)^{2} − (M + 2^{−(n+1)/2}x)^{2}]\}
for − 2^{(n+1)/2}M < x < −r̄.  (5.10′)

We estimate the integral in (5.4) with the help of (5.5), (5.6), (5.6′), (5.10) and (5.10′). Let us first consider the case −2^{(n+1)/2}M < x < −r̄ and integrate in the domain \{u > 0, x + \sqrt{2}u < −r̄\}. This integral can be estimated in the following way:

∫_{\{u>0,x+\sqrt{2}u<−r̄\}} P (x, u) du

≤ \frac{ε^{7/4}\sqrt{π}}{β} exp \{−\frac{1}{2β} r̄^{2} − ¯ ᾱ2^{n/2} [(M − 2^{−(n+1)/2}r̄)^{2} − (M + 2^{−(n+1)/2}x)^{2}]\}

∫_{\{u>0,x+\sqrt{2}u<−r̄\}} exp \{−2^{−n/2}(1 − ¯ ᾱ)u^{2}\} du .  (5.11)

We give an upper bound on the right-hand side of (5.11) by replacing ¯ ᾱ with ᾱ in it and multiplying the expression by exp{−(¯ᾱ − ᾱ)|r̄+ x|}. The integral in this expression can be estimated by the rather rough bound |r̄ + x|. These estimates show that the right-hand side of (5.11) is much less than the expression at the right-hand side of (5.9′). To estimate the integral ∫ P (x, u) du in the case −2^{(n+1)/2}M < x < −r̄ in the domain \{x + \sqrt{2}u > −r̄\} observe that some calculation yields that

exp \{−\frac{1}{2β} x^{2}\} = exp \{−\frac{1}{2β} r̄^{2} − ¯ ᾱ2^{n/2} [(M − 2^{−(n+1)/2}r̄)^{2} − (M + 2^{−(n+1)/2}x)^{2}]\}
exp \{− (\frac{1}{2β} + 2^{−(n+2)/2}¯ᾱ) (x + r̄)^{2}\} ,  (5.12)

because of the definition of r̄.

Because of this identity the estimates (5.10) and (5.10′) enable us to estimate the integral ∫ P (x, u) du in this case similarly to the estimation of (5.11), only in this case the last term in (5.12) helps us to bound the pre-exponential term. Similar calculations enable us to bound the integral (5.4) for x > −r̄ and to deduce the estimates (5.9) and (5.9′). Lemma 3 is proved.

Formulas (5.8) and (5.8′) hold for f(x) = f_{0}(x) with β = 2β̂_{0} = 20, α = \frac{1}{100}, M = M_{0} and r = \sqrt{2}αβM. If the conditions of Lemma 3 are satisfied for f_{n}(x) with M = M_{n}, β = 2β̂_{n} and some α_{n} and r_{n} then Lemma 3 gives an estimate on Q̄_{n,M_{n}}f_{n}(x). An argument similar to that given after Lemma 2 gives an estimate when the operator Q̄_{n,M_{n}} is replaced by Q_{n,M_{n}}. In such a way we get by induction the estimates (5.8) and (5.8′) for f_{n}(x) with β = 2β̂_{n}, an increasing sequence α_{n} which tends to 1 − ε^{1/8} and a number r_{n} which is a small perturbation of the expression given in (5.8′′). Since \frac{β̂_{n}}{n} has a positive limit as n → ∞, the number r = r_{n} which appears in the estimates (5.8) and (5.8′) for f_{n}(x) during this induction has the order n. By rewriting these estimates for q̄_{n}(x) with the help of (3.1) we obtain the estimates (2.7) and (2.7′) (with ε^{1/8} instead of ε).

6. The proof of Theorem B

The proof of Proposition B is the same as that of Lemma 1 in Part II of [2], hence we omit it. The proof of Theorem B is also very similar to the method of Part II in [2], only the number c must be replaced by \sqrt{2} and M by the constant M̄ defined in (1.9) everywhere. The main difference is that now we have a weaker control about the tail behaviour of the density function of the average spin p_{n}(x). As a consequence, we can prove some estimates only in a weaker form. Nevertheless, they are sufficient for our purposes.

Let us discuss this question in more detail. Introduce the functions p̄_{n}(x) and g_{n}(x), x ∈ R^{1} as

p̄_{n}(x) = K_{n} exp \{\frac{a_{0}}{2a_{1}} 2^{n/2}M^{2}\} p_{n}(x̃),  x̃ = (x, 0) ∈ R^{2},  (6.1)
g_{n}(x) = 2^{−n/2}p̄_{n} (M̄ + 2^{−n/2}x) ,  (6.2)

where p_{n}(x) is defined after formula (1.6), the number M̄ in (1.9), and K_{n} is the same norming constant as in (1.7). By formula (1.7)

g_{n}(x) = 2^{−n/2} exp \{−\frac{a_{0}}{2T} x (2M̄ + 2^{−n/2}x)\} q̄_{n} (M + 2^{−n/2}\sqrt{\frac{a_{1}}{T}} x) ,

hence Theorem A yields that

g_{n}(x) = \frac{1}{\sqrt{n}} exp \{−\frac{a_{0}}{2T} x (M̄ + 2^{−n/2}x)\} [\frac{\sqrt{2}M}{\sqrt{π}} exp \{−\frac{2M̄^{2}}{n} x^{2}\} + R_{n}(x)] (6.3)

with

|R_{n}(x)| ≤ \frac{K}{\sqrt{n}}.  (6.3′)

On the other hand, we get by rewriting Proposition A for g_{n}(x) that there are some numbers B > 0, D > 0 and R_{n}, −C_{1}n < R_{n} < −C_{2}n with some C_{1} > C_{2} > 0 such that

g_{n}(x) ≤ \frac{K}{\sqrt{n}} exp \{−\frac{a_{0}}{2T} x (2M̄ + 2^{−n/2}x) − \frac{B}{n} x^{2}\}  for x > R_{n}  (6.4)

and

\begin{aligned}g_{n}(x) & ≤ \frac{K}{\sqrt{n}} exp\{−\frac{a_{0}}{2T} R_{n} (2M̄ + 2^{−n/2}R_{n}) − \frac{B}{n} R_{n}^{2} \\ & − D(R_{n} − x) (2M̄ + 2^{−n/2}(R_{n} + x))\} for − 2^{−n/2}M̄ < x < R_{n}. \end{aligned} (6.4′)

(We have to choose B = \frac{βa_{1}}{T}, R_{n} = 2^{n/2}\sqrt{\frac{T}{a_{1}}} (r_{n} − M) and D = (1 − ε − \frac{a_{0}}{2a_{1}} ) \frac{a_{1}}{T} in Proposition A. We may assume that D > 0 by choosing ε in (2.7) sufficiently small.)

The estimates (6.4) and (6.4′) are the natural counterparts of the estimates (4.11′) and (4.11′′) in Part II of [2]. The function f_{n}(x) defined by formula (4.11) of that work is the analogue of our function g_{n}(x).

The bound given on g_{n}(x) decreases at infinity slower than its counterpart in [2] because of the multiplying term 1/n in formula (6.4). Another, and even more important difference between the two cases is that in the points x ∼ −const. n relations (6.4) and (6.4′) give no better bound on the function g_{n}(x) than exp{Cn} with some positive C > 0. As a consequence, in several estimates we have to multiply the upper bound by an exponential term instead of a constant, as it is the case in [2]. But this estimates suffice for us, because in the final estimates we have a double exponential term which compensates this effect.

Applying the same argument as in [2] we get that Theorem B follows from an analogue of Proposition 1′ in Part II of [2] which is obtained if c is replaced by \sqrt{2} and M by M̄ in this result. For the sake of convinience we also make the following modification. From now on we shall work with the function K_{n} exp \{\frac{a_{0}}{2a_{1}} 2^{n/2}M^{2}\} p_{n}(x) instead of the original function p_{n}(x) and we denote it in the same way. This modification influences only the norming constant L_{n} in the Radon–Nikodym derivative.

The proof of this modified version of Proposition 1′ of [2] is very similar to the original one. We have to estimate certain integral expressions in the domains Ω_{n}^{i}, i = 1 2, 3, defined in (2.14)–(2.14′′). We rewrite these integrals in polar coordinate system and first estimate the integrals on a circle of fixed radius r. This can be done in the same way as in [2]. Then the integrals with respect to r can be estimated with the help of formulas (6.4) and (6.4′) instead of formulas (4.11′) and (4.11′′) in [2]. We get in such a way slightly weaker estimates than those in [2], but they suffice for our purposes. Lemmas 2 and 3 of Part II of [2] remain valid after the replacement of c and M by \sqrt{2} and M̄ in the following weaker form: In Lemma 3 the multiplying term K and in Part a) of Lemma 2 the multiplying term c^{n} before the exponent must be replaced by K^{n}, where K is some appropriate constant depending on t and T. Also the estimates of Section 5 of Part II of [2] remain valid. The only place where the argument of the proof must be slightly changed is Part a) where S_{n}^{1}f(x) is estimated for x ∈ Ω_{n}^{1}. The argument of [2] works if we show that the expressions J_{n,ε̄}(x_{1}) defined by the formula

J_{n,ε̄}(x^{(1)}) = ∫_{|t|<ε̄2^{0.3n}} exp \{\frac{tx^{(1)}}{T} + \sqrt{2}\frac{ḡ_{n+1}}{2} t\} g_{n}(t) dt  (6.5)

with some sufficiently small ε̄ > 0 satisfy the following relations:

J_{n,ε̄}(x^{(1)}) = (1 + O(2^{−0.1n})) J_{n,ε̄}(M)  if x ∈ Ω_{n}^{1} ,  (6.6)

and

J_{n,ε̄}(M̄) > K_{1} > 0.  (6.7)

Relation (6.7) simply follows from (6.3) if we restrict the domain of integration in (6.5) to the domain |t| < \frac{1}{3M} \sqrt{n log n}. (The corresponding estimate (5.11) in Part II of [2] also contained an upper bound on J_{n,ε̄}(M̄), but we do not need this bound.) Then relation (6.5) follows from the following observations: The ratio of the integrands in the expressions J_{n,ε̄}(x^{(1)}) and J_{n,ε̄}(M̄) are closer to 1 than const.2^{−0.05n} if |t| < 2^{0.05n} and x^{(1)} ∈ Ω_{n}^{1} and therefore |x^{(1)} − M̄| < 2^{−0.2n}, and the contribution of the domain |t| > 2^{0.05n} to these integrals is less than exp{−const.2^{0.05n}}. The remaining part of the proof works with some natural modification of the proof given in [2], hence we omit it.

7. The proof of Theorems 1 and 2

To prove Theorem 1 first we show that for all q, 2^{−0.1} < q < 1 there are some thresholds n_{0} and N_{0}(n, q) such that if n ≥ n_{0} and N ≥ N_{0}(n, q) then

\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})  (7.1)

with

f_{n,N}^{h_{N}}(x) = L_{n} exp {ḡ2^{n/2}(x^{(1)} − M̄) + Ā2^{n/2}x^{(2)2} + ε_{n}(x)}  for x ∈ Ω_{n}^{1} , (7.2)

where

sup_{x∈Ω_{n}^{1}} |ε_{n}(x)| ≤ q^{n} ,  (7.2′)

f_{n,N}^{h_{N}}(x) ≤ L_{n} exp \{ḡ2^{n/2}(|x| − M̄) − (\frac{ḡ}{2M̄} − Ā)2^{0.1n} + q^{n}\}  for x ∈ Ω_{n}^{2} (7.3)

f_{n,N}^{h_{N}}(x) ≤ L_{n} exp \{\frac{ḡ2^{n/2}}{M̄} (|x|^{2} − M̄^{2})\}  if x > M̄ + 2^{−0.2n}  (7.4)

f_{n,N}^{h_{N}}(x) ≤ L_{n} exp \{\frac{ḡ2^{n/2}}{2M̄} (|x|^{2} − M̄^{2})\}  if 0 < x < M̄ − 2^{−0.2n}  (7.4′)

with the numbers ḡ and Ā appearing in Proposition B and some appropriate norming constant L_{n} which satisfies the relation

C_{1} < 2^{−n/4}L̄_{n} < C_{2}  with some 0 < C_{1} < C_{2} < ∞ ,  (7.5)

where L̄_{n} = L_{n}K_{n} exp \{\frac{a_{0}}{2T} 2^{n/2}M̄^{2}\} with the same norming constant K_{n} as in (1.9).

In the proof of this statement we argue just as in Section 6 of Part II of [2]. Because of Theorem B and Proposition B the constants g_{n} and A_{n} can be replaced by ḡ2^{n/2} and Ā2^{n/2} in (2.15)–(2.17) by slightly changing the error terms. To show that L_{n} = L_{n}(N, h_{N}) can be chosen independently of N and h_{N} we observe that a calculation analogous to that in Section 6 of [2] yields that

μ_{n,N}^{h_{N}} (Ω_{n}^{2} ∪ Ω_{n}^{3}) ≤ L_{n}(N, h_{N}) exp {−K2^{0.3n}}  (7.6)

and for

T_{n} = ∫_{Ω_{n}^{1}} exp {ḡ2^{n/2}(x^{(1)} − M̄) + Ā2^{n/2}x^{(2)2}} p_{n}(x) dx .  (7.7)

the relation

μ_{n,N}^{h_{n}}(Ω_{1}^{n}) = L_{n}(N, h_{N})T_{n} (1 + O(q^{n})) ,  2^{−0.1} < q < 1,  (7.8)

holds. The estimate

C_{1}2^{−n/4} < K_{n} exp \{\frac{a_{0}}{2T} 2^{n/2}M̄^{2}\} T_{n} < C_{2}2^{−n/4} with some 0 < C_{1} < C_{2} < ∞ . (7.9)

also holds true.

The proof of (7.9) is similar to that of (6.9) in [2], only one has to observe that ḡ equals \frac{a_{0}M̄}{T}, i.e. −1 times the coefficient of x in (6.3)–6.4′), and this causes some cancellation if we express p_{n}(x) through g_{n}(x) in the integral (7.7). Since μ_{n.N}^{h_{n}}(R^{2}) = 1, relations (7.6) and (7.8) imply that L_{n} can be chosen as T_{n}^{−1}, and then (7.9) implies (7.5). Theorem 1 can be proved with the help of this information in the following way:

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 {ḡ2^{−n/2} ∑_{j=1}^{2^{n}}(x_{j}^{(1)} − M̄) + Ā2^{−3n/2}(∑_{j=1}^{2^{n}} x_{j}^{(2)})^{2}} ∏_{j=1}^{2^{n}} p(x_{j}) dx_{j}

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

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

with some K > 0 independent of the set A.

Theorem 1 can be proved with the help of the above relation similarly to [2]. Moreover, this argument also yields the following

Corollary of Theorem 1. Let μ̄_{n} denote the projection of of the measure μ̄ constructed in Theorem 1 to (R^{2})^{2^{n}}. There is some function f̄_{n}(x) such that

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

Let n > n_{0} with some threshold n_{0} > 0. Then relations (7.1)—(7.5) remain valid if f_{n,N}^{h_{N}} is replaced by f̄_{n}(x) in them.

Now we turn to the proof of Theorem 2. Let us introduce the Hamiltonian H_{k} in the volume (R^{2})^{2^{k}} by the formula

H_{k}(x_{1}, …, x_{2^{k}}) = − ∑_{i=1}^{2^{k}−1} ∑_{j=i+1}^{2^{k}} d(i, j)^{−3/2}x_{i}x_{j} .

Let σ = {σ(j) = (σ_{1}(j), σ_{2}(j)), j ∈ Z} be a μ̄ distributed vector and consider the random vector {(R_{n}σ^{(1)}(j), R_{n}σ^{(2)}(j)) , 1 ≤ j ≤ 2^{k}} defined by formulas (1.3)–(1.5) with A_{n} = 2^{n/2}\sqrt{n} and B_{n} = 2^{3n/4}. The argument at the beginning of Section 7 in Part II of [2] also shows that the density function h_{n,k}(x_{1}, …, x_{2^{k}}) of this vector can be expressed in the following way:

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}(2^{n/4}x̃_{1}, …, 2^{n/4}x̃_{2^{k}})\} ∏_{j=1}^{2^{k}} p_{n}(x̃_{j})

(7.10)

with

x̃ = x̃(x) = (M̄ + 2^{−n/2}\sqrt{n}x^{(1)}, 2^{−n/4}x^{(2)})  for x = (x^{(1)}, x^{(2)}) .  (7.10′)

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

W̄_{n} = \{(x^{(1)}, x^{(2)}), M̄ − \frac{\sqrt{T}}{8M̄} 2^{−n/2}\sqrt{n log n} < |x| < M̄ + \frac{\sqrt{T}}{8M̄} 2^{−n/2}\sqrt{n log n},
|x^{(2)}| < 2^{−n/4}n^{1/5}, x^{(1)} > 0}
W_{n} = {(x^{(1)}, x^{(2)}), x̃(x) ∈ W̄ } ,

We claim that for all j = 1, 2, …, 2^{k}

P ( (R_{n}σ^{(1)}(j), R_{n}σ^{(2)}(j) ) ∉ W_{n}) ≤ n^{−1/100}  if n ≥ n_{0}.  (7.11)

and

h_{n,k}(x_{1}, …, x_{2^{k}}) = h_{k}(x_{1}, …, x_{2^{k}}) (1 + O(n^{−1/9}))
if x_{j} ∈ W_{n} for all j = 1, 2, …, 2^{k},  (7.12)

where h_{k}(x_{1}, …, x_{2^{k}}) is the function defined in (1.11) (with s = 2).

Relations (7.11) and (7.12) together imply Theorem 2. Relation (7.12) can be proved with the help of the following estimates:

p_{n}(x) = K_{n} [exp \{−\frac{2^{n+1}M^{2}a_{1}}{nT} (|x| − M̄)^{2}\} + O (\frac{1}{\sqrt{n}})] exp \{−\frac{a_{0}}{2T} 2^{n/2}x^{2}\}

= K_{n} exp \{−\frac{2^{n+1}M^{2}a_{1}}{nT} (|x| − M̄)^{2} − \frac{a_{0}}{2T} 2^{n/2} (x^{(1)2} + x^{(2)2}) + O (n^{−1/9})\}

= K̄_{n} exp\{−\frac{2^{n+1}M^{2}a_{1}}{nT} (x^{(1)} − M̄)^{2} − \frac{a_{0}}{2T} 2^{n/2} (2M̄(x^{(1)} − M̄) + x^{(2)2})

+ O (n^{−1/9})}  if x ∈ W̄_{n},  (7.13)

since in this case we can put the O(·) term into the exponent by appropriately decreasig the power of n in it,

\frac{2^{n}}{n} (|x| − M̄)^{2} = \frac{2^{n}}{n} (x^{(1)} − M̄)^{2} + O (n^{−3/10}\sqrt{log n})  for x ∈ W̄_{n} ,  (7.14)

and

2^{n/2}x^{(1)2} = 2^{n/2}M̄^{2} + 2 · 2^{n/2}M̄(x^{(1)} − M̄) + O (2^{−n/2}n log n)  for x ∈ W̄_{n}. (7.15)

We also have

H_{k}(2^{n/4}x_{1}, …, 2^{n/4}x_{2^{k}}) = C_{k,n} − 2^{n/2}[ ∑_{i=1}^{2^{k}−1} ∑_{j=i+1}^{2^{k}} d(i, j)^{−3/2}x_{i}^{(2)}x_{j}^{(2)}

+ a_{0}M̄ ∑_{i=1}^{2^{k}} (1 − 2^{−k/2}) (x_{i}^{(1)} − M̄)] + O (2^{−n/2}n log n)

if x_{i} ∈ W̄_{n}, j = 1, …, 2^{k},  (7.16)

since ∑_{j=1}^{2^{k}} d(i, j)^{−3/2} = a_{0}(1−2^{−k/2}) for all 1 ≤ i ≤ 2^{k}, and 2^{n/2}(x_{i}^{(1)}−M̄)(x_{j}^{(1)}−M̄) = O(2^{−n/2}n log n) in this case.

Because of the Corollary of Theorem 1 and the relation ḡ = \frac{a_{0}M̄}{T}

f̄_{n+k} (2^{−k} ∑_{j=1}^{2^{k}} x_{j}) = C_{n,k} exp\{2^{n/2}(\frac{a_{0}M̄}{T} 2^{−k/2} ∑_{j=1}^{2^{k}}(x^{(1)} − M̄)

+ Ā2^{−3k/2}(∑_{j=1}^{2^{k}} x_{j}^{(2)})^{2}) + O(q^{n})}

if x_{j} ∈ W̄_{n}, j = 1, …, 2^{k}.  (7.17)

Relation (7.12) follows from (7.10), (7.13), (7.16) and (7.17). Relation (7.11) can be proved in the same way as it is done in Section 7 of Part II of [2], only the relations ||x|−M| < c^{−0.4n} and |x^{(2)}| < c^{−0.45n} must be replaced by ||x|−M̄| < \frac{\sqrt{T}}{8M̄} 2^{−n/2}\sqrt{n log n} and |x^{(2)}| < 2^{−n/4}n^{1/5} in the definition of the set Ω̃_{n}^{1}. We get a weaker bound in (7.11) than the corresponding estimate in [2], but it is sufficient for our purposes. Theorem 2 is proved.

References:

[1] Bleher, P.M., Major, P.: Renormalization of Dyson’s hierarchical vector-valued Φ^{4}-model at low temperatures. Commun. Math. Phys. 95, 487–532 (1984)

[2] Bleher, P.M., Major, P.: The large-scale limit of Dyson’s hierarchical vector-valued model at low temperatures. The non-Gaussian case. Annales de l’Institut Henri Poincaré, Physique Théorique 49, Vol. 1 (1988)