\magnification=\magstep1
\hsize=16.5truecm
\input amstex
\documentstyle{amsppt}
 
\define\Om#1{\Omega_n^#1}
\define\fn{f_{n,N}^{h_N}(x)}
\define\mn{\mu_{n,N}^{h_N}}
\define\mk{\mu_{k,N}^{h_N}}
\define\gb{\bar g }
\define\lhn{L_n(N,h_N)}
\define\ab{\bar A}
\define\e{\varepsilon}
\define\rn{(R^2)^{2^n}}
\define\rk{(R^2)^{2^k}}
\define\co#1{c^{#1n}}
\define\im#1{\int_{\Om#1}}
\define\fformg{\vf\in\Gamma_n(r,x)}
\define\s#1{S_n^{#1}f(x)}
\define\obb{\hat\Omega_n^2}
\define\ebb{\hat\varepsilon}
\define\Omm#1{\Omega_{n+1}^#1}
\define\eb{\bar\e }
\define\ebx{\bar\e_n(x)}
\define\vf{\varphi}
\define\formg{\{\vf\in\Gamma_n(r,x)\}}
\define\x#1{x^{(#1)}}
\define\y#1{y^{(#1)}}
\define\xo#1{x^{(2)#1}}
\define\xy{\frac{x+y}2}
\define\ca{(\frac{c^n}T+\frac{A_{n+1}}2)}
\define\cga{(\frac{c^n}{2T}+\frac{g_{n+1}}{4M}-\frac{A_{n+1}}4)}
\define\je{J_{n,\eb}}
\define\jeb{\bar J_{n,\eb}}
\define\gnx{g_n(\x1-M)+A_n\xo2}
\define\ga{\frac{g_n}{2M}-A_n}
\define\te{T_n^{\e}f(x)}
\define\gan{\frac{g_{n+1}}{2M}-A_{n+1}}
\define\ssb{\bar\sigma}
\define\zz{\bold Z}
\define\mb{\bar \mu}
\define\zzd{\bold Z^d}
\define\yn{Y_k(N)}
\define\de{\delta}
 
\leftheadtext\nofrills{{\rightline{\rm P. M. Bleher and P. Major}}}
\rightheadtext\nofrills{\leftline{\rm The large-scale limit of Dyson's
hierarchical model}}
 
\topmatter
\title The large-scale limit of Dyson's hierarchical\\ vector-valued
model at low temperatures. \\ The non-Gaussian case. \\
\rm Part II. Description of the large-scale limit.
\endtitle
 
\author $^1$Bleher, P. M. and $^2$Major, P. \endauthor
\affil $^1$Keldysh Institute of Applied Mathematics,
 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 \endaffil
\endtopmatter
\subheading{1. Introduction}
 In this paper we investigate the large-scale limit of the equilibrium
state of Dyson's hierarchical vector valued  $p$  dimensional,
$p\ge2$, model with parameter  $c$, $1<c<\sqrt2$, at low temperatures.
More precisely, in Theorem 1 we construct a probability measure
 $\bar{\mu}=\bar{\mu}(T)$  on  ${(R^p)}^{\bold Z}$  with
 $\bold Z=\{\,1,2,...\,\}$  which is an equilibrium state of the
model. In Theorem 2 we determine the large-scale limit of a
$\bar{\mu}$   distributed random field together with the right
scaling, i.e. we prove that if
$$
\sigma\,=\left\{\,\sigma(j)=\left(
         \sigma^{(1)}(j),\dots,\sigma^{(p)}(j)
\right) \,\in R^p ,\, j\in \bold Z \right\}
$$
is a  $\bar{\mu}$  distributed random field then the finite dimensional
distributions of the random fields
$$
\align
\Cal R_n\sigma&=\left\{\left(\Cal R_n\sigma^{(1)}(j),\dots ,\Cal R_n
\sigma^{(p)}(j)\right) \in R^p, \, j\in \bold Z\,\right\}, \tag1.1\\
\Cal R_n\sigma^{(1)}(j)&=c^n\,2^{-n} \sum_{k=(j-1)2^n+1}^{j2^n}
\left[\sigma^{(1)}(k)-E\sigma^{(1)}(k)\right],\quad  j\in\bold Z\,,
\tag1.2 \\
\Cal R_n\sigma^{(s)}(j)&=c^{n/2}2^{-n} \sum_{k=(j-1)2^n+1}^{j2^n}
\sigma^{(s)}(k)\,,\quad  j\in\bold Z , \quad s=2,\dots,p  \tag1.3
\endalign
$$
tend to those of a limit random field, and describe the finite dimensional
distributions of this limit field.\par
The distributions of the fields  $\Cal R_n\sigma$  defined in
(1.1)--(1.3)
 are called the renormalizations of the distribution  $\bar{\mu}$
  of the underlying field  $\sigma$. More precisely, they are
 its renormalization with parameters  $\alpha=1-\frac{\log c}{\log 2}$
  in the first coordinate and  $\bar{\alpha}=1-\frac12\frac
 {\log c}{\log 2}$  in the coordinates  $s=2,\dots,p$, because we
 multiplied by  $2^{-n\alpha}$  in (1.2), by  $2^{-n\bar{\alpha}}$
 in (1.3), and the number of summands is  $2^n$  in these formulas.
 If the finite dimensional distributions of the fields  $\Cal R_n
 \sigma$  converge to those of a limit field then this limit field,
 more precisely its distribution, is called the large-scale limit of
 the measure  $\bar{\mu}$. \par
Given some  $h\in R^1$, $h\ge 0$, and positive integer  $N$  let
the Gibbs measure  $\mu^h_N=\mu^h_N(T,t)$  be defined on  $
(R^p)^{2^N}$  with density function
$$
\align
p^h_N(x_1,\dots,x_{2^N})&=
p^h_N(x_1,\dots,x_{2^N},t,T)\,,\\
&\qquad x_j=(x_j^{(1)},\dots,x_j
^{(p)})\in R^p\,,\quad j=1,\dots,2^N\,,
\endalign
$$
$$
\aligned
&p^h_N(x_1,\dots,x_{2^N})
\\&\qquad=Z^{-1}_N(T,t,h)\exp\biggl\{-\frac1T\biggl(
-\sum^{2^N-1}_{i=1}\sum_{j=i+1}^{2^N}U(i,j)x_ix_j-h\sum_{j=1}^
{2^N}x_j^{(1)}\biggr)\biggr\}\prod^{2^N}_{j=1}p(x_j,t)\,,
\endaligned  \tag 1.4
$$
where
$$
Z_N=\int\exp\biggl\{-\frac1T\biggl(-\sum^{2^N-1}_{i=1}
\sum_{j=i+1}^{2^N}U(i,j)x_ix_j-h\sum_{j=1}^{2^N}x_j^{(1)}\biggr)\biggr\}
\prod_{j=1}^{2^N}p(x_j,t)\,dx_j
$$
is the grand partition function, and  $p(x,t)$  is defined in
(1.3) of Part I. Let  $p^h_N(x)$  denote the density function
of the average  $2^{-N}\sum^{2^N}_{j=1}\sigma (j)$  of the
 $\mu^h_N$  distributed random vector  $(\sigma (1),\dots,
\sigma (2^n))$. Put  $\mu_N=\mu^h_N$, \ $ p(x_1,\dots,
x_{2^N})=p^h_N(x_1,\dots,x_{2^N})$  and  $p_N(x)=p^h_N(x)$
in the case  $h=0$.\par
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
$$
\align
g(x)=\left(\frac2{c\sqrt\pi}\right)^{p-1}\int_{R^p}\exp
(-v^2)&g\biggl(\frac xc+u+\frac{v^2}2\biggr)g\biggl(\frac xc-u+
\frac{v^2}2\biggr)\,dudv\,, \tag1.5\\&\qquad x,\;u\in R^1\,,\quad v\in
R^{p-1}\,,
\endalign
$$
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  $\Cal A=\left\{g,
\int e^{tx}|g(x)|dx<\infty\;\, \text{if }\; t<t_0(g),\;
t_0(g)>0
\right\}$. 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  $\bar p_n(z)=
\bar p_n(z,t,T)$, \  $z\in R^1$, such that  $p_n(x)=\bar p_n
(|x|)$  for all  $x\in R^p$. Now we formulate the following
 
\proclaim{Theorem A}\it If  $1<c<\sqrt2$  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:\par
There are some  $M=M(c,T,t)>0$  and  $n_0=n_0(c,T,t)>0$
such that for  $n>n_0$
$$
\align
&c^{-n}p_n(x,T)=c^{-n}\bar p_n(|x|,T)
         \tag1.6\\&\quad=B\exp\left\{-\frac{a_0c^n}T
M(|x|-M)\right\}g\left(\frac{a_1c^n}T M(|x|-M)\right)(1+r_n(x))
\endalign
$$
for  $-\eta nc^{-n}<|x|-M<\eta n^{1/\alpha}c^{-n}$  with some
 $B=B(c,T,t)>0$, \ $ \eta=\eta(c,T,t)>0$, and the error term
 $r_n(x)$  satisfies the inequality
$$
|r_n(x)|\le Kq^n\tag1.7
$$
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),  $a_0=\frac2{2-c}$, \ $a_1=a_0+1$  and  $\alpha=\frac
{\log 2}{\log c}$.
$$
c^{-n}\bar p_n(x,T)\le
         Kq^n\exp\left\{-L(c^n|x-M|)^{2+\delta}\right\}\quad\text
{if } x>M+\eta n^{1/\alpha}c^{-n}\tag1.8
$$
with some  $\delta >0$,\ $ K>0$  and  $L>0$  which depend
on  $c$, $ T$  and  $t$. The solution of the equation
(1.6) satisfies the inequality
$$
0<g(x)<C\exp(-Ax^\alpha)\quad \text{for }  x>0\tag1.9
$$
with some  $C>0$, \ $ A>0$.   We  have
$$
c^{-n}\bar p_n(x,T)\le C_1\exp \{-C_2c^n|x-M|\}
\quad\text {for} \quad 0<x<M \tag1.10
$$
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  $|R(t,T)|\le \text {const}.$, and such that  $R(t,T)\to 0$
  and  $T\to 0$.\endproclaim
Given some integers  $N\ge k\ge 0$  we define the probability
measure  $\mu^h_{k,N}$  as  the projection of the measure
 $\mu^h_N$  to the first  $2^k$  coordinates, i.e.  $\mu^
h_{k,N}$  is a probability measure on  $(R^p)^{2^k}$, and for
all measurable  $A\subset (R^p)^{2^k}$ $ \mu^h_{k,N}(A)=
\mu^h_N(A)=\mu^h_N\left( A\times (R^p)^{2^N-2^k}\right)$. Our first
result is the following
\proclaim{Theorem 1} \it Let the conditions of Theorem A be satisfied.
Consider an arbitrary sequence of real numbers  $h_N $, $ N=
0,1,2,\dots $  such that
$$
\frac2{2-c} \frac MT \left(\frac c2 \right)^N \le \frac{h_N}T \le D
\left(\frac c2 \right)^N \tag 1.11
$$
with some  $ \infty > D > \frac2{2-c}  \frac MT$, where  $M$  and
 $T$  are the same as in Theorem A. Then the measures  $\mu_N^{h_N}$
tend to a probability measure  $\bar \mu =\bar \mu(t,T,c)$  on
$(R^p)^{\bold Z}$. More precisely, for all  $k \ge 0$  the measures
 $\mu^{h_N}_{k,N}$  converge to the projection of  $\bar \mu$  to
the first  $2^k$  coordinates in variational metric as  $N \to
\infty$. The measure  $\mu$  does not depend on the choice of
sequences  ~$h_N$. \endproclaim
 Then we prove the following
\proclaim {Theorem 2}\it  Let  $\sigma =\left\{\sigma
(n)=\left(\sigma^{(1)}(n),
\dots,\sigma^{(p)}(n)\right) \in R^p,\; n\in\bold Z\right\}$ \ be a
 $\bar \mu$  distributed random field with the distribution  $\bar
\mu$  defined in Theorem 1. Then the finite dimensional distributions
of the random fields  $R_n\sigma$  defined in (1.1), (1.2), (1.3) tend
to those of a random field  $Y=\left(Y(n)=\left(Y^{(1)}(n),\dots,
Y^{(p)}(n)\right)\in R^p, \; n \in \bold Z\right)$. For all
 $k \ge 0$  the density function
$h_k\left(x_1,\dots,x_{2^k}\right)$, \
$ x_j=\left(x_j^{(1)},\dots,x_j^{(p)}\right)\in R^p$,  of the
random vector  $(Y(1),\dots,Y(2^k))$  is given by the formula
$$
\aligned
&h_k\left(x_1,\dots,x_{2^k}\right)=C(k)\exp\biggl\{-\frac1T
\sum_{s=2}^p\biggl(\frac1{2-c}\sum_{j=1}^{2^k} x_j^{(s)2}-\frac
{2-c}c \biggl(\frac c4 \biggr)^k\biggl(\sum_{j=1}^{2^k}
         x_j^{(s)}\biggr)^2\\
&\qquad+\sum^{2^k-1}_{i=1}\sum_{j=i+1}^{2^k} U(i,j)x_i^{(s)}x_j^{(s)}
\biggr)\biggr\}\prod _{j=1}^{2^k}
         g\biggl(\frac{4-c}{(2-c)T}\biggl(Mx_j
^{(1)} +\frac12 \sum_{s=2}^p x_j^{(s)2}\biggr)\biggr)\,,
\endaligned \tag1.12
$$
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.\endproclaim
\par In  Appendix E we prove the following
\proclaim{Theorem B}\it The measure \ $\bar{\mu}=\bar{\mu}(T,t,c)$ \
constructed in Theorem 1 is a Gibbs state with Hamiltonian \ $\Cal H$ \
and free measure \ $\nu$ \ defined in formulas (1.1)--(1.3) of Part I
at temperature \ $T$. \endproclaim
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.
\par Let us discuss the role of condition (1.11) in Theorem 1. The
lower bound
$$
h_N>\frac2{2-c}M\left(\frac2c\right)^N \tag 1.11${}^{\prime}$
$$
is essential in Theorem 1, it is needed to get a pure state with
magnetization in the direction  $e_1=(1,0,\dots,0)$  for the
limit measure  $\bar{\mu}$. If it were violated we would get a
Gibbs state with Hamiltonian  $\Cal H$  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  $\bar{\mu}$  would be obtained for any sequence
 $h_N$, \ $h_N>0$  satisfying (1.11${}^{\prime}$) or with the help of
the double limiting procedure  $\mu^h=\lim_{N\to\infty}\mu_N^h$,
\ $h>0$, \ $\bar{\mu}=\lim_{h\to 0} \mu^h $. This second
way was chosen to construct the equilibrium state in the case
 $\sqrt2<c<2$  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 $Y^{(\ell)}(j)$, \ $\ell=
2,\dots,p$  appearing in Theorem 2 are jointly Gaussian. We describe
the structure of this limit field in more detail. The random fields
$\{\Cal R_n\sigma^{(s)}(j), \; j\in\bold Z\}$, $s=2,\dots,p$,
and  $\{M\Cal R_n\sigma^{(1)}(j)+\frac12 \sum\limits^p_{s=2} \Cal R_n
\sigma^{(s)} (j)^2,\; j\in\bold Z\}$  tend to independent
random fields as  $n\to\infty$. The limit of the random fields
$\{\Cal R_n(\sigma^{(s)}(j)),\;j\in\bold Z\}$ is the
(disregarding
a multiplying factor) unique Gaussian self-similar field with
self-similarity parameter  $1-\frac12\frac{\log c}{\log 2}$,
whose distribution is invariant under all permutations of the index
set  $\bold Z$  which preserves the hierarchical distance
$d(i,j)$. The random fields
$\{M\Cal R_n\sigma^{(1)}(j)+\frac12 \sum^p_{s=2} \Cal R_n
\sigma^{(s)} (j)\,,\; j\in\bold Z\}$ \ 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).
\par The above result can also be interpreted in the following way:
Given a  $\bar{\mu}$  distributed random field  $\sigma(n)$, \
$ n\in\bold Z$, define the absolute value of the appropriately
normalized partial sums
 $|\Cal R_n|\sigma (j)=c^n2^{-n}\left(|\Cal R_n \sum_{k=(j-1)2^n+1}
^{j2^n}\sigma (k)|-M\right)$, \ $j\in \bold Z$.
Then the random fields  $\Cal R_n\sigma^{(s)}(j)$, \ $j=2,\dots,p$,
and the random fields  $|\Cal R_n|\sigma(j)$ tend in distribution to
independent random fields. The limit of $\Cal R_n\sigma^
{(s)}(j)$  \ $j=2,\dots,p$ is Gaussian, and the limit of  $|\Cal
         R_n|\sigma(j)$
consists of independent random variables. This follows immediately
from the above description of the limit behaviour of the fields
$\Cal R_n\sigma$  together with the observation      that
$|\Cal R_n|\sigma (j)-(\Cal R_n\sigma^{(1)}(j)+\frac1{2M}
\sum_{s=2}^p\Cal R_n\sigma^{(s)}(j)^2)\Rightarrow 0$
stochastically as $n \to \infty $.
\par 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  $\sqrt2<c<2$, but this is a degenerate case.
It follows from the results of [5] that if  $\{\sigma (j),\,
 j \in \bold Z \}$ is a $\bar{\mu}=\bar{\mu}(c)$,
$\sqrt2<c<2$ distributed random field with the equilibrium state
$\bar{\mu}$ constructed in [5] then the random fields
$$
|\Cal R_n|(\sigma)(j)=2^{-n/2}\biggl(\biggl|\sum_{k=(j-1)2^n+1}^{j2^n}
\sigma(k)\biggr|-M\biggr)\,,\quad j\in \bold Z,
$$
have the same limit as the random fields
$$
\Cal R_n\sigma^{(1)}(j)=2^{-n/2}\sum_{k=(j-1)2^n+1}^{j2^n}
\left(\sigma^{(1)}(k)-M\right)\,
$$
as $n\to\infty$, since in this case
$|\Cal R_n|\sigma (j)-\Cal R_n\sigma^{(1)}(j)
\Rightarrow 0$.
This limit consists of independent (Gaussian) random variables
which is also independent of the limit of the random fields
$\Cal R_n\sigma^{(j)}$, \ $s=2,\dots,p$.
\par 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
 $\frac{d\mu^{h_N}_{n,N}}{d\mu_n}$.
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
 $\frac{d\mu^{h_N}_{n,N}}{d\mu_n}$
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].
 
 
\subheading{2. On the basic estimates needed in the proof.
Reduction to integral equations}
 We need a good asymptotic formula for the Radon--Nikodym
derivative
 $\frac{d\mu^{h_N}_{n,N}}{d\mu_n}$.
It can be expressed exactly with the help of the following formulas:
$$
\frac{d\mu^{h_N}_{n,N}}{d\mu_n}\left(x_1,\dots,x_{2^n}\right)=
f^{h_N}_{n,N}\biggl(2^{-n}\sum^{2^n}_{j=1}x_j\biggr),\quad n \le N
\tag2.1
$$
$$
\align
f^{h_N}_{N,N}(x)&=K(N,h_N)\exp
\left(\frac{2^Nh_Nx^{(1)}}T\right)\tag2.2\\
f^{h_N}_{n,N}(x)&=K(n,N,h_N)S_nf^{h_N}_{n+1,N}(x)\tag2.3
\endalign
$$
with
$$
S_nf(x)=\int_{R^p}\exp\biggl(\frac{c^n}Txy\biggr)f\biggl(\frac{x+y}2
\biggr)p_n(y)\,dy \tag2.3${}^{\prime}$
$$
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${}^{\prime}$) 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.
\par Let us define the sequences   $g_n=g_n(N,h_N)$  and
\ $A_n=A_n(N,h_N)$ \ by the recursive relations
$$
\align
g_N&=g_N(N,h_N)=\frac{2^Nh_N}T\,\tag2.4\\
g_n&=g_n(N,h_N)=\frac{g_{n+1}}2  + \frac{c^n}TM\qquad \text {for }
n<N \tag2.4${}^{\prime}$\\
A_N&=A_N(N,h_N)=0\tag2.5\\
A_n&=A_n(N,h_N)=\frac{A_{n+1}}4+\frac{(\frac{c^n}T
+\frac{A_{n+1}}2)^2}
{\frac{2c^n}T+\frac{g_{n+1}}M-A_{n+1}} \qquad \text {for }
n<N\,,\tag2.5${}^{\prime}$ \endalign
$$
where  $M$  and  $T$  are the same as in Theorem A. In Section 7
of [6] we have claimed that
$$
f_n(x)=f^h_{n,N}(x)\sim
         K_n\exp\left\{g_n(x^{(1)}-M)+A_n\sum^p_{s=2}x^{(s)2}
\right\}\,,
$$
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
$$
\align
\Omega^1_n&=\{x\in R^2,\,\bigl||x|-M\bigr|<c^{-0.4n}\,,\;|x^{(2)}|
<c^{-0.4n}\,,\;x^{(1)}>0\}\tag2.6\\
\Omega^2_n&     =\left\{x\in
         R^2,\,\bigl||x|-M\bigr|<c^{-0.4n}\right\}-
\Omega^1_n \tag2.6${}^{\prime}$\\
\Omega^3_n&=\left\{x\in R^2,\,\bigl||x|-M\bigr|\ge
         c^{-0.4n}\right\}\,.\tag2.6${}^{\prime\prime}$
\endalign
$$
Clearly  $\Omega^1_n\cup\Omega^2_n\cup\Omega^3_n =R^2$. Now we
formulate the following
\proclaim {Proposition 1}\it 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
 $N\ge n\ge n_0$  then the Radon--Nikodym derivative
 $f_n(x)=f_{n,N}^{h_N}(x)$  appearing in (2.1) satisfies the
following relations:
 
\flushpar
 a) In the domain  $\Omega^1_n$
$$
f_n(x)=L_n\exp\left\{g_n\left(x^{(1)}-M\right)+A_nx^{(2)2}+
\varepsilon_n(x)\right\}\tag2.7
$$
with
$$
\sup_{x\in\Omega^1_n}|\varepsilon_n(x)|\le q^n\,.
$$
 b) In the domain  $\Omega_n^2$
$$
0\le f_n(x)\le L_n \exp\left\{g_n(|x|-M)-\biggl(\frac{g_n}{2M} -
A_n\biggr)c^{-0,8n}+q^n\right\}\,.\tag2.8
$$
 c) In the domain  $\Omega^3_n$
$$
0\le f_n(x)\le L_n \exp\left\{\frac{g_n}{2M}(|x|^2-M^2)
\right\}\,,\tag2.9
$$
where the numbers  $A_n$  and  $g_n$  are defined in (2.4)--(2.5${}^{\prime}$),
and  $L_n=L_n(N,h_N)$  is an appropriate norming constant.
\endproclaim
 We also prove the following result which is a slight modification
of Lemma 1 in [5].
\proclaim{Lemma 1}\it Let us choose some integer  $N$  and  $h_N>0$.
Define the sequences   $g_n$  and  $A_n$, \ $0\le n\le N$,
by formulas (2.4)--(2.5${}^{\prime}$) and put  $\bar g_n=c^{-n}g_n$, \
$\bar A_n=c^{-n}A_n$. If  $h_N$  satisfies relation (1.11) then
 $\bar g_N\ge \bar g_{N-1}\ge \dots \ge \bar g_0 \ge \bar g$
and  $0=\bar A_N\le \bar A_{N-1} \le \dots \le \bar A_0 \le \bar A$
 with  $\bar g=\frac2{2-c}\frac MT$, and $\bar A=\frac{2-c}{cT}$.
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
 $|\bar g_n-\bar g|<4^{-n}$, $|\bar A_n-\bar A|<4^{-n}$.
\endproclaim
 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  $\Omega^1_n$  is the typical region,
where we have a good asymptotic formula, in  $\Omega^2_n$  and
 $\Omega^3_n$  we have only given an upper bound. Actually we
are interested in the density function
$$
p_n\biggl(2^{-n}\sum^{2^n}_{j=1}x_j \biggr)f^{h_N}_{n,N}\biggl(
2^{-n}\sum^{2^n}_{j=1}x_j\biggr)
$$
of the measure  $\mu^{h_N}_{n,N}$. The tail behaviour of the
functions  $p_n(x)$  and  $f_n(x)$  together show that
 $2^{-n}\sum^{2^n}_{j=1}x_j$  is contained in  $\Omega^3_n$
with a negligible small  $\mu^{h_N}_{n,N}$  probability. It is
contained in  $\Omega^2_n$  also with a small probability, since
in this domain  $f_n(x)$ is small. To see it, let us observe that
by Lemma 1
$$
 \frac{g_n}{2M}-A_n\geq c^n\left(\frac{\bar g}{2M}-\bar A\right)=
 \frac{c^n}T\left(\frac1{2-c}-\frac{2-c}c\right)=\frac{c^n}T
 \frac{(4-c)(c-1)}{2-c}>0 \,,
 $$
hence the term  $-\left(\frac{g_n}{2M}-A_n\right)c^{-0.8n}$
in the exponent of (2.8), makes this upper bound (2.8) sufficiently
small for our purposes.
\par 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).
 
 \demo{Remark} If  $x\in\Omega_n^1$  then
$$
\align
|x|&=\left(x^{(1)2}+x^{(2)2}\right)^{1/2}=x^{(1)}\left(1+
\frac{x^{(2)2}}{x^{(1)2}}\right)^{1/2}=x^{(1)}+\frac{x^{(2)2}}
{2x^{(1)}}+O(x^{(2)4})\\&=x^{(1)}+\frac{x^{(2)2}}{2M}+O\left(x^{(2)4}+
x^{(2)2}|x^{(1)}-M|\right)=x^{(1)}+\frac{x^{(2)2}}{2M}+O(c^{-1.2n})\,,
\endalign
$$
hence
$$
\align
&\exp\left\{g_n(|x|-M)-\left(\frac{g_n}{2M}-A_n\right)x^{(2)2}
\right\}\\
&\qquad=\exp\left\{g_n(x^{(1)}-M)+A_nx^{(2)2}+O(c^{-0.2n})\right\}\,.
\endalign
$$
\enddemo
The above calculation shows that on the boundary of the domains
 $\Omega^1_n$  and  $\Omega^2_n$  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  $\Omega^2_n$.
\par 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  $\bar{\mu}_n$
denote the projection of the measure  $\bar{\mu}$  constructed in
Theorem 1 to the first  $2^n$  coordinates, i.e. let  $\bar{\mu}
_n$  be the measure on  $(R^2)^{2^n}$  defined by the relation
 $\bar{\mu}_n(A)=\bar{\mu}\left(A\times(R^2)^\infty\right)$  for
all measurable sets  $A\subset (R^2)^{2^n}$. The following result
holds true:
\proclaim {Proposition 2}\it For all  $q$, \ $c^{-0.2}<q<1$, there
is some  $n_0=n_0(c,T,M,q)$  such that for all  $n\geq n_0$
the measure  $\bar{\mu}_n$  is absolute continuous with respect
to the measure  $\mu_n$, and its Radon--Nikodym derivative
satisfies the relations:
$$
\frac{d\bar{\mu}_n}{d\mu_n}\left(x_1,\dots,x_{2^n}\right)=
\bar f_n\left(2^{-n}\sum_{i=1}^{2^n}x_i\right)\tag2.10
$$
and
 
\flushpar
 a) For  $x\in\Omega^1_n$
$$
\bar f_n(x)=L_n\exp\left\{c^n\bar g(x^{(1)}-M)+c^n\bar Ax^{(2)2}+
\varepsilon_n(x)\right\}\tag2.11
$$
with
$$
\sup_{x\in\Omega^1_n}|\varepsilon_n(x)|\leq q^n\,.\tag2.11${}^{\prime}$
$$
 b) For  $x\in\Omega^2_n$
$$
0\leq \bar f_n(x)\leq L_n\exp\left\{c^n\bar g(|x|-M)-c^{0.2n}
\left(\frac{\bar g}{2M}-\bar A\right)+q^n\right\}\,.\tag2.12
$$
 c) For  $x\in\Omega^3_n$
$$
\align
0\leq \bar f_n(x)&\leq L_n\exp\left\{\frac{\bar gc^n}{4M}(x^2-
M^2)\right\}\qquad \text {if }\, 0<x<M-c^{-0.4n}\,,\tag2.13\\
0\leq \bar f_n(x)&\leq L_n\exp\left\{\frac{\bar gc^n}M(x^2-
M^2)\right\}\qquad \text {if }\, x>M+c^{-0.4n}\,,\tag2.13${}^{\prime}$
\endalign
$$
with  $\bar g=\frac2{2-c}\frac MT$, \ $\bar A=\frac
{2-c}{cT}$  and an appropriate norming constant  $L_n$. This
norming constant satisfies the relation
$$
C_1<c^{-n/2}L_n<C_2 \qquad\text{with some } \;0<C_1<C_2<\infty \,.
$$
\endproclaim
 Theorem 2 can be deduced from Proposition 2 and Theorem A.
\par Let us finally remark that the function  $f_n(x)=f^{h_N}_
{n,N}(x)$  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
\proclaim {Proposition 1${}^{\prime}$}\it For all  $q$, \
$c^{-0.2}<q<1$,
there exists some  $n_0=n_0(T,M,c,D,q)$  such that if for
$n\geq n_0$  the function  $f(x)$  satisfies the following
relations with some  $\bar gc^{n+1}<g_{n+1}\leq Dc^{n+1}$, \
$0\leq A_{n+1}\leq \bar Ac^{n+1}$, \ $\bar g=
\frac2{2-c} \frac MT$, \ $\bar A=\frac{2-c}{cT}$, \  $D>\bar g$:
 
\flushpar
a) For  $x\in\Omega^1_{n+1}$
$$
\align
f(x)=&\exp\left\{g_{n+1}(x^{(1)}-M)+A_{n+1}x^{(2)2}+\varepsilon
_{n+1}(x)\right\}\tag2.15 \\
&\sup_{x\in\Omega^1_{n+1}}|\varepsilon_{n+1}(x)|\leq q^{n+1}\,.
\tag2.15${}^{\prime}$\endalign
$$
b) For  $x\in\Omega^2_{n+1}$
$$
0\leq f(x)\leq\exp\left\{g_{n+1}(|x|-M)+\left(\frac{g_{n+1}}{2M}-
A_{n+1}\right)c^{-0.8(n+1)}+q^{n+1}\right\}\,;\tag2.16
$$
c) For  $x\in\Omega^3_{n+1}$
$$
0\leq f(x)\leq \exp\left\{\frac{g_{n+1}}{2M}(|x|^2-M^2)\right\}
\,\; \tag2.17
$$
then the function \ $S_nf(x)$ \ defined by (2.3${}^{\prime}$)
satisfies, with
the constants \ $g_n$ \ and \ $A_n$ \ defined by (2.4${}^{\prime}$)
and (2.5${}^{\prime}$)
with the above \ $g_{n+1}$ \ and \ $A_{n+1}$, the following relations
with some appropriate norming constant \ $L_n$:
 
\flushpar
 a) In the domain \ $\Omega^1_n$
$$
S_nf(x)=L_n\exp\left\{g_n(x^{(1)}-M)+A_nx^{(2)2}+\varepsilon_n(x)
\right\}\tag2.18
$$
with
$$
\sup_{x\in\Omega^1_n}|\varepsilon_n(x)|\leq q^n\,.\tag2.18${}^{\prime}$
$$
b) In the domain  $\Omega^2_n$
$$
0\leq S_nf(x)\leq L_n\exp\left\{g_n(|x|-M)-\left(\frac{g_n}{2M}
-A_n\right)c^{-0.8n}+q^n\right\}\,.\tag2.19
$$
c) In the domain  $\Omega^3_n$
$$
0\leq S_nf(x)\leq L_n\exp\left\{\frac{g_n}{2M}(|x|^2-M^2)\right\}\,.
\tag2.20
$$
\endproclaim
 
\subheading{3. The proof of Lemma 1}
 The proof is a modification of that given for Lemma 1 in [5].
Simple calculation shows that  $\bar g_n-\bar g =(\frac c2)^{N-n}
(\bar g_N-\bar g)$. The statements of Lemma 1 about the sequence
 $g_n$  follow from this identity. To investigate  $\bar A_n$
let us introduce the function
$$
T(x,g)=\frac c4 x+\frac{\left(\frac1T+\frac c2 x\right)^2}
{\frac2T+\frac{cg}M-cx}\,,\quad x\in R^1\,,\quad g\in R^1
\,.
$$
Clearly,  $\bar A_n=T(\bar A_{n+1},\bar g_{n+1})$. On the other hand
 $T(\bar A,\bar g)=\bar A$, and some calculation shows that  $T$
has the following monotonicity properties:  $T(x,g)<T(x,g^{\prime})
$  if  $0<x<\bar A$ and  $g>g^{\prime}>\bar g$; and  $T(x^
{\prime},g)>T(x,g)$  if  $0<x<x^{\prime}<A$ and  $g>\bar g$.
(These properties follow e.g. from the relations
$$
\frac{\partial}{\partial g}T(x,g)=
-\frac cM  \frac{\left(\frac1T+\frac c2 x\right)^2}
{\left(\frac2T+\frac cMg-cx\right)^2}<0\,,
$$
and
$$
\frac{\partial}{\partial x}T(x,g)=
\frac{c\left(\frac2T+\frac{cg}{2M}\right)^2}
{\left(\frac2T+\frac cMg-cx\right)^2}>0\,,
$$
and the fact that  $T(x,g)$  has no singularity in the domain
 $\{(x,g),\,0<x<\bar A,\, g>\bar g\}$.)
We have  $0<\bar A_{N-1}<\bar A$,  since  $\bar A_{N-1}=
T(0,\bar g_N)>0$ and  $\bar A_{N-1}=T(0,\bar g_N)<T(\bar A,\bar g)
=\bar A$. Then we get by induction that  $0\leq \bar A_{n+1}<
\bar A_n<\bar A$  implies that  $0<\bar A_n<\bar A_{n-1}<\bar A$.
Indeed,  $\bar A_{n-1}=T(\bar A_n,\bar g_n)<T(\bar A,\bar g)=
\bar A$, and  $\bar A_{n-1}=T(\bar A_n,\bar g_n)>T(\bar A_{n+1},
\bar g_{n+1})=\bar A_n$,  as we have claimed.
\par The conditions  $N>N_0$  and  $N>n^B$  with sufficiently
large  $N_0$  and  $B$  imply that  $|\bar g
_{\ell}-\bar g|<10^{-n}\ $ for all  $0<\ell \leq \sqrt N$,
and \ $\bar A-5^{-n}<A^*<\bar A$, where  $A^*$  is the smaller
solution of the equation  $T(x,g^*)=x$  with  $g^*=\bar g_{\sqrt N}$.
Indeed, the last equation is a small perturbation of the equation
 $T(x,\bar g)=x$, which has two solutions  $A_1=\bar A$  and
 $A_2=\frac 1{(2-c)T} >\bar A$. 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  $\bar g_n$  and  $\bar A_n$  and the function
 $T(x,g)$  imply that  $\bar A>\bar A_n\geq T^{\sqrt N-n}_{g^*}
(0)$, where  $T^k_{g^*}$  denotes the  $k$-th iteration of
the function  $T(x,g^*)$  with fixed  $g^*$  in the variable
 $x$. Indeed,  $\bar A>\bar A_n$, $ 0 <\bar A_{\sqrt N}
<\bar A$, and we get by induction that for all  $\ell \geq 0 $
$ \bar A_{\sqrt N-\ell}\geq T^\ell_{g^*}(0)$, which implies
the required statement with  $\ell = \sqrt N-n$.
\par To complete the proof of Lemma 1 it is enough to show that
 $T^n_{g^*}(0)$  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
 $\frac {\partial T(x,g^*)}{\partial x}\leq \alpha <1$
for  $x=A^*$  if  $(A^*,g^*)$  is in a small neighbourhood
of the point  $(\bar A,\bar g)$. But this follows from the
continuity of the function
 $\frac {\partial T(x,g)}{\partial x}$,
and the fact that its value in the point  $(\bar A,\bar g)$
equals  $c^{-1}<1$. Lemma 1 is proved.
 
\subheading{4. Some preparatory remarks to the proof of
Proposition 1${}^{\prime}$}
 We shall prove the following estimates under the conditions
of Proposition 1${}^{\prime}$.
\par Put
$$
S^i_nf(x)=\int_{\{y,\,\frac{x+y}2\in\Omega^i_{n+1}\}}
\exp\left(\frac{c^n}Txy\right)f\left(\frac{x+y}2\right)p_n(y)\,dy\,,
\quad i=1,2,3\,.
$$
Then we have with some appropriate  $\varepsilon^{\prime}=
\varepsilon^{\prime}(c)$, \ $ \varepsilon^{\prime}>0$, and
the same  $q$  as in Proposition ~1${}^{\prime}$:
 
\medpagebreak
\flushpar
 In the domain  $x\in\Omega^1_n$
$$
S^1_nf(x)=L_n\exp\left\{g_n(x^{(1)}-M)+A_nx^{(2)2}+\bar \varepsilon
_n(x)+\ebb_n(x)\right\} \tag4.1
$$
with
$$
\sup_{x\in\Omega^1_n}|\bar \varepsilon_n(x)|\leq q^{n+1}\,,
\quad \sup_{x\in\Omega^1_n}| \ebb_n(x)|
\leq Kc^{-0.2n}\,,\tag4.1${}^{\prime}$
$$
where  $K$  does not depend on  $n$,
$$
\align
&L_n \geq c^{-n}\exp\left(\frac{c^n}T M^2\right)\,, \tag4.1${}^{\prime\prime}$\\
&S^2_nf(x)\leq L_n\exp\left\{g_n(x^{(1)}-M)+A_nx^{(2)2}- \varepsilon
^{\prime}c^{0.2n}\right\}\,,\tag4.2\\
&S^3_nf(x)\leq L_n\exp\left\{g_n(x^{(1)}-M)+A_nx^{(2)2}-\frac16 c^{n/2}
\right\}\,.\tag4.3
\endalign
$$
 
\medpagebreak
\flushpar
 In the domain  $x\in\Omega^2_n$
$$
\align
S^1_nf(x)&\leq L_n\exp\left\{g_n(|x|-M)-\left(\frac{g_n}{2M}-A_n
\right)c^{-0.8n}+q^{n+1}+Kc^{-0.2n}\right\}\,,\tag4.4\\
S^2_nf(x)&\leq L_n\exp\left\{g_n(|x|-M)-\left(\frac{g_n}{2M}-A_n
\right)c^{-0.8n}-\varepsilon^{\prime}c^{0.2n}\right\}\,,\tag4.5\\
S^3_nf(x)&\leq L_n\exp\left\{g_n(|x|-M)-\left(\frac{g_n}{2M}-A_n
\right)c^{-0.8n}-\frac16 c^{n/2}\right\}\,.\tag4.6\\
\endalign
$$
 
\medpagebreak
\flushpar
 In the domain  $x\in\Omega^3_n$
$$
S_nf(x)\leq L_n\exp\left\{\frac{g_n}{2M}(|x|^2-M^2)\right\}\,.
\tag4.7
$$
We show that these estimates imply Proposition 1${}^{\prime}$. Indeed, for
$x\in\Omega^1_n$
$$
S_nf(x)=S^{(1)}_nf(x)+S_n^{(2)}f(x)+S_n^{(3)}f(x)=
L_n\exp\{g_n(x^{(1)}-M)+A_nx^{(2)2}+\varepsilon_n(x)\}
$$
with
$$
\sup_{x\in\Omega^1_n}|\varepsilon_n(x)|\leq \sup_{x\in\Omega^1_n}
|\bar \varepsilon_n(x)|+\sup_{x\in\Omega^1_n}|\ebb
_n(x)|+2\exp(-\varepsilon^{\prime}c^{0.2n})+
2\exp\left(-\frac16 c^{n/2}\right)\,.
$$
(Here we have exploited that  $e^t<1+2|t|$  for small  ~$t$.)
Hence
$$
\sup_{x\in\Omega^1_n}|\varepsilon_n(x)|\leq q^{n+1}+Kc^{-0.2n}+
2\exp(-\varepsilon^{\prime}c^{0.2n})+
2\exp\left(-\frac16 c^{n/2}\right)\leq q^n
$$
if  $c^{-0.2}<q<1$, and  $n\geq n_0(q,D,\varepsilon^{\prime})$.
\par For  $x\in\Omega^2_n$  we have analogously
$$
\align
S_nf(x)&=S^1_nf(x)+S^2_nf(x)+
S^3_nf(x)\\&\leq L_n\exp\Biggl\{g_n(|x|-M)-\left(\frac{g_n}{2M}-A_n
\right)c^{-0.8n}+q^{n+1}+Kc^{-0.2n}\\&\qquad \qquad
+2\exp(-\varepsilon^{\prime}c^{0.2n})
+2\exp\left(-\frac16 c^{n/2}\right)\Biggr\}\\
&\leq L_n\exp\left\{g_n(|x|-M)-\left(\frac{g_n}{2M}-A_n
\right)c^{-0.8n}+q^n\right\}\,, \endalign
$$
as we have claimed in Proposition 1${}^{\prime}$. For  $x\in\Omega^3_n$  (4.7)
contains the needed estimate.
\par The above estimates will be proved in the next Section. In this
Section we prove two lemmas which we need during the proof. Put
$$
S^{\varepsilon}_nf(x)=\int_{R^2-V^{\varepsilon}_n(x)}
\exp \biggl(\frac{c^n}Txy\biggr)f_n\biggl(\frac{x+y}2\biggr)\,p_n(y)\,dy\tag4.8
$$
with
$$
V^{\varepsilon}_n(x)=\left\{y\in R^2\,,\quad
         \bigl||y|-M\bigr|\leq\varepsilon
c^{-0.4n}\,,\quad xy \geq |x||y|-\varepsilon
         c^{-0.4n}\right\}\,,\tag4.8${}^{\prime}$
$$
where   $xy$  denotes scalar product.
\proclaim {Lemma 2} \it There is some  $\varepsilon_0=\varepsilon_0(c)$
and  $n_0=n_0(T,M,D,c,\varepsilon)$ such that if  $n \geq n_0$, \
$0<\varepsilon < \varepsilon_0$, \ $0\leq g_{n+1}<Dc^{n+1}$ and
$$
0\leq f(x) \leq \exp\left\{\frac{g_{n+1}}{2M}(|x|^2-M^2)\right\}
\qquad \text{for all } x\in R^2 \tag4.9
$$
then
$$
0\leq S_nf(x)\leq c^n\exp\left\{\frac{c^n}T M^2+
\frac{g_n}{2M}(|x|^2-M^2)-\frac{c^n}{3T}(|x|-M)^2\right\}\,,
\quad x\in R^2 \leqno \text{a)}
$$
and
$$
 0\leq S^{\varepsilon}_nf(x)\leq
         \exp\left\{\frac{c^n}T M^2+g_n(|x|-M)-c^{n/2}\right\}
\qquad \text{if }\; \bigl||x|-M\bigr|<c^{-0.4n}\leqno \text{b)}
$$
with \ $g_n = \frac {g_{n+1}}2 +\frac {c^n}T M$. \endproclaim
\proclaim {Lemma 3}\it There is some  $n_0=n_0(T,M,D,c)$  such
that if for  $n\geq n_0$
$$
\matrix \format\r &\l&\qquad\l \\
0\leq f(x)&\;\leq \exp\{g_{n+1}(|x|-M)\}& \text {for }\,
\bigl||x|-M\bigr|<c^{-0.4(n+1)}\\    \vspace{1\jot}
f(x)&\;=0& \text{for } \,
\bigl||x|-M\bigr|\geq c^{-0.4(n+1)}
\endmatrix
$$
then
$$
0\leq S_nf(x)\leq K\exp\left\{\frac{c^n}T M^2+g_n(|x|-M)\right\}
\qquad\text{for }\; ||x|-M|\leq c^{-0.4n}
$$
with
$$
g_n=\frac{g_{n+1}}2 +\frac{c^n}TM
$$
and some   $K=K(T,M,D,c)>0$. \endproclaim
 \demo{Proof of Lemma 2}
 
\flushpar
 Part a). We have
$$
0\leq S_nf(x)\leq \int\exp\left\{\frac{c^n}T xy + \frac{g_{n+1}}
{2M}\left(\left|\frac{x+y}2\right|^2-M^2\right)\right\}p_n(y)\,dy\,.
$$
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
$$
0\leq S_nf(x)\leq2\pi\int_0^{\infty}\exp\left\{\frac{c^n}T|x|r+
\frac{g_{n+1}}
{2M}\left[\left(\frac{|x|+r}2\right)^2-M^2\right]\right\}r\bar p_n
(r)\,dr\,.\tag4.10
$$
Let us split up the integral into two parts,  $\int^M_0$  and
 $\int^{\infty}_M$. Put
$$
f_n(t)=c^{-n}\bar p_n(M+c^{-n}t)\,.\tag4.11
$$
It follows from (1.10) that
$$
f_n(t)\leq C_1\exp(-C_2|t|)\qquad \text{for } -c^nM<t<0
\tag4.11${}^{\prime}$ $$
and from (1.8), (1.7) and (1.9) that
$$
f_n(t)\leq C_3\exp(-t^2) \qquad \text{for } t>0
\tag4.11${}^{\prime\prime}$
$$
with some appropriate  $C_1>0$,  $C_2>0$  and  $C_3>0$.
(Relations (1.7) and (1.9) are needed in the domain
 $0<|x|<\eta n^{1/\alpha}c^{-n}$.)
 
First we estimate the integral
 $\int ^M_0$. For $0\leq r\leq M$  we have
$$
\frac {c^n}T|x|r +\frac{g_{n+1}}{2M}\left(\left(\frac{|x|+r}2
\right)^2-M^2\right)\leq \frac{c^n}T|x|M+\frac{g_{n+1}}{2M}
\left(\left(\frac{|x|+M}2\right)^2-M^2\right)\,.
$$
Hence (4.11) and (4.11${}^{\prime}$) imply that
$$
\align
\int^M_0\dots\, dr&\leq C_1
\exp\left\{\frac{c^n}T|x|M+\frac{g_{n+1}}2
\left[\left(\frac {|x|+M}2\right)^2-M^2\right]\right\}\\
&\qquad\int^0_{-c^nM}(M+c^{-n}t)e^{-C_2|t|}\,dt\\
&\leq C_4M\exp\left\{
\frac{c^n}T|x|M+\frac{g_{n+1}}{2M}\left[\left(\frac{|x|+M}2
\right)^2-M^2\right]\right\}\,.
\endalign
$$
Simple calculation yields the identity
$$
\align
&\frac{c^n}T|x|M+\frac{g_{n+1}}{2M}\left[\left(\frac{|x|+M}2
\right)^2-M^2\right]\\&\qquad=\frac{c^n}T
 M^2+\frac{g_n}{2M}(|x|^2-M^2)-
\left(\frac{c^n}{2T}+\frac {g_{n+1}}{8M}\right)(|x|-M)^2\,.
\endalign
$$
Then, since  $C_4 M<c^{n/2}$  we get that
$$
\aligned
\int^M_0\dots \,dr\leq c^{n/2}
 \exp\left\{\frac{c^n}TM^2+\frac{g_n}{2M}
(|x|^2-M^2)
+\left(\frac{c^n}{2T}
+\frac {g_{n+1}}{8M}\right)(|x|-M)^2\right\}\,.
\endaligned
\tag4.12
$$
Let us estimate  $\int^\infty_M\dots\, dr$. We make the change of
variables \ $r=M+c^{-n}t$, introduce
\ $f_n(t)=c^{-n}\bar p_n(M+c^{-n}t)\,,\quad\bar g_n=c^{-n}g_n$ \
and \ $\bar g_{n+1}=c^{-(n+1)}g_{n+1}$.
Since
$$
\aligned
&\qquad\frac{c^n}T|x|r+\frac{g_{n+1}}{2M}\left[\left(\frac{|x|+r}2
\right)^2-M^2\right]\\
&=\frac{c^n}T|x|M+\frac{g_{n+1}}{2M}
\left[\left(\frac {|x|+M}2\right)^2-M^2\right]\\
&\qquad\frac{c^n}T|x|(r-M)+\frac{g_{n+1}}{8M}(r-M)(2|x|+r+M)\\
&=\frac{c^n}T M^2+\frac{g_n}{2M}(|x|^2-M^2)-
\left(\frac{c^n}{2T}+\frac{g_{n+1}}{8M}\right)(|x|-M)^2\\
&\qquad+\left(\frac{c^n}T+\frac{g_{n+1}}{4M}\right)|x|(r-M)+
\frac {g_{n+1}}4(r-M)+\frac{g_{n+1}}{8M}
(r-M)^2
\endaligned \tag4.13
$$
hence
$$
\int^\infty_M\dots \,dr =\exp\left\{\frac{c^n}TM^2+\frac{g_n}{2M}
(|x|^2-M^2)-\left(\frac{c^n}{2T}+\frac{g_{n+1}}{8M}\right)
(|x|-M)^2\right\}J_n(|x|) \tag4.14
$$
with
$$
\aligned
J_n(|x|)&=\int^{\infty}_0\exp\biggl\{\left(\frac1T+\frac c{4M}
\bar g_{n+1}\right)|x|t\\
&\qquad\qquad+\frac c4\bar g_{n+1}t+c\frac {c^{-n}}{8M}\bar
  g_{n+1}t^2\biggr\}
(M+c^{-n}t)f_n(t)\,dt\\
&=\int^{\infty}_0\exp\biggl\{\left(\frac1T+\frac c{4M}\bar g_{n+1}
\right)(|x|-M)t\\
&\qquad\qquad+\left(\frac MT +\frac c2\bar g_{n+1}\right)
t+\frac{c^{-n+1}}{8M}\bar g_{n+1}t^2\biggr\}
(M+c^{-n}t)f_n(t)\,dt\,.
\endaligned   \tag 4.14${}^{\prime}$
$$
Relation (4.11${}^{\prime\prime}$) implies that
$$
\align
J_n(|x|)&\leq C_5\int^{\infty}_0\exp\left\{\left(\frac1T+\frac c{4M}
\bar g_{n+1}\right)(|x|-M)t-\frac{t^2}2\right\}\, dt \\
&\le C_6\{\exp C_7(|x|-M)^2\}\,.
\endalign
$$
Thus
$$
\align
\int_M^{\infty}\dots \,dr\leq C_6\exp&\biggl\{\frac {c^n}T
M^2+\frac{g_n}{2M}
(|x|^2-M^2)\\
&\qquad-\biggl(\frac{c^n}{2T}+\frac {g_{n+1}}{8M}\biggr)
(|x|-M)^2+C_7(|x|-M)^2\biggr\}\,.
\endalign
$$
Since $\frac {c^n}{2T}+\frac{g_{n+1}}{8M}-C_7\geq \frac {c^n}{3T}$
and $C_6\leq c^{n/2}$ for large $n$, hence
$$
\int^{\infty}_M\dots dr\leq c^{n/2}\exp\biggl\{\frac{c^n}T M^2+
\frac{g_n}{2M}(|x|^2-M^2)-\frac{c^n}{3T}(|x|-M)^2\biggr\}\,.
$$
This inequality together with (4.12) imply that
$$
\align
S_nf(x)&\leq 4\pi c^{n/2}\exp\biggl\{\frac{c^n}T M^2+
\frac{g_n}{2M}(|x|^2-M^2)-\frac {c^n}{3T}(|x|-M)^2\biggr\}\leq
\tag 4.15\\
&\leq c^n\exp\biggl\{\frac{c^n}T M^2+\frac{g_n}{2M}(|x|^2-M^2)
-\frac{c^n}{3T}(|x|-M)^2\biggr\}
\endalign
$$
as we have claimed.
 
\medpagebreak
\flushpar
 Part b). Let us introduce
$$
R^{\e}_nf(x)=\int_{\{y,\,||y|-M|\geq\e c^{-0.4n}\}}\exp\biggl(
\frac{c^n}T xy\biggr)f\biggl(\frac{x+y}2\biggr)p_n(y)\,dy
$$
and
$$
Q^{\e}_nf(x)=\int_{\{y,\,||y|-M|<\e c^{-0.4n},\, xy\leq
|x||y|-\e c^{-0.4n}\}}\dots\,dy\,.
$$
Clearly, $S^\e _n f(x)=R^{\e}_nf(x)+Q^{\e}_nf(x)$, and by
~(4.10)
$$
\align
0\leq R^{\e}_n f(x)&\leq
2\pi\biggl[\int^{M-\e c^{-0.4n}}_0+\int^{\infty}_
{M+\e c^{-0.4n}}\biggr]\\
&\qquad\qquad\qquad\exp\biggl\{\frac{c^n}T|x|r+
\frac{g_{n+1}}{2M}\biggl(\biggl(\frac{|x|+r}2\biggr)^2
-M^2\biggr)\biggr\}r\bar p_n(r)\,dr\,.
\endalign
$$
Moreover, similarly to part a), we get by using (4.13) and the
observation
$$
\align
&\biggl(\frac{c^n}T+\frac{g_{n+1}}{4M}\biggr)|x|(r-M)+
\frac{g_{n+1}}4(r-M)+\frac{g_{n+1}}{8M}(r-M)^2\\
&<\frac{g_{n+1}}{4}(r-M)\bigl(1+\frac {r-M}{2M}\bigr)
<-2c^{n/2} \quad \text{if }
-M<r-M<-c^{-0.4n} \endalign
$$
that
$$
\aligned
&\int^{M-\e c^{-0.4n}}_0\dots\,dr\\&\qquad \leq C_1
\exp\left\{-2c^{n/2}+\frac{c^n}T
M^2+\frac{g_n}{2M}(|x|^2-M^2)-\biggl(\frac{c^n}{2T}
+\frac{g_{n+1}}{8M}\biggr)(|x|-M)^2\right\}\\&\qquad
\qquad\qquad\qquad\qquad \int^{M-\e c^{-0.4n}}_0 r\bar p_n(r)\,dr\\
&\qquad\leq c^{-n/2}\exp\biggl\{-2c^{n/2}+
\frac {c^n}T M^2+\frac {g_n}{2M}(|x|^2-M^2)\\
&\qquad\qquad-\biggl(\frac{c^n}{2T}+
\frac{g_{n+1}}{8M}\biggr)(|x|-M)^2\biggr\}\\
&\qquad\leq c^{-n/2}\exp\biggl\{-2c^{n/2}+\frac{c^n}T M^2 +\frac
{g_n}{2M}(|x|^2-M^2)\biggr\}\,,
\endaligned\tag 4.16
$$
and by (4.13) and (4.11${}^{\prime\prime}$), similarly to (4.14), (4.14${}^{\prime}$)
$$
\int^{\infty}_{M+\e c^{-0.4n}}\dots\,dr\leq\exp\biggl\{\frac{c^n}TM^2
+\frac{g_n}{2M}(|x|^2-M^2)-\biggl(\frac{c^n}{2T}+\frac{g_{n+1}}
{8M}\biggr)(|x|-M)^2\biggr\}J^{\e}_n(|x|)
$$
with
$$
J^{\e}_n(|x|)=C_5\int^{\infty}_{\e c^{0.6n}} c^n\exp\biggl\{
\biggl(\frac1T+\frac c{4M}\bar g_{n+1}\biggr)(|x|-M)t+\biggl(
\frac MT+\frac c2\bar g_{n+1}\biggr)t -\frac{t^2}2\biggr\}\,dt\,.
$$
Since $||x|-M|\leq c^{-0.4n}$, and $\bar g_{n+1} < D$
$$
\sup_{t\geq 0} \biggl\{\biggl(\frac1T+\frac c{4M}\bar g_{n+1}\biggr)
(|x|-M)t+\biggl(\frac MT +\frac c2 \bar g_{n+1}\biggr)t
-\frac{t^2}6\biggr\}\leq C^{\prime}\,,
$$
and  therefore
$$
J^{\e}_n(|x|)\leq C_6\int^{\infty}_{\e c^{0.6n}}c^n\exp
\Bigl(-\frac{t^2}3\Bigr)\,dt\leq\exp(-\e^{\prime} c^{1.2n})
\leq c^{-n}\exp(-2c^{n/2})
$$
with some $\e^{\prime}=\e^{\prime}(\e) >0$. Hence
$$
\int^{\infty}_{M+\e c^{-0.4n}}\dots \,dr \leq c^{-n}\exp\biggl\{
\frac{c^n}T M^2+\frac{g_n}{2M}(|x|^2-M^2)-2c^{n/2}\biggr\}\,.
$$
The last inequality together with (4.16) imply that
$$
R^{\e}_nf(x)\leq 4\pi c^{-n/2}\exp\biggl\{-2c^{n/2}+\frac{c^n}TM^2+
\frac {g_n}{2M}(|x|^2-M^2)\biggr\}\,.\tag4.17
$$
Now we estimate $Q^{\e}_nf(x)$. We have
$$
0\leq Q^{\e}_nf(x)\leq \int \Sb \{y,||y|-M|\leq
c^{-0.4n},\\ {}
\;\;\;xy\leq|x||y|-\e c^{-0.4n}\;\,\}\endSb \exp\biggl\{\frac{c^n}T
xy+
\frac{g_{n+1}}{2M}\biggl[\left(\frac{x+y}2\right)^2-M^2\biggr]\biggr\}
p_n(y)\,dy\,.
$$
Since for $|y|=r$ \ $xy\leq r|x|-\e c^{-0.4n}$ and
$\left(\frac{x+y}2\right)^2\leq\bigl(\frac{|x|+r}2\bigr)^2$
in the last integral, hence
$$
\align
Q^{\e}_nf(x)&\leq 2\pi\int^{\infty}_0\exp\biggl\{\frac{c^n}T
\left(|x|r-\e c^{-0.4n}\right)+\frac{g_{n+1}}{2M}\biggl[\left(\frac
{|x|+r}2\right)^2-M^2\biggr]\biggr\}r\bar p_n(r)\,dr\\
&\leq2\pi\exp\biggl(-\frac{\e c^{0.6n}}T\biggr)\int^{\infty}_0
\exp\biggl\{\frac{c^n}T|x|r+\frac{g_{n+1}}{2M}
\biggl[\biggl(\frac{|x|+r}2\biggr)^2-M^2\biggr]\biggr\}\\
&\hskip 10truecm r\bar p_n(r)\,dr  \,.
\endalign
$$
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
$$
\align
Q^{\e}_nf(x)&\leq 2\pi c^n\exp\biggl\{-\frac{\e}Tc^{0.6n}+
\frac{c^n}TM^2+\frac{g_n}{2M}\left(|x|^2-M^2\right)\biggr\}\\ &\leq
\frac12\exp\biggl\{-2c^{n/2}+\frac{c^n}TM^2+\frac{g_n}{2M}
\left(|x|^2-M^2\right)\biggr\} \,.
\endalign
$$
This inequality together with (4.17) imply that
$$
S^{\e}_nf(x)\leq\exp\biggl\{-2c^{n/2}+\frac{c^n}T M^2+
\frac{g_n}{2M}\left(|x|^2-M^2\right)\biggr\}\,.
$$
Since $\frac1{2M}(|x|^2-M^2)=(|x|-M)+\frac1{2M}(|x|-M)^2$ and
$\frac{g_n}{2M}(|x|-M)^2\leq Dc^{0.2n}$ if $||x|-M|\leq c^{-0.4n}$,
hence the last inequality implies that under the conditions of Part
         ~b)
$$
\align
S^{\e}_nf(x)&\leq\exp\biggl\{-2c^{n/2}+\frac{c^n}TM^2+\frac{g_n}{2M}
(|x|-M)+Kc^{0.2n}\biggr\}\\
&\leq\exp\biggl\{\frac {c^n}T M^2+g_n(|x|-M)-c^{n/2}\biggr\}\,,
\endalign
$$
as we have claimed. \enddemo
 \demo {Proof of Lemma 3}
$$
\align
0\leq S_nf(x)&\leq \int_{\{y,\;||\frac{x+y}2|-M|\leq c^{-0.4(n+1)}\}}
\exp\left\{\frac{c^n}Txy+g_{n+1}\left(\left|\frac{x+y}2
\right|-M\right)\right\}\\
&\qquad\qquad\qquad p_n(y)\,dy\,.
\endalign
$$
Since
$$
\max_{|y|=r}\left\{\frac{c^n}Txy+g_{n+1}\left(\left|\frac
{x+y}2\right|-M\right)\right\}\leq\frac{c^n}T|x|r+g_{n+1}
\left[\left(\frac{|x|+r}2\right)-M\right]\,,$$
hence
$$
S_nf(x)\leq 2\pi\int^{\infty}_0\exp\left\{\frac{c^n}T
|x|r+g_{n+1}\left(\frac{|x|+r}2-M\right)\right\}r\bar p_n(r)\,dr\,.
$$
Writing $\frac{c^n}T|x|r=\frac{c^n}TM^2+\frac{c^n}TM(|x|-M+r-M)+
\frac{c^n}T(|x|-M)(r-M)$ we get that
$$
\align
S_nf(x)\leq 2\pi\exp&\left\{\frac{c^n}TM^2+g_n(|x|-M)\right\}\\
&\int^{\infty}_0\exp\left\{g_n(r-M)+\frac{c^n}T(|x|-M)(r-M)
\right\}r\bar p_n(r)\,dr\,.
\endalign
$$
The change of variables $r=M+c^{-n}t$ and the introduction of
$\bar g_n=c^{-n}g_n$ yields that
$$
\align
S_nf(x)&\leq 2\pi\exp\left\{\frac {c^n}TM^2+g_n(|x|-M)\right\}\\
&\qquad\int^{\infty}_{-c^nM}\exp\left\{\bar g_nt +
\frac{(|x|-M)}T t\right\}(M+c^{-n}t)f_n(t)\,dt\,.
\endalign
$$
Since $0 <\bar g_n <D$, and $||x|-M|<c^{-0.4n}$ relations (4.11${}^{\prime}$)
and (4.11${}^{\prime\prime}$) imply that for large ~$n$
$$
\int^{\infty}_{-c^nM}\exp\left\{\bar g_nt+\frac 1T(|x|-M)t\right\}
(M+c^{-n}t)f_n(t)\,dt < \bar K
$$
with some $\bar K > 0$ independent of $n$. Hence
$$
\align
S_nf(x)&\leq 2\pi\bar K\exp\left\{\frac{c^n}T M^2+g_n(|x|-M)\right\}\\
&\leq K\exp\left\{\frac{c^n}T M^2+g_n(|x|-M)\right\}\,,
\endalign
$$
as we have claimed.
\enddemo
 
\subheading{5. The proof of Proposition 1${}^{\prime}$}
In this Section
we
prove the estimates (4.1)--(4.7) which imply Proposition 1${}^{\prime}$.
 
 
\medpagebreak
a) The estimation of $\s1$ for $x\in\Om1$.
 
It follows from (2.15) and (2.15${}^{\prime}$) that
$$
\align
\s1&=\int_{\{y,\,\frac{x+y}2\in\Omm1\}}\exp
\Biggl\{\frac{c^n}Txy+
g_{n+1}\biggl(\frac{x^{(1)}+y^{(1)}}2-M\biggr)+A_{n+1}\\
&\qquad\biggl(\frac {x^{(2)}+y^{(2)}}2\biggr)^2
+\e_{n+1}\biggl(\frac{x+y}2\biggr)\Biggr\}
p_n(y)\,dy\,.
\endalign
$$
Hence
$$
\align
\s1&=\exp(\ebx)\int_{\{y,\,\frac{x+y}2\in\Omm1\}}\exp
\Biggl\{\frac{c^n}Txy+
g_{n+1}\left(\frac{x^{(1)}+y^{(1)}}2-M\right)\\
&\qquad+A_{n+1}\biggl(\frac {x^{(2)}+y^{(2)}}2\biggr)^2\Biggr\}
p_n(y)\,dy\,.
\endalign
$$
with some
$$
\sup_{x\in\Om1}|\ebx|\le q^{n+1}\,. \tag 5.1
$$
Let us rewrite the last expression in polar coordinate
system. We get that
$$
\s1=\exp(\ebx)\int_0^{\infty}I_n(r)\bar p_n(r)\,dr \tag 5.2
$$
with
$$ \align
I_n(r)=\int_{\formg}r\exp
\Biggl\{\frac{c^n}T\left(x^{(1)}y^{(1)} +
x^{(2)}y^{(2)}\right)&+
g_{n+1}\left(\frac{x^{(1)}+y^{(1)}}2-M\right)\\&+A_{n+1}
\left(\frac
{x^{(2)}+y^{(2)}}2\right)^2\Biggr\}\,d\vf
\endalign
$$
where $y^{(1)}=r\cos\vf$, \ $y^{(2)}=r\sin\vf$, \ $-\pi<\vf<\pi$,
$y=(y^{(1)},y^{(2)})$ and $\Gamma_n(r,x)=\allowmathbreak
\{\vf,\,\frac{x+y}2\in\Omm1\}$.
We shall express $I_n(r)$ as an asymptotically Gaussian
integral with respect to $\vf$. For this aim we give some
bounds on $x^{(1)}$, \ $x^{(2)}$, \ $y^{(1)}$ and  $y^{(2)}$
if $x\in\Om1$ and $\frac{x+y}2\in\Omm1$. We have
$$
\aligned
&|x^{(2)}|<\co{-0.4}\\
&|x^{(1)}-M|<\frac32\co{-0.4}
\endaligned
\aligned
\qquad\qquad \text{if}\quad x\in\Om1\,.
\endaligned   \tag 5.3
$$
The second relation in (5.3) holds, since
$$
\align
&|x^{(1)}-M|=\bigl|(|x|^2-x^{(2)2})^{1/2}-M\bigr|=
\left|\,|x|-\frac{x^{(2)2}}{2|x|}+O(x^{(2)4})-M\right|\\
&\qquad\le \bigl||x|-M\bigr|+\left|
\frac{x^{(2)2}}{2|x|}+O(x^{(2)4})\right|\le
\co{-0.4}+K\co{-0.8}\le\frac32\co{-0.4}\,.
\endalign
$$
Similarly, if $\frac{x+y}2\in \Omm1$ then $|\frac{x^{(2)}
+y^{(2)}}2|<\co{-0.4}$, and
$|\frac{x^{(1)}+y^{(1)}}2-M|<2\co{-0.4}$. Hence
$$
|\y2|\le |-\x2|+|\x2+\y2|\le 3\co{-0.4}, \tag 5.4
$$
$$
|\y1-M|=|M-\x1|+|\x1+\y1-2M|
\le6\co{-0.4}\, \tag 5.4${}^{\prime}$
$$
and
$$
\align
|r-M|=||y|-M|&=\left|(y^{(1)2}+
y^{(2)2})^{1/2}-M\right|\tag 5.4${}^{\prime\prime}$\\
&=\left|y^{(1)}+\frac{y^{(2)2}}{2\y1}+O(y^{(2)4})-M\right|
   \le 10\co{-0.4}
\endalign
$$
if $x\in \Om1$ and $\xy\in\Omm1$. In particular, (5.4${}^{\prime\prime}$)
implies that
$$I_n(r)=0 \qquad\text{if }
|r-M|>10\co{-0.4},\text { and } x\in\Om1\,.
\tag 5.5 $$
 
Furthermore, $|\vf|\le2|\sin\vf|  =\frac2r|\y2|=O(\co{-0.4})$
and $\y1=r(1-\cos\vf
)\allowmathbreak=r(1-\vf^2/2)+O(\co{-1.6})$,  \
$\y2=r\sin\vf\allowmathbreak=r\vf+O(\co{-1.2})$ and
$r=M+O(\co{-0.4})$ if $\fformg$. Hence
$$
\align
I_n(r)&=\left(1+O(\co{-0.2})\right)
\int_{\formg}M\exp\Biggl\{\frac{c^n}T
\left(\x1r\biggl(1-\frac{\vf^2}2\biggr)+\x2r\vf\right)\\
&\qquad
+g_{n+1}\left(\frac{\x1+r(1-\frac{\vf^2}2)}2-M\right)
+A_{n+1}\left(\frac{\x2+r\vf}2\right)^2\Biggr\} \,d\vf\\
&=\left(1+O(\co{-0.2})\right)M\exp
\left\{\frac{c^n}Tr\x1
+\frac{g_{n+1}}2\biggl(\frac{\x1+r}2-M\biggr)
+\frac{A_{n+1}}4 \xo2\right\}\\
&\qquad
\int_{\formg}\exp\biggl\{-\biggl(\frac{c^n}{2T}\x1
+\frac{g_{n+1}}4-r
\frac{A_{n+1}}4\biggr)r\vf^2\\
&\hskip6truecm +
\biggl(\frac{c^n}T+\frac{A_{n+1}}2 \biggr)\x2 r\vf\biggr\}\,d\varphi\,.
\endalign
$$
Moreover, since $c^n\x1
r\vf^2=c^nM^2\vf^2\allowmathbreak+O(\co{-0.2})$,
$g_{n+1}r\vf^2\allowmathbreak=g_{n+1}M\vf^2\allowmathbreak
+O(\co{-0.2})$, $A_{n+1}r^2\vf^2\allowmathbreak
=A_{n+1}M^2\vf^2\allowmathbreak+O(\co{-0.2})$
and $\ca\x2 r\vf \allowmathbreak=\ca \x2M\vf
\allowmathbreak+O(\co{-0.2})$
in our case (observe that $g_{n+1}\le Dc^{n+1}$ and
$A_{n+1}\le\ab
c^{n+1}$), hence we make an error of order $O(\co{-0.2})$ by
substituting $\x1$ and $r$ by $M$ in the last integral, i.e.
$$
\align
&I_n(r)=\left(1+O(\co{-0.2})\right)  M
\exp\left\{\frac{c^n}Tr\x1+\frac
{g_{n+1}}2\biggl(\frac{\x1+r}2-M\biggr)
+\frac{A_{n+1}}4\xo2\right\}\\
&\qquad\int_{\formg}\exp\left\{\cga M^2\vf^2+\ca
M\x2\vf\right\}\,d\vf               \,,
\endalign
$$
or equivalently
$$
\align
I_n(r)&=\left(1+O(\co{-0.2})\right)
\exp\biggl\{\frac{c^n}Tr\x1+\frac
{g_{n+1}}2\biggl(\frac{\x1+r}2-M\biggr)\\
&\qquad+\frac{A_{n+1}}4\xo2+
\frac{\ca^2\xo2}{4\cga}\biggr\}\\
&\qquad\int_{\formg}M\exp\left\{-M^2\cga
(\vf-\gamma_n\x2)^2\right\}\,d\vf   \,,
\endalign
$$
with
$$
\gamma_n=\frac{\ca}{\cga}\,.
$$
By (2.5${}^{\prime}$) we get from this relation that
$$
\align   &\tag 5.6\\
I_n(r)&=\left(1+O(\co{-0.2})\right)
\exp\left\{\frac{c^n}Tr\x1+\frac
{g_{n+1}}2\biggl(\frac{\x1+r}2-M\biggr)+A_n\xo2
 \right\}\\&\qquad\int_{\formg}M\exp\left\{-M^2\cga
\left(\vf-\gamma_n\x2\right)^2\right\}\,d\vf\,.
\endalign
$$
Since
 $$
\cga\ge
c^n\left(\frac1{2T}+c\frac{\gb}{4M}-c\frac{\ab}4\right)=Kc^n
\tag5.7
$$
with $K= \frac1{2T}+c\frac{\gb}{4M}-c\frac{\ab}4=
\frac{c(4-c)}{4(2-c)T}>0$ relation (5.6) implies that
$$
\align
I_n(r)&\le\sqrt\pi\left(1+O(\co{-0.2})\right)
\cga^{-1/2}
\tag5.8\\ &\qquad\exp\left\{\frac{c^n}Tr\x1+\frac
{g_{n+1}}2\left(\frac{\x1+r}2-M\right)+A_n\xo2
 \right\}
\endalign
$$
Moreover, we claim that
$$
\aligned
I_n(r)&=\left(1+O(\co{-0.2})\right)\sqrt   \pi\
\cga^{-1/2}\\ &\quad\exp\left\{\frac{c^n}Tr\x1+\frac
{g_{n+1}}2\left(\frac{\x1+r}2-M\right)+A_n\xo2
 \right\}\quad\text{if}\;|r-M|<\eb\co{-0.4}
\endaligned \tag 5.9
$$
with some approppriate $\eb>0$. Because of (5.6) and (5.7) to
prove (5.9) it is enough to show that there is some
$\e=\e_0(\eb)>0$ such that
$$
\bigl\{\vf\:|\vf-\gamma_n
\x2|<\e_0\co{-0.4}\bigr\}\subset\Gamma_n(x,r)
\qquad \text{if }\;|r-M|<\eb\co{-0.4}\,. \tag 5.10
$$
Since $A_{n+1}\le \ab c^{n+1}$ and $g_{n+1}\ge \gb c^{n+1}$
$$
\gamma_n=\frac{\ca}{2\cga}\le\frac{1
+\frac{2-c}2}{(1+\frac
c{2-c}-\frac{2-c}2)M}=\frac{2-c}{cM}\,,
$$
and we prove (5.10) by showing that
$$
\left\{\frac{\x1+r\cos\vf}2,\,
\frac{\x2+r\sin\vf}2\right\}\in\Omm1\tag 5.10${}^{\prime}$
$$
if $|\vf-\gamma_n\x2|<\e_0\co{-0.4}$, $|r-M|\le \eb\co{-0.4}$
and $x\in\Om1$. But in this case
$$
\align
&\left|\frac{\x2+r\sin\vf}2\right| \le
\frac12\left(|\x2|+r|\vf|\right)\le\frac12
\left(|\x2|+r(\gamma_n|\x2|+|\vf-\gamma_n\x2|)\right)\\
&\qquad\le \frac12\biggl[\co{-0.4}+(M+\bar\e\co{-0.4})
\biggl(\frac{2-c}{cM}\co{-0.4}
+\e_0\co{-0.4}\biggr)\biggr]\\
&\qquad\le \frac 1{\sqrt
c}\co{-0.4}\le c^{-0.4(n+1)}
\endalign
$$
if $\e_0>0$ and $\eb>0$ are sufficiently small. We also get
with the help of (5.3${}^{\prime}$) that
$$
\align
&\left|\frac{\x1+r\cos \vf}2-M\right|\le
\left|\frac{\x1-M}2\right|+\frac
r2\left|\cos\vf-1\right|+\left|\frac{r-M}2\right|\\
&\qquad\le\left|\frac{\x1-M}2\right|
+\frac{r\vf^2}2+\left|\frac{r-M}2\right|\\
&\qquad\le\left|\frac{\x1-M}2\right|+\frac r2\left(\gamma_n|\x2|
+\e_0\co{-0.4}\right)^2+ \left|\frac{r-M}2\right| \\
&\qquad\le\left(\frac34+\eb\right)\co{-0.4}+K\co{-0.8}\le
c^{-0.1}c^{-0.4(n+1)}
\endalign
$$
if $\e_0>0$ and $\eb>0$ are sufficiently small. The above
estimates imply (5.10${}^{\prime}$)  hence also (5.10). Now we can
estimate the term $\int_0^{\infty}I_n(r)\bar p_n(r)\,dr$.
Relation (5.9) yields that
$$
\align
\int_{|r-M|<\eb \co{-0.4}}I_n(r)\bar p_n(r)\,dr
&=\left(1+O(\co{-0.2})\right)\sqrt{\pi}\cga^{-1/2}\je\\
&\qquad\exp\left\{\frac{c^n}T\x1 M
+\frac{g_{n+1}}4\left(\x1-M\right) +A_n\xo2\right\}
\endalign
$$
with
$$
\align
\je=\je(\x1)&=\int_{|r-M|<\eb\co{-0.4}}
\exp\left\{\frac{c^n}T\x1 (r-M)+\frac{g_{n+1}}4(r-M)
\right\}\bar p_n(r)\,dr\\
&=\int_{|t|<\eb\co{0.6}}\exp\left\{
\frac{t\x1}T+c\frac{\gb_{n+1}}4 t\right\}f_n(t)\,dt
\endalign
$$
with the function $f$ defined in (4.11). On the other hand,
by (5.8) and (5.5)
$$
\align
&\int_{|r-M|>\eb \co{-0.4}}I_n(r)\bar p_n(r)\,dr\\
&\qquad\le\left(1+O(\co{-0.2})\right)\sqrt{\pi}\cga^{-1/2}\\
&\qquad\qquad\exp \left\{\frac{c^n}T\x1 M+\frac{g_{n+1}}4(\x1-M)
+A_n\xo2\right\}\jeb
\endalign
$$
with
$$
\jeb=\int_{10\co{0.6}>|t|>\eb\co{0.6}}\exp\left\{\frac{t\x1}T+
c\frac{\gb_{n+1}}4t\right\}f_n(t)\,dt\,.
$$
 
Let us remark that $\je=\je(\x1)$ depends on $\x1$. We show
that this dependence is very weak. Namely, since
$$
\left|\frac {d\;\;{}}{d\x1}\je(\x1)\right|=
\int_{|t|<\eb\co{0.6}}\exp\left\{\frac{t\x1}T+
c\frac{\gb_{n+1}}4t\right\}\frac{|t|}Tf_n(t)\,dt\le C<\infty
$$
by (4.11${}^{\prime}$) and (4.11${}^{\prime\prime}$), and for $\x1=M$  the expression
$$
\je(M)=
\int_{|t|<\eb\co{0.6}}\exp\left\{\frac{tM}T+
c\frac{\gb_{n+1}}4t\right\}f_n(t)\,dt
$$
satisfies the relation
$$
0<K_1<\je(M)<K_2<\infty \tag 5.11
 $$
because of (4.11${}^{\prime}$), (4.11${}^{\prime\prime}$), the inequality
$0\le\gb_{n+1}\le D$ and the relation $f_n(t)\ge const.>0$
for $|t|<1$ that follows from Theorem A. Hence
$$
\je(\x1)=\left(1+O(\co{-0.4})\right)\je(M)\qquad\text{if
}\;x\in\Om1\,. $$
Similarly,
$$
0\le\jeb=O\left(\exp(-K\co{0.6})\right)\,.
$$
The above relations imply that
$$
\align\int    I_n(r)\bar p_n(r)\,dr
&=\left(1+O(\co{-0.2})\right)\sqrt{\pi}\cga^{-1/2}\je(M)\\
&\qquad\exp\left\{\frac{c^n}T M^2+g_n(\x1-M)
+A_n\xo2\right\} \,.
\endalign
$$
The last formula together with (5.1) and (5.2) imply (4.1)
with
$$
L_n=\sqrt{\pi}\cga^{-1/2}\je(M)
\exp\left(\frac{c^n}TM^2\right)\,. $$
Since $\cga<const.c^n$ relation (5.11) and the last formula
imply (4.1${}^{\prime\prime}$).
 
\medpagebreak
b) The estimation of $\s1$ for $x\in\Om2$.
 
We divide $\Om2$ to two subsets $\bar \Om2$ and $\obb$,
where we apply different arguments. Put
$$           \align
&\Omega^1_{n,\e}=
\left\{x,\bigl||x|-M\bigr|\le\co{-0.4},|\x2|\le
(1+\e)\co{-0.4},\,\x1>0\right\},\;\e>0,    \\
&\bar\Om2=     \Omega^1_{n,\e} -\Om1\,,
\endalign $$
 and
$$
\obb = \left\{x,\bigl||x|-M\bigr|\le\co{-0.4}\right\}-
\Omega^1_{n,\e}\,.
 $$
Clearly, $\Om2=\bar\Om2\cup\obb$. The domain
$\Omega^1_{n,\e}$
is a slight enlargement of $\Om1$. It is not difficult to see
by analizing the proof of relation (4.1) that for
sufficiently small $\e>0$
$$
\s1=L_n\exp\left\{g_n(\x1-M) +A_n\xo2+\eb(x)+\ebb(x)\right\}
$$
if $x\in\Omega^1_{n,\e}$ with some $|\ebx|<q^{n+1}$ and
$\ebb_n(x)<K\co{-0.2}$. Since $\x1-M=\allowmathbreak |x|-M
-\frac{\xo2}{2M}+O(\co{-1.2})$ and $|\x2|\ge\co{-0.4}$ for
$x\in\bar\Om2$, the above relation implies that
$$
\align
\s1&=L_n\exp\biggl\{g_n(|x|-M)
-\biggl(\frac{g_n}{2M}-A_n\biggr)\xo2\\
&\qquad\qquad\qquad\qquad +\eb_n(x)+
\ebb_n(x)+O\left(\co{-0.2}\right)biggr\}\\
&\le L_n\exp\left\{g_n(|x|-M)-\left
(\frac{g_n}{2M}-A_n\right)\co{-0.8}+q^{n+1} +K\co{-0.2}
\right\}
\endalign
$$
in this case, what we had to show. For $x\in\obb$ we define
$$
\s{\e}=\int_{R^2-V^{\e}_n(x)}\exp\biggl(\frac{c^n}Txy\biggr)
f\biggl(\xy\biggr)p_n(y)\,dy
$$
and
$$
T^{\e}_nf(x)=\int_{\{y,\;\xy\in\Omm1,\;y\in V^{\e}_n(x)\}}
\exp\left(\frac{c^n}Txy\right) f\left(\xy\right)p_n(y)\,dy\,,
$$
where $V^{\e}_n(x)$ is defined in (4.8${}^{\prime}$), and $\e>0$ is
appropriately chosen. The function $\s{\e}$  will be bounded
with
the help of Part b) of Lemma 2, and $T^{\e}_nf(x)$ similarly
to $\s1$ in the case $x\in\Om1$. To apply Lemma 2 first we
show that under the conditions of Proposition~1${}^{\prime}$
$$
0\le
f(x)\le2\exp\left\{\frac{g_{n+1}}{2M}\left(|x|^2-M^2\right)
\right\}\qquad\text {for all }\;x\in R^2\,.\tag 5.12
$$
For $x\in\Omm1$
$$
\x1-M=|x|-M-\frac{\xo2}{2|x|}+O
\left(\xo4\right)=|x|-\frac{\xo2}{2M}-M
+O\left(\co{-1.2}\right)\,,
$$
hence
$$
0\le
f(x)\le\exp\left\{g_{n+1}(|x|-M)-
\biggl(\frac{g_{n+1}}{2M}-A_{n+1}\biggr)
\xo2+\e_{n+1}(x)+O(\co{-0.2})\right\}\,,
$$
and since $\frac{g_{n+1}}{2M}-A_{n+1}>0$ hence
$$
0\le f(x)\le\frac32\exp\{g_{n+1}(|x|-M)\}\,.
$$
This inequality together with the relation
$$
|x|-M=\frac1{2M}
\left[(|x|^2-M^2)-(|x|-M)^2\right]\le\frac1{2M}\left(
|x|^2-M^2\right) $$
imply (5.12) for $x\in\Omm1$.
Similarly, for $x\in \Omm2$  the relation
$$  \align
0\le
f(x)&\le\exp\left\{g_{n+1}(|x|-M)
-\biggl(\frac{g_{n+1}}{2M}-A_{n+1} \biggr)
c^{-0.8(n+1)}+q^{n+1}
\right\}\\ &\le\frac32\exp\{g_n(|x|-M)\}   \endalign
$$
implies (5.12), and this relation also holds for $x\in\Omm3$
by relation (2.17).
 
By (5.12) part b) of Lemma 2 can be applied for $\frac12
f(x)$, and it yields that
$$
\s{\e}\le
2\exp\left\{\frac{c^n}TM^2+g_n(|x|-M)-c^{n/2}\right\}
\qquad \text{if }x\in\Om2\;\;(\text{or }x\in\Om1)\,.
$$
Since $L_n\ge c^{-n}\exp\{\frac{c^n}TM^2\}$, and
$(\ga)\co{-0.8}=O(\co{0.2})\allowmathbreak\le\frac13c^{n/2}$,
the last inequality implies that
$$
\align
\s{\e}\le
L_n\exp&\left\{g_n(|x|-M)-
\biggl(\ga\biggr)\co{-0.8}-\frac12c^{n/2}\right\}
\tag 5.13\\&\qquad\qquad\qquad \qquad\qquad\text{if
}x\in\Om2\;\;(\text{or }x\in\Om1)\,. \endalign
 $$
To estimate $T_n^{\e}f(x)$ first we show that if $x\in\Om2$,
\ $y\in V_n^{\e}(x)$ and $\xy\in\Omm1$ then
$$
\align
&|\x2|<2\sqrt{\e}\co{-0.2},\;\; |\y2|<2\sqrt{\e}\co{-0.2},\;\;
 |\x1-M|\le2
\sqrt{\e}\co{-0.2},\tag 5.14   \\
&|\y1-M| \le2\sqrt{\e}\co{-0.2},\;\;\;\bigl||y|-M\bigr|\le
\e\co{-0.4}\,.
\endalign
$$
Indeed, in this case $(x,y)\ge |x| |y|-\e\co{-0.4}$, and
$$
\align
0&\le |x-y|^2=|x|^2+|y|^2-2(x,y)\le
|x|^2+|y|^2-2|x||y|+\e\co{-0.4}  \\
&=(|x|-|y|)^2+\e\co{-0.4}   \le2(|x|-M)^2+2(|y|-M)^2
+\e\co{-0.4} \\
&\le2(1+\e)\co{-0.8}+\e\co{-0.4} \le2{\e}\co{-0.4}\,.
\endalign
$$
Hence $|\frac{\x2-\y2}2|\le\sqrt{\e} \co{-0.2}$ and $\frac12
|(\x1-M)-(\y1-M)|\allowmathbreak\le \sqrt{\e}\co{-0.2}$.
Since
$\xy\in\Omm1$, \ $|\frac{\x2+\y2 }2|\allowmathbreak\le
c^{-0.4 (n+1)}\allowmathbreak\le \sqrt{\e}\co{-0.2}$ and
$\frac12|(\x1-M)+(\y1-M)|\le\allowmathbreak\frac32c^{-0.4(n+1)}
\allowmathbreak\le\sqrt{\e}\co{-0.2}$ by (5.3). These
relations
together with the definition of $V_n^{\e}(x)$ imply (5.14)
that enables us to estimate $\te$ similarly to $\s1$ for
$x\in\Om1$.
 
We get that
$$
\te\le(1+q^{n+1})\int_0^{\infty} I_n(r)\bar p_n(r)\,dr
\tag 5.15
$$
with
$$
\aligned
I_n(r)&=\int_{\formg}r\exp\Biggl\{\frac{c^n}T
\left(\x1\y1+\x2\y2 \right)\\
&\qquad+g_{n+1}\biggl(\frac{\x1+\y1}2-M\biggr)
A_{n+1}\biggl(\frac{\x2+\y2}2 \biggr)^2\Biggr\}\,d\vf \,,
\endaligned \tag 5.16
$$
where $\y1=r\cos\vf$, \ $\y2=r\sin\vf$, \ $y=(\y1,\y2)$ and
$$
\Gamma_n
(r,x)=\{\vf\:\xy\in\Omm1, y\in V_n^{\e}(x)\}  .
$$
 
Observe that by (5.14) $|r-M|<\e\co{-0.4}$, \ $ |\vf|\le2|\sin
\vf|\allowmathbreak
=\frac2r|\y2|\allowmathbreak\le\frac2r\sqrt{\e}\co{-0.2}$.
Let
us make the change of variables $z=\sin\vf$ in the integral
$I_n(r)$. We have $\y2=rz$,     \
$\y1=r(1-\frac{y^{(2)2}}{r^2})^{1/2}\allowmathbreak=
r(1-\frac{y^{(2)2}}{2r^2})+O(y^{(2)4})$, \ $z\le
2\frac{\sqrt{\e}}r\co{-0.2}$, hence $|\y1-r(1-\frac{z^2}2)|
\allowmathbreak \le K\e^2\co{-0.8}$ with some $K>0$
independent of $\e$. These relations imply that if $\fformg$
then
$$
\align
&\frac{c^n}T\left(\x1\y1+\x2\y2  \right)
+g_{n+1}\biggl(\frac{\x1+\y1}2-M\biggr)+
A_{n+1}\biggl(\frac{\x2+\y2}2\biggr)^2  \\
&\quad \le \frac{c^n}Tr\left(\x1\biggl(
1-\frac{z^2}2\biggr)+\x2 z\right)+g_{n+1}
\left(\frac{\x1+r(1-\frac{z^2}2)}2-M\right)\\
&\qquad\qquad+A_{n+1}\biggl(\frac{\x2+rz}2\biggr)^2+K\e^2\co{0.2}
\endalign
$$
with some $K>0$. Since $\frac{dz}{d\vf}>\frac12$ and
$|r-M|<\e\co{-0.4}$ if $\Gamma_n(r,x)$ is not empty, the
above inequality together with (5.16) imply that
$$
I_n(r)=0\qquad\text{if }
|r-M|\ge\e\co{-0.4}\tag 5.17
$$
and
$$
I_n(r)\le3\exp(K\e^2\co{0.2})\hat I_n(r)\qquad \text{for }\;
|r-M|<\e\co{-0.4}\tag 5.17${}^{\prime}$
$$
with
$$
\align
\hat I_n(r)&=\int_{-\infty}^{\infty}M\exp\Biggl\{
\frac{c^n}Tr\left(\x1\biggl(1-\frac{z^2}2\biggr)+\x2
z\right)\\
&\qquad+g_{n+1}\left(\frac{\x1 +r(1-\frac{z^2}2)}2-M\right)
+A_{n+1}\biggl(\frac{\x1+rz}2\biggr)^2\Biggr\}\,dz\,.
\endalign
$$
The expression $\hat I_n(r)$ can be calculated explicitly,
and we get that for $|r-M|<\e\co{-0.4}$
$$
\align
\hat
I_n=\sqrt{\pi}M\biggl(\frac{c^n}{2T}r\x1+&\frac{g_{n+1}}4r-
\frac{A_{n+1}}4r^2\biggr)^{-1/2}    \\
&\exp\left\{\frac{c^n\x1r}T+g_{n+1}\left(\frac{\x1
+r}2-M\right)+ A_n(\x1,r)\xo2\right\}
\endalign
$$
with
$$
A_n(\x1,r)=\frac{A_{n+1}}4
+\frac{\ca^2}{2(\frac{c^n\x1}{Tr}+
 \frac{g_{n+1}}{2r}-\frac{A_{n+1}}2)}\,.
$$
Hence
$$
\align
\hat I_n(r)&\le Kc^{-n/2}
\exp\left\{\frac{c^n\x1 r}T
+g_{n+1}\left(\frac{\x1+r}2-M\right)+
A_n(\x1,r)\xo2\right\}          \\
&=Kc^{-n/2}
\exp\biggl\{\frac{c^n}TM^2+g_n(\x1-M)+A_n(\x1,r)\xo2\\
&\qquad\qquad+\biggl(\frac{c^n\x1}T+\frac{g_{n+1}}2\biggr)
(r-M)\biggr\}\,.
\endalign
$$
Observe that $A_n=A_n(M,M)$, and if $x\in\obb$, \ $y\in
V_n^{\e}(x)$ and $\xy\in\Om1$ then
$$
\align
&|A_n(\x1,r)-A_n|\le K_0c^n(|\x1-M|+|r-M|)\\
&\left|\x1-M-\biggl(|x|-M-\frac{\xo2}{2M}\biggr)\right|\le
K_1\xo2(\xo2+|\x1-M|)\,,
\endalign
$$
hence (5.17${}^{\prime}$) implies that
$$
\aligned
I_n(r)&\le
\bar Kc^{-n/2}
\exp\Biggl\{K\e^2\co{0.2}+\frac{c^n}TM^2+g_n(\x1-M)+A_n\xo2\\
&\qquad+ K_0c^n(|\x1-M|+|r-M|) \xo2
+\left(\frac{c^n\x1}T+\frac {g_{n+1}}2\right)(r-M)\Biggr\}\\
&\le\bar Kc^{-n/2}
\exp\Biggl\{K\e^2\co{0.2}+\frac{c^n}TM^2+g_n(|x|-M)-\left(
\frac{g_n}{2M}-A_n\right)\xo2\\
&\qquad+\left[K_0c^n(|\x1-M|+|r-M|)+K_1g_n(\xo2+|\x1-M|)\right]\xo2\\
&\qquad+\left(\frac{c^n\x1}T+\frac{g_{n+1}}2\right)(r-M)\Biggr\}
\endaligned \tag5.18
$$
if $|r-M|\le\e\co{-0.4}$. Since $\frac {g_n}{2M}-A_n>\alpha
c^n$ with some $\alpha>0$ and $\xo2>(1+\e)\co{-0.4}$ if
$x\in\obb$
$$
\align
K\e^2\co{0.2}&+
\left[K_0c^n(|\x1-M|+|r-M|)+K_1g_n(\xo2+|\x1-M|)\right]\xo2
\\&-\left(\frac{g_n}{2M}-A_n\right)\xo2\\
&=K\e^2\co{0.2}
-\left(\frac{g_n}{2M}-A_n+O\left(\co{0.8}\right)\right)
\xo2\\
&\le K\e^2\co{0.2}
-\left(\frac{g_n}{2M}-A_n+O\left(\co{0.8}\right)\right )
(1+\e)^2\co{-0.8}  \\
&\le-\left(\frac{g_n}{2M}-A_n\right)(1+\e)\co{-0.8} \endalign
$$
if $\e>0$ is chosen  sufficiently small. This relation
together with (5.18) imply that
$$
\align
I_n(r)&\le
Kc^{-n/2}\exp\biggl\{\frac{c^n}TM^2+g_n(|x|-M)-(1+\e)
\biggl(\frac{g_n}{2M}-A_n\biggr)\co{-0.8}\\
&\qquad+\biggl(\frac{c^n\x1}T+\frac{g_{n+1}}2\biggr)(r-M)\biggr\}
\quad\text{if }|r-M|<\e \co{-0.4}\,.
\endalign
$$
With the help of this inequality, the estimate we have on the
function $\bar p_n(r)$ and (5.17)
we can bound the integral in (5.15) from above. We get that
$$
\te\le L_n\exp\left\{g_n(|x|-M)-\biggl(1+\frac{\e}2\biggr)
\biggl(\frac{g_n}{2M}-A_n\biggr)\co{-0.8}\right\}\,.
$$
Since $\s1 \le\s{\e}+\te$ the last inequality together with
(5.13) imply (4.4) for $x\in\obb$.
 
\medpagebreak
c) The estimation of $\s2$ for $x\in\Om1$ and $x\in\Om2$.
 
We have
$$
\align
0\le\s2&=
\int_{\{y,\xy\in\Omm2\}}
\exp\biggl(\frac{c^n}Txy\biggr)f\biggl(\xy\biggr)p_n(y)\,dy
\\&\le\exp
\left\{-\biggl(\gan\biggr)c^{-0.8(n+1)}+q^{n+1}\right\}\\
&\qquad\quad\int_{\{y,\xy\in\Omm2\}}
\exp\left\{\frac{c^n}Txy+g_{n+1}\biggl(\frac{|x+y|}2-M\biggr)
\right\}p_n(y)\,dy\,. \endalign
$$
It follows from Lemma 3 that for $x\in\Om1\cup\Om2$
$$
\align
0\le\s2\le K\exp&\biggl\{-\biggl(\gan\biggr)
c^{-0.8(n+1)}\\
&\qquad +q^{n+1} +\frac{c^n}TM^2+g_n(|x|-M)\biggr\}\,.
\endalign
$$
By Lemma 1 $g_{n+1}c^{-(n+1)}\ge g_nc^n$, \ $A_nc^{-n}\ge
A_{n+1}c^{-(n+1)}$ i.e. $g_{n+1}\ge cg_{n}$ and  $cA_n\ge
A_{n+1}$. Hence
$$
\biggl(\gan\biggr) c^{-0.8(n+1)}\ge
\biggl(\ga\biggr)\co{-0.8}c^{0.2}\ge
(1+\e)\biggl(\ga\biggr)\co{-0.8} $$
if $0<\e<c^{0.2}-1$. Since
$\e(\ga)\co{-0.8}>\e^{\prime}\co{0.2}$ with some appropriate
$\e^{\prime}(\e)>0$ we get that
$$
\biggl(\gan\biggr)c^{-0.8(n+1)} \ge
\biggl(\ga\biggr)\co{-0.8}+\e^{\prime}\co{0.2} $$
Therefore
$$
\align            & \tag 5.19\\
\s2&\le
K\exp\left\{\frac{c^n}TM^2+g_n(|x|-M)-\biggl(\ga\biggr)\co{-0.8}
+q^{n+1}-\e^{\prime}\co{0.2}\right\}  \\
&\le L_n\exp\left\{g_n(|x|-M)-\biggl(\ga\biggr)\co{-0.8}
-\frac{\e^{\prime}}2\co{0.2} \right\} \,.
\endalign
$$
This is estimate (4.5) (with $\frac{\e^{\prime}}2$ instead of
$\e^{\prime}$). For $x\in\Om1$
$$
\gnx=g_n(|x|-M)-\biggl(\ga\biggr) \xo2+O(\co{-0.2})\,,
$$
hence (5.19) implies that for $x\in\Om1$
$$
\s2\le
L_n\exp\left\{\gnx-\frac{\e^{\prime}}4\co{0.2}\right\}\,, $$
and this is relation (4.2) (with $\frac{\e^{\prime}}4$).
 
\medpagebreak
d) The estimation of $\s3$ for $x\in\Om1$ and $x\in \Om2$.
 
Clearly
$$
\s3 =\s{\e}+
\int_{\{y,\;\xy\in\Omm3\}\cap V_n^{\e}(x)}
\exp\biggl(\frac{c^n}Txy\biggr)f\biggl(\xy\biggr)p_n(y)\,dy
\,, $$
where $\s{\e}$ and $V_n^{\e}$ are defined in (4.8) and
(4.8${}^{\prime}$). The term $\s{\e}$ is bounded in (5.13). On the other
hand, we claim that there is some $\e_0=\e_0(c)$ such that if
$0<\e<\e_0$ and $x\in \Om1\cup\Om2$ then the set
  $ \{y,\xy\in\Omm3\}  \cap V_n^{\e}(x)$ is empty, hence the
last integral is zero. We have to show that if
$\bigl||x|-M\bigr|<\co{-0.4}$,      \
$y\in V_n^{\e}$, i.e.
$\bigl||y|-M\bigr|<\e\co{-0.4}$
and $xy>|x||y|-\e\co{-0.4}$ then $\xy\notin\Omm3$, i.e.
$-c^{-0.4(n+1)}\allowmathbreak\le|\frac{x+y}2|-M\allowmathbreak
\le c^{-0.4(n+1)}$.
 
Estimation from above:
$$
\left|\frac{x+y}2\right|-M\le
\frac{|x|-M}2+\frac{|y|-M}2\le\frac{1+\e}2\co{-0.4}
\le c^{-0.4(n+1)}
$$
if $\frac{1+\e}2\le c^{-0.4}$, what holds for sufficiently
small $\e$.
 
Estimation from below:
$$
\align
&\left|\frac{x+y}2\right|^2-M^2=\frac{|x|^2+|y|^2+2xy}4-M^2\\
&\qquad\ge\frac{|x|^2+|y|^2+2|x||y|-2\e\co{-0.4}}4-M^2=
\biggl(\frac{|x|+|y|}2\biggr)^2-M^2-\frac{\e}2\co{-0.4}\\
&\qquad\ge\left(\frac{M-\co{-0.4}+M-\e\co{-0.4}}2\right)^2-M^2-
\frac{\e}2\co{-0.4}\\
&\qquad\ge -(1+\e)M\co{-0.4}-\frac{\e}2\co{-0.4}=
-M\left(1+\e+\frac{\e}{2M}\right)\co{-0.4}\,.
\endalign
$$
Hence
$$
\align
\biggl|\frac{x+y}2\biggr|&\ge \left(M^2-M\biggl(1+\e
+\frac{\e}{2M}\biggr)\co{-0.4}\right)^{1/2}\\
&\ge M-\frac{(1+\e+\frac{\e}{2M})}2\co{-0.4}-K\co{-0.8}\\
&=M-\left[\biggl(\frac12+\frac{\e}2\biggl(1+\frac1{2M}
\biggr)\biggr) +K\co{-0.4}\right]\co{-0.4} \ge
M-c^{-0.4(n+1)} \endalign
$$
if  $\frac12+\frac{\e}2(1+\frac1{2M})+K\co{-0.4} \le
c^{-0.4}$, what we had to show.
 
Hence $\s3=\s{\e}$, and (5.13) implies (4.6). To prove (4.3)
we still have to remark that for $x\in \Om1$
$$
\align
&g_n(|x|-M)-\biggl(\ga\biggr) \co{-0.8}-\frac12c^{n/2}  \\
&\le g_n(|x|-M)-\biggl(\ga\biggr) \xo{2}-\frac12c^{n/2} \le
g_n(\x1-M)+A_n \xo2-\frac16c^{n/2}\,.
\endalign
$$
 
 \medpagebreak
e) The estimation of $S_nf(x)$ for $x\in\Om3$.
 
We get from (5.12) and Part a) of Lemma 2 that
$$
0\le S_nf(x) \le
 2c^n\exp\left\{\frac{c^n}TM^2+ \frac{g_n}{2M}(|x|^2-M^2)
-\frac{c^n}{3T}(|x|^2-M^2)\right  \}\,.
$$
For $x\in\Om3$  \
$\frac{c^n}{3T}(|x|^2-M^2)\ge\frac{c^{0.2n}}{3T}$, hence
(4.1${}^{\prime\prime}$) implies that
$$
\align
0\le S_nf(x) &\le
2c^n\exp\left\{\frac{c^n}TM^2+ \frac{g_n}{2M}(|x|^2-M^2)
-\frac{\co{0.2}}{3T}\right\}\\
&\le L_n\exp \left\{\frac{g_n}{2M}(|x|^2-M^2)\right\}\,,
\endalign
$$
i.e. relation (4.7) holds, as we have claimed. Proposition
1${}^{\prime}$ is proved.
 
\subheading{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
$n\ge n_0$ such that if
$n\ge n_0$ and $N\ge N_0(n,q)$  then
$$
\frac{d\mn}{d\mu_n}(x_1,\dots,x_{2^n})=
f_{n,N}^{h_N}\biggl(2^{-n}\sum_{j=1}^{2^n}x_j\biggl) \tag 6.1
$$
with
$$
\fn= L_n\exp\left\{\gb c^n(x^{(1)}-M)+\ab c^{n} x^{(2)2}+
\e_n(x)\right\}\qquad \text{for } x\in \Om1\,, \tag 6.2
$$     where
$$
 \sup_{x\in\Om1}|\e_n(x)|\le q^n\,,\tag 6.2${}^{\prime}$
$$
$$
\fn\le L_n\exp\left\{\gb c^n(|x|-M)-\biggl(\frac
  {\gb}{2M}-\ab\biggr)\co{0.2}
+q^n\right\}\qquad\text{for  } x\in\Om2\tag 6.3
$$
$$
\fn\le L_n\exp\left \{\frac{\gb
c^n}{M}(|x|^2-M^2)\right\}\qquad\text{if }\;x>M+\co{-0.4}
\tag 6.4 $$
$$
\fn\le L_n\exp \left\{\frac{\gb
c^n}{2M}(|x|^2-M^2)\right\}\qquad\text{if }\;0<x<M-\co{-0.4}
\tag 6.4${}^{\prime}$
$$
with some appropriate norming constant $L_n$ which satisfies
the relation
$$
C_1<c^{-n/2}L_n<C_2\qquad\text{with some
}\;0<C_1<C_2<\infty\,.\tag 6.5
$$
Indeed, Proposition 1 and Lemma 1 imply (6.1)--(6.4${}^{\prime}$) with
some norming constant $L_n=\lhn$. (In the domain $\Om3$ we
have divided the cases $|x|^2-M^2>0$ and $|x|^2-M^2<0$, since
here we apply that $\gb c^n<g_n<2\gb c^n$. ) 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
$$
1=\mn(\rn)= \int\fn p_n(x)\,dx=\int_{\Om1}  + \int_{\Om2}+
\int_{\Om3}\,.\tag 6.6
$$
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
$$ \align
&\im3\fn p_n(x)\,dx\le\int_{0}^{M-\co{-0.4}} \lhn\exp
\left\{\frac{\gb c^n}{2M}\left(r^2-M^2\right)\right\}r\bar
p_n(r)\,dr\\
&\qquad\qquad+\int_{M+ \co{-0.4}}^{\infty}\lhn\exp\left\{\frac{\gb
c^n}{M}\left(r^2-M^2\right)\right\}r\bar p_n(r)\,dr     \\
&\qquad=\lhn\biggl[\int_{-c^nM}^{-\co{0.6}}\exp\left\{
\frac{\gb}{2M}\biggl(2Mt+\frac
{t^2}{c^n}\biggr)\right\}\left(M+c^{-n}t\right)f_n(t)\,dt\\
&\qquad\qquad+\int_{\co{0.6}}^{\infty}\exp\left\{
\frac{\gb}{M}\biggl(2Mt+\frac
{t^2}{c^n}\biggr)\right\}\left(M+c^{-n}t\right)f_n(t)\,dt\biggr]\,.
\endalign
$$
Relations (4.11${}^{\prime}$) and (4.11${}^{\prime\prime}$)
imply that $$
\im3=\lhn O\left(\exp(-K\co{0.6})\right)\qquad \text{with
some }K>0\,.\tag 6.7
$$
Similarly,
$$
\im2\le\lhn
\exp\left\{-\frac12\biggl(\frac{\gb}{2M}
-\ab\biggr)\co{0.2}\right\}\,.\tag 6.7${}^{\prime}$ $$
(Observe that $\frac{\gb}{2M}-\ab>0$.)
 
Define the number $T_n$,
$$
T_n=\im1\exp\left\{\gb c^n(x^{(1)}-M)+\ab c^n
x^{(2)2}\right\}p_n(x)\,dx\,.\tag 6.8
$$
It follows from Theorem A that
$$
C_1c^{-n/2}<T_n< C_2c^{-n/2}\qquad  \text{with some
}0<C_1<C_2<\infty\,.\tag 6.9
$$
Indeed, since the expression in the exponent of (6.8) can be
written in the form
$$
\gb c^{n}(x^{(1)}-M)+\ab c^{n}x^{(2)2}=\gb
c^n(|x|-M)-\biggl(\frac{\gb}{2M}
-\ab\biggr)c^nx^{(2)2}+O\left(\co{-0.2}\right)\,, $$
 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${}^{\prime}$) shows that
$$
\left|\im1\fn p_n(x)\,dx -\lhn T_n\right|\le\lhn T_nq^n\,.
\tag 6.10 $$
Relations (6.6), (6.7), (6.7${}^{\prime}$) and (6.10) imply that
$$
1=\lhn T_n \left(1+\e_n+O(\exp(-\co{0.1}))\right)\,,\qquad
\text{and }\e_n\le q^n\,.
$$
The last relation implies that relations (6.1)--(6.4${}^{\prime}$) remain
valid if we choose $L_n=T^{-1}_n$, 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 $k\ge 0$, and define for all $n\ge k$ and
 measurable sets $A\subset \rk$ the
cylindrical set $A(n)=A\times (R^2)^{2^n-2^k}\subset \rn$.
Put
$$
\align
\tilde \mu_n(A)=L_n\int_{\tilde A(n)}&\exp\left\{\gb c^n2^{-n}
\sum_{j=1}^{2^n}(x^{(1)}_j-M)+\ab
c^n4^{-n}\biggl(\sum_{j=1}^{2^n}
x_j^{(2)}\biggr)^2\right\}\\
&\qquad\qquad\qquad\mu_n(\,dx_1\dots ,dx_{2^n})
\endalign
$$
with $\tilde
A(n)=A(n)\cap\{(x_1,\dots,x_{2^n}),\;2^{-n}\sum_{j=1}^{2^n}x_j\in
\Om1\}$.
We claim that if $n>n_0$ and $N>N_0(n,q)$ then
$$
\left|\tilde\mu_n(A(n))-\mn (A(n))\right|\le Kq^n\tag 6.11
$$
with some $K>0$ independent of the set $A$. Indeed,
$$
\align
&\left|\tilde\mu_n(A(n))-\mn (A(n))\right|\le
\int_{\Om2\cup\Om3}\fn p_n(x)\,dx\\
&\qquad+\int_{\tilde A(n)}
\mu_n(\,dx_1\dots, dx_{2^n})
\biggl|f^{h_N}_{n,N}\biggl(2^{-n}
\sum_{j=1}^{2^n}x_j\biggr)\\
&\qquad\qquad-L_n\exp\left\{\gb
c^n2^{-n}\sum_{j=1}^{2^n} (x_j^{(1)}-M)+\ab c^n 4^{-n}
\biggl(\sum_{j=1}^{2^n}x_j^{(2)}\biggr)^2\right\}\biggr|
=I_1+I_2\,.
\endalign
$$
It follows from (6.5), (6.7) and (6.7${}^{\prime}$) that
$$
I_1\le \exp\left(-\co{0.1}\right)\,.\tag 6.12
$$
 
On the other hand we increase the term $I_2$ by enlarging the
 domain of integration to the set $\{(x_1,\dots,x_{2^n}),\;
    \,2^{-n}\sum_{j=1}^{2^n}x_j\in \Om1\}$.
Hence
$$
I_2\le \im1\left|\fn -L_n\exp\left\{\gb c^n
(x^{(1)}-M)+\ab c^nx^{(2)2}\right\}\right|p_n(x)\,dx\,.
$$
The last inequality together with (6.2) and (6.2${}^{\prime}$)
 imply that
$$
I_2\le2 q^n\mn\left((R^2)^{2^n}\right)=2q^n\;.
$$
The last inequality together with
(6.12) imply (6.11). Since for all $k\ge0$ and measurable
sets $A\in \rk$, \ $k\le n\le N$ we have $\mn(A(n))=\mk(A)$,
relation
(6.11)  implies that for all $\e>0$ there is some $N_0(\e)$
such that for   $N>N_0(\e)$ and $N^{\prime}>    N_0(\e)$ the
relation
$|\mn(A)-\mu^{h_{N^{\prime}}}_{k,N^{\prime}}(A)|<\e$ holds
true.
Let us emphasize that the threshold $N(\e)$ does not depend
on the set $A$. Hence the last relation means that the limit
$\bar\mu_k(A)=\lim_{N\to\infty}\mk(A)$ exists, and the
convergence is uniform in $A$. This implies that
$\mk\to\bar\mu_k$
in variational metric. To complete the proof of Theorem 1 we
have
to show that the measure $\bar\mu_k$ 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 $\bar\mu_k(A)=
\lim_{n\to\infty}\tilde\mu(A(n))$, and the right hand side of
the last expression does not depend on $h_N$.
\demo{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 $\mn$ converge in variational metric
to the projection $\bar\mu_n$ of the measure $\bar\mu$ to
$\rn$ as $N\to \infty$. Since all measures are absolute
continuous with respect to the measure $\mu_n$ the above
convergence is equivalent to the convergence of the
Radon--Nikodym derivatives
$\frac{d\mn}{d\mu_n}=f_{n,N}^{h_N}$ to
$\frac{d\bar\mu_n}{d\mu_n}=\bar f$ in $L_1$ norm in the space
$(\rn,\mu_n)$ as $N\to\infty$. Since all functions $\fn$
satisfy (6.1)--(6.5)
for $N>N_0(n,q)$, their limit $\bar f$ also has this
property. Hence Proposition 2 holds true.\enddemo
 
\subheading{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  $Q_n=Q_n(k)$, \  $Q_n\:
(R^2)^{2^{n+k}}\to (R^2)^{2^k}$ in the following way:
For all $(x_1,\dots,x_{2^{n+k}})$, \ $x_j\in R^2$,
$j=1,\dots,2^{n+k}$
$$
Q_n (x_1,\dots,x_{2^{n+k}})=(y_1,\dots,y_{2^k}),\qquad y_j=
2^{-n}\!\!\sum_{l=(j-1)2^n+1}^{j2^n}\!\! x_l,\qquad
j=1,\dots,2^k. $$
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 $\nu$
is a probability measure on $(R^2)^{2^{n+k}}$  and $(\eta(1),
\dots,\eta(2^{n+k}))$ is a $\nu $ distributed vector then
$Q_n\nu$ is the distribution of the random vector $Q_n
(\eta(1),\dots,\eta(2^{n+k}))$. In Theorem 2 we have to
study
an appropriately rescaled version of the measure $Q_n\bar \mu
_{n+k}$. It is not difficult to see that relation (2.10)
implies that
$$
\frac{dQ_n\bar \mu_{n+k}} {dQ_n \mu_{n+k}}
(x_1,\dots,x_{2^{k}})=
\bar f_{n+k}\biggl(2^{-k}\sum_{j=1} ^{2^k}x_j\biggr).\tag
7.1 $$
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.
\proclaim{Theorem C}\it The above defined measure
$Q_n\mu_{n+k}=Q_n\mu_{n+k}
(T,t)$ has a density function $h_k(x_1,\dots,x_{2^k})$ of the
form
$$
\align
h_k(x_1,\dots,x_{2^k})=L(T,t,n,k)\exp&\left\{-\frac1T\Cal
 H_k\left(c^{n/2}x_1,\dots,c^{n/2}x_{2^k}\right)\right\}
\prod_{j=1}^{2^k}p_n(x_j)\,\\
&\qquad\qquad\qquad x_j\in R^2,\quad j=1,\dots,2^k.
\endalign
$$
Here $\Cal H_k$ is the Hamiltonian function defined in (1.2${}^{\prime}$)
of Part I, $p_n(x)$ is the function appearing in Theorem A,
and $L$ is an appropriate norming constant.\endproclaim
Formula (7.1) and Theorem C enable us to express the density
function of the random vector $\{\Cal R_n \sigma^{(1)}(j), \,
\Cal R_n \sigma^{(2)}(j),\;1\le j\le 2^k\}$ with the help of
the functions $p_n(x)$ and $f_n(x)$, where  the sequence
$\{\sigma(j),\;j\in \bold Z\}$ is $\bar\mu$ distributed, and
$\Cal R_n\sigma^{(1)}$, $\Cal R_n\sigma^{(2)}$ are defined in
(1.2) and (1.3) of Part II. This density equals to
$$  \align    & \tag 7.2   \\
&h_{n,k}(x_1,\dots,x_{2^k})\\&\qquad=L_{n,k}\bar
f_{n+k}\biggl(2^{-k}\sum_{j=1}^{2^k}\tilde
x_j\biggr)\exp\left\{-\frac1T\Cal
H_k\biggl(c^{n/2}\tilde x_1,\dots,c^{n/2}\tilde
x_{2^k}\biggr)\right\}
\prod_{j=1}^{2^k}p_n(\tilde x_j)\,
\endalign
$$
with
$$
\tilde x =
\tilde
x(x)=\left(M+c^{-n}x^{(1)},c^{-n/2}x^{(2)}\right)\qquad
\text{for }\;x=\left(x^{(1)},x^{(2)}\right)\,.   \tag 7.2${}^{\prime}$
$$
Let us define the sets $W_n\subset R^2$  and $ \bar
W_n\subset R^2$  by the formulas
$$
\align
&\bar W_n=\left\{(x^{(1)},x^{(2)}),\;M-
\frac {n\eta} {c^{n}}<|x|<M+\frac{\eta
n^{1/\alpha}}{c^{n}},\;|x^{(2)}|<c^{-0.45n},\,x^{(1)}>0\right\}
\\
&W_n=\left\{(x^{(1)},x^{(2)}),\;\tilde x(x)\in \bar
W\right\}\,,\endalign $$
where $\eta$ and $\alpha$  are the same constants as in
Theorem
A, and $\tilde x(x)$ is defined in (7.2${}^{\prime}$). We shall show that
there is  some $n_0>0$ and $0<q<1$ such that
$$
P\left(\,\bigl(\Cal R_n\sigma^{(1)}(j),\;\Cal
R_n\sigma^{(2)}(j)\,\bigr)\notin W_n\right)\le
q^n\qquad \text{if }\,n\ge n_0\tag 7.3
$$
for a $\bar\mu$ distributed random field $\sigma(j)$,  \
$j\in\bold
Z$, and give a good asymptotic formula for the expression
$h_{n,k}(x_1,\dots,x_{2^k})$ defined in (7.2) if $x_j\in W_n$
for all $1\le j\le 2^k$. First we prove ~(7.3).
$$    \align
&P\left(\bigl(\Cal R_n\sigma^{(1)}(j),\;\Cal
R_n\sigma^{(2)}(j)\,\bigr)\notin
W_n\,\right)=P\biggl(2^{-n}\sum_{l=1}^{2^k}\sigma(k)\notin\bar
W_n\biggr) \\&\qquad= \int_{R^2-\bar W_n}\bar
f_n(x)p_n(x)\,dx= \int_{\Omega_n^1-\bar W_n}+
\int_{\Omega_n ^2} +
\int_{\Omega_n^3}=I_1+I_2+I_3\,.
\endalign
$$
We get similarly to the estimates (6.7) and (6.7${}^{\prime}$) that
$$  \align
 &I_3\le \exp\left(-Cc^{0.6n}\right),\\&I_2\le
const.\exp\left\{-\biggl(\frac {\bar g}{2M}-\bar
A\biggr)c^{0.2n}\right\}\le \exp\left(-Cc^{0.2n} \right)
\endalign $$
The term $I_1$ has to be estimated a little more carefully.
 
Define
$$
\tilde
\Omega_n^1=\left\{(x^{(1)},x^{(2)}),\;\bigl||x|-M\bigr|<c^{-0.4n},\;
|x^{(2)}|<c^{-0.45n},\;x^{(1)}>0\right\}\,,
$$
and write
$$
I_1=I_{1,1}+I_{1,2}=\int_{\Omega_n^1-\tilde\Omega_n^1}+
\int_{\tilde\Omega_n^1-\bar W_n}\,.
$$
We get, similarly to the estimation of $I_3$ and $I_2$ that
$I_{1,1}\le\exp(-Kc^{0.1n})$, and we can write by (2.11) and
(2.11${}^{\prime}$) that
$$\align
I_{1,2}&\le 2L_n\int_{\tilde\Omega^1_n-\bar W_n}
 \exp\left\{\bar gc^n(x^{(1)}-M)+\bar
Ac^nx^{(2)2}\right\}p_n(x)\,dx
\\&\le 3L_n\int_{\tilde\Omega^1_n-\bar W_n}
\exp\left\{\bar gc^n(|x|-M)-\biggl(\frac{\bar
g_n}{2M}-A\biggr) c^nx^{(2)2}\right\}p_n(x)\,dx \,.
\endalign
$$
Then integrating first by $x^{(2)}$ we get that
$$
I_{1,2}\le KL_nc^{-n/2}\biggl[\int_{M-c^{-0.4n}}^{M-\eta
nc^{-n}}+\int_{M+\eta
n^{1/\alpha}c^{-n}}^{M+c^{-0.4n}}\biggr]
\exp\left(\bar gc^n(r-M)\right)\bar p_n(r)\,dr\le q^n
$$
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 $x_j\in W_n$,
\ $j=1,2,\dots,2^k$, we make some preparatory remarks. Put
$$\ell_n(x)=c^n(|\tilde
x(x)|-M)=c^n\left\{\left[(M+c^{-n}x^{(1)})^2
+c^{-n}x^{(2)2}\right]^{1/2}-M\right\} $$
We have
$$
\align
\ell_n(x)=x^{(1)}+\frac{x^{(2)2}}{2M}&+O\left(c^{-n}
\Bigl(x^{(2)4}+|x^{(1)}| x^{(2)2}\Bigr)\right)\tag 7.4\\
&=x^{(1)}+\frac{x^{(2)2}}{2M}+O\bigl(c^{-0.8n}\bigr)\qquad
\text{if} \quad x\in W_n\,,
\endalign
$$
because, as it is not difficult to see,
$c^{-n/2}|x^{(2)}|\allowmathbreak<c^{-0.45n}$, and $
M-2c^{-0.9n}\allowmathbreak<M+c^{-n}x^{(1)}\allowmathbreak<M+2c^{-0.9n}
$ if $x\in W_n$. We show with the help of Theorem A and (7.4)
that
$$
p_n(\tilde
x)=\exp\left\{-\frac{a_0}T\biggl(Mx^{(1)}
+\frac{x^{(2)2}}2\biggr)\right\}
g\left(\frac{a_1}T\biggl(Mx^{(1)}+\frac
{x^{(2)2}}2\biggr)\right)\left(1+O(q^n)\right) \tag 7.5 $$
with some $ 0<q<1$ for $x\in W_n$ if $\eta>0$ is chosen
sufficiently small in Theorem A. Indeed, by Theorem A
$$
p_n(\tilde x)=\exp
\left\{-\frac{a_0M}T\ell_n(x)\right\}
g\left(\frac{a_1M}T\ell_n(x)\right)\left(1+O(q^n)\right)\,,
$$ $$
\exp\left\{-\frac{a_0M}T\ell_n(x)\right\}=
\exp
\left\{-\frac{a_0}T\biggl(Mx^{(1)}
+\frac{x^{(2)2}}2\biggr)+O\left(c^{-0.8n}\right)\right\}\,,
$$ and
$$
\left|g\biggl(\frac{a_1}TM\ell_n(x)\biggr)
-g\biggl(\frac{a_1}T(Mx^{(1)}
+\frac{x^{(2)2}}2)\biggr)\right|=O(c^{-0.8n})     \tag 7.6
$$
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 $-2\eta
n<x^{(1)}+\frac{x^{(2)2}}{2M}<2\eta n^{1/\alpha}$ if $x\in
W_n$, which holds because of the definition of $W_n$, and the
inequality
$$
\left|c^n\bigl(|\tilde
x(x)|-M\bigr)
-\biggl(x^{(1)}+\frac{x^{(2)2}}{2M}\biggr)\right|
\le Kc^{-0.8n} $$
we have
$$
g\left(\frac{a_1}T\biggl(Mx^{(1)}+\frac{x^{(2)2}}2\biggr)\right)>c^{-0.3n}
$$
if $x\in W_n$, and $\eta$ is chosen in Theorem A sufficiently
small. Hence (7.6) can be rewritten as
$$
g\left(\frac{a_1}TM\ell_n(x)\right)
=g\biggl(\frac{a_1}T\biggl(Mx^{(1)}+\frac
{x^{(2)2}}2)\biggr)\left(1+O(c^{-n/2}\biggr)\right)\,.
$$
The above relations imply (7.5).
 
We also claim that
$$
\align
&\Cal H_k\left(c^{n/2}\tilde x_1,\dots,c^{n/2}\tilde
x_{2^k}\right)
\tag 7.7\\&\qquad=-\sum_{i=1}^{2^k-1}\sum_{j=i+1}^{2^k}
U(i,j)\left[M(x_i^{(1)}+x_j^{(1)})+x_i^{(2)}x_j^{(2)}+c^nM^2\right]
+O(q^n)
\endalign
$$
and
$$
\bar f_{n+k}\biggl(2^{-k}\sum_{j=1}^{2^k}\tilde
x_j\biggr)=L_{n+k}\exp
\left\{\frac{c^k}{2^k}\bar g\sum_{j=1}^{2^k}x_j^{(1)}+
\frac{c^k}{4^k}\bar
A\biggl(\sum_{j=1}^{2^k}x_j^{(2)}\biggr)^2
+O(q^n)\right\}\tag 7.8 $$
if $x_j\in W_n$, $j=1,\dots,2^k$.
 
Indeed,
$$
\align
&\Cal H_k\left(c^{n/2}\tilde x_1,\dots,c^{n/2}\tilde
x_{2^k}\right)
\\&\qquad=-\sum_{i=1}^{2^k-1}\sum_{j=i+1}^{2^k}
U(i,j)\left[c^n(M+c^{-n}x_i^{(1)})(M+c^{-n}x_j^{(1)})
+x_i^{(2)}x_j^{(2)}\right]\,,
\endalign
$$
hence to prove (7.7) it is enough to remark that in the last
expression the terms $c^{-n}x^{(1)}_ix^{(1)}_j$ are
negligibly small, since
$c^{-n}x^{(1)}_ix^{(1)}_j=O(c^{-0.8n})$ if
$x_i\in W_n$ and $x_j\in W_n$.
 
To prove (7.8) we have to show that
$2^{-k}\sum_{j=1}^{2^k}\tilde x_j\in \Omega^1_{n+k}$ if
$\tilde x_j\in \bar W_n$ for all $j=1,\dots,2^k$ and then
apply Proposition 2. We can write with the notation $\tilde
x_j=(\tilde x^{(1)}_j,\tilde x^{(2)}_j)$ that
$|2^{-k}\sum_{j=1}^{2^k}\tilde x_j^{(2)}
|<c^{-0.45n}\allowmathbreak\le c^{-0.4n}4^{-k}$, and
$|2^{-k}\sum_{j=1}^{2^k}\tilde x_j^{(1)}-M|\le2c^{-0.45n}
\allowmathbreak\le4^{-k}c^{-0.4n}$ if $n$ is sufficiently
large
($k$ is fixed, $n\to\infty$) and $\tilde x_j\in W_n$ for
$j=1,\dots,2^k$. These relations imply that
$2^{-k}\sum_{j=1}^{2^k}\tilde x_j\in \Omega^1_{n+k}$.
We get, by putting (7.5), (7.7) and (7.8) into (7.2) that
$$
\align
&h_{n,k}(x_1,\dots,x_{2^k})=\bar L_{n,k}\exp\Biggl\{
\frac{c^k}{2^k}\bar g\sum_{j=1}^{2^k}
x_1^{(j)}+\frac{c^k}{4^k}\bar
A\biggl(\sum_{j=1}^{2^k}x_j^{(2)}\biggr)^2 \\
&\quad-\frac1T\sum_{i=1}^{2^k-1}\sum_{j=i+1}^{2^k}
U(i,j)\left(Mx_i^{(1)}+Mx_j^{(1)}+x_i^{(2)}
x_j^{(2)}\right)-\sum_{j=1}^{2^k}
\frac{a_0}T\biggl(Mx_j^{(1)}+\frac{x_j^{(2)2}}2\biggr)
\Biggr\}\\ &\qquad \qquad\prod_{j=1}^{2^k}g\biggr(\frac{a_1}T
\biggl(Mx_j^{(1)}+\frac{x_j^{(2)2}}2
\biggr)\biggr)\left(1+O(q^n)\right)
\endalign
$$
with some $0<q<1$ if $x_j\in W_n$, $j=1,2,\dots,2^k$.
 
Simple calculation shows that $-\sum_{i=1}^{2^k}U(i,j)=\frac
{1-(c/2)^k}{1-c/2}$, hence  the coefficient of $x_j^{(1)}$,
\ $(c/2)^k\bar g+\frac MT\sum_{j=1}^{2^k}U(i,j)-\frac{a_0M}T$
equals zero, and
$$
h_{n,k}(x_1,\dots,x_{2^k})=
h_k(x_1,\dots,x_{2^k})\bigl(1+O(q^n)\bigr) \tag 7.9
$$
if $x_j\in W_n$, \ $j=1,2,\dots,2^k$, 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.
 
\subheading{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 $\varphi\:\zz\to R^1$, models with the
Hamiltonian function
$$
\Cal H(\sigma)=-\sum_{i\in \zz}\sum\Sb j\in \zz\\ j>i\endSb
\varphi(d(i,j))\s(i)\sigma(j),\qquad \sigma=\{\sigma(i),\;i\in \zz\}\,,
\tag 8.1
$$
where $d(\cdot,\cdot)$ denotes the hierarchical distance on
$\zz$ given in formula (1.1) of Part I. In this work we have
considered models in the special case $\varphi(x)=|x|^{-a}$
with $a= 2-\frac{\log c}{\log2}$. One question we  are going
to discuss here is that which are the functions $\varphi$ 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
$$
M_{n+1}=M_n-\frac1{4^nM_n\varphi (2^n)} \tag 8.2
 $$
holds true. The existence or non-existence of phase
transition depends on whether $M=\lim_{n\to\infty}M_n$ 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
$\sum\frac1{4^n\varphi(2^n)}$ is convergent. Dyson has
formulated the same conjecture in [13] and proved its
convergent part in the special case when $\sigma(i)\in R^3$. He
has also solved the problem for scalar-valued models. He
proved that there is a phase transition  at low temperatures
if $\varphi (n)\ge C\frac{\log\log n}{n^2}$
with some $C>0$, and there is none if
$\varphi (n)\frac{\log\log n}{n^2}\to 0$. Moreover, in the
boundary case   $\varphi (n)=C\frac{\log\log n}{n^2}$
the following Thouless effect occurs: There is some critical
parameter $T_{cr.} $ such that
$M(T)=\lim_{n\to\infty}M_n(T)>0$
for $T\le T_{cr.}$ 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 $\varphi (n)=C\frac{\log\log n}{n^2}$.
On the other hand, the Thouless effect occurs in some other
models too, like in the one-dimensional Ising model with
$\frac1 {|x-y|^2}$ 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 $\varphi$ in scalar-valued models too. But this
formula is essentially different from his vector-valued
counterpart, namely
$$
M_{n+1}\sim M_n\left[1-\exp\biggl\{-\frac1T
M_n^24^n\varphi(2^n)\biggr\}\right]\,.  \tag 8.3
$$
In the particular
case $\varphi(n)=\frac{\log\log n}{n^2}$ we have
  $$
_{n+1}\sim M_n\left[1-\exp \biggl\{-\frac1T M_n^2\log
n\biggr\}\right]\,. \tag 8.3${}^{\prime}$
  $$
Formula (8.3${}^{\prime}$) 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 $M(T)=\allowmathbreak\lim_{n\to\infty} M_n(T)=0$, hence
either $M(T)\ge \sqrt T$
or $M(T)=0$. Since $M(T)\ne 0$ 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
$$
\Cal H(\sigma)=-\sum\Sb |i-j|=1\\i,j\in \zzd\endSb
\sigma(i)\sigma(j)-\sum_{i\in \zzd}p(\sigma(i))
$$
with
$$
p(x)=-\frac t4|x|^4-\frac{|x|^2}2,\qquad t>0\,,
$$
and the Lebesgue measure on $R^p$ with some $p\ge 2$ as the
free measure of the model. The expression $\sigma(i)\sigma(j)$ in the
above formulas denotes scalar product.
 
 Let $\{X(k)=
(X_k^{(1)},\dots,X_k^{(p)}),k\in\zzd\}$
be a random field with the distribution of a (pure)
equlibrium state with the above Hamiltonian function at a
certain temperature $T$. If $d\ge3$ 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
$e_1=(1,0,\dots,0)$, i.e. $E\,X_k^{(1)}=M>0$, and
$E\,X_k^{(s)}=0$ for $s=2,\dots,p$.
 
Define the ``renormalized'' random fields
$\{Y_k(N)= \allowmathbreak(\,Y_k(N)^1,\dots,Y_k(N)^p\,)\}$,
$N=1,2,\dots$ by the formulas
$$
Y_k(N)^1=A(N)^{-1}\sum_{j\in D_k(N)}
(X_j^{(1)}-EX_j^{(1)}   )  \tag 8.4
$$
$$
Y_k(N)^s=B(N)^{-1}\sum_{j\in D_k(N)}
X_j^{(s)},\qquad  s=2,\dots,p,   \tag 8.4${}^{\prime}$
$$
with
$$
\align
D_k(N)=\Bigl\{j=(j^{(1)},\dots,j^{(d)})\in\zzd,\quad
&k^{(r)}N+1\le j^{(r)}\le (k^{(r)}+1)N,\\
&\qquad r=1\dots,d\Bigr\}\,.
\endalign
$$
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 $\zzd$ with $d=\frac2{1-\log_2c}$.
(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 $c>\sqrt2$, $c=\sqrt2$ and $c<\sqrt2$ 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
$\{Y_k=(Y_k^{(1)},\dots, Y_k^{(p)}),\;k\in\zzd\}$ is such
that the fields $\{Y_k^{(j)},\;k\in\zzd\}$, \ $j=1,2,\dots,p$
are independent, $\{Y_k^{(1)},\;k\in\zzd\}$ consists of
independent identically distributed Gaussian random variables
with zero mean,   $\{Y_k^{(s)},\;k\in\zzd\}$, \
$s=2,\dots,p$,
are massless free Gaussian fields, i.e. they have the same
distribution as the field
$$
Y_k=C\int\frac{\exp(ikx)}{|x|}
\prod_{j=1}^d\biggl\{\frac{\exp(ix^{(j)})-1}
{ix^{(j)}}  \biggr\}
W(\,dx),\qquad k\in \zzd,
$$
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${}^{\prime}$ )
with $A(N)=
N^2\sqrt{\log N}$ and $B(N)=N^{(d+2)/2}=N^3$ have a Gaussian
limit as $N\to\infty$ which consists of independent
components $s=1,\dots,p$, similarly to the case $d>4$. The
limit of the fields $Y_k^{(s)}$, $s=2,\dots,p$, is a massless
free field.
 
The result of the present work motivates the following
conjecture for $d=3$.
\proclaim{Conjecture}\it 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
$\{Y_k=(Y_k^{(1)},\dots,Y_k^{(p)}),\;k\in\zz^3\}$ defined by
the formulas
$$
Y_k^{(s)}=C\int\frac{\exp(ikx)}{|x|}
\prod_{j=1}^3\frac{\exp(ix^{(j)})-1}
{ix^{(j)}}
 W_s(\,dx),\quad k\in \zz^3\,,  \;
s=2,\dots,p\tag8.5 $$
and
$$    \align  &  \tag8.6\\
Y_1^{(s)}=-\frac{C^2}{2M}
\sum_{s=2}^p\iint\frac{\exp(ik(x+y))}{|x||y|}
\prod_{j=1}^3\biggl\{\frac{\exp[i(x^{(j)}+y^{(j)})]-1}
                   {i(x^{(j)}+y^{(j)})}\biggr\}
              &W_s(\,dx)W_s(\,dy),\\&k\in \zz^3\,,
\endalign $$
where $C>0$ is an appropriate positive constant, and
$W_s(\,dx)$, $s=1,\dots,p$ 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 $Y_1^{(s)}$.) \endproclaim
 
The fields $Y_s^{(k)}$, \ $s=2,\dots,p$  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 $1<c<
\sqrt2$.
 
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 $Y_k^{(1)}+
\frac1{2M}\sum_{s=2}^pY_k^{(s)2}$ 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 $\de(t)$ denote the Dirac-delta  function in the point
$t$, and consider the generalized field which at $\de(t)$
takes the value
$Y(\de(t))=\bigl(\,Y^{(1)}(\de(t)),\dots,Y^{(s)}(\de(t))\,\bigr)$,
$$\align &Y^{(s)}(\de(t))=  C
\int \frac{\exp(itx)}{|x|}W^{(s)}(\,dx)\,,
\qquad s=2,\dots,p\, ,   \\
 &Y^{(1)}(\de(t))=-\frac{C^2}{2M}\sum_{s=2}^p
\iint \frac{\exp(it(x+y))}{|x||y|}W^{(s)}(\,dx)
\,W^{(s)}(\,dy)\,.
  \endalign
           $$
Actually this definition is not correct, since the above
stochactic integrals are meaningless because of the
divergence of the integrals $\int_{R^3}\frac{dx}{|x|^2}$
and $\int_{R^3}\int_{R^3} \frac{dx\,dy}{|x|^2|y|^2}$. But
the integral
$$
 Y^{(1)}(\varphi)= \int Y^{(1)}(\de(t))
\varphi (t)\,dt=-\frac{C^2}{2M}\sum_{s=2}^p
\iint \frac{\tilde\varphi(x+y)}{|x||y|}W^{(s)}(\,dx)
\,W^{(s)}(\,dy)\,.
$$
and
$$  Y^{(s)}(\varphi)= \int Y^{(s)}(\de(t))\varphi(t)\,dt= C
\int \frac{\tilde\varphi(x)}{|x|}W^{(s)}(\,dx), \quad2\le
s\le p\,,
$$
are meaningful for nice funtions $\varphi$. In particular,
they are meaningful for the indicator functions of the unit
cubes $\prod\limits_{i=1}^3[k_i,k_i+1)$ which we denote by
$\varphi_k$ 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
$Y(\varphi(k))$.
 
A formal application of It\^o's formula (see e.g. [16])
would supply the relation
$Y^{(1)}(\de(t))-\frac1{2M}\sum_{s=2}^pY^{(s)}(\de(t))^2=const.$,
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 $Y(\de(\cdot))$. 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 $\{X_k,\;k\in \zz^1\}$ 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 $p$, \ $(\sigma(i)\in R^p)$, 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
$d=2$, \ $p\ge3$ 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 two-dimensional 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.
 
\heading Appendix\endheading
 
\subheading  {Appendix  A.  The proof  of  the  basic
recursive
relations (2.1) and (2.1${}^{\prime}$) in Part I}
Formula (2.1${}^{\prime}$)
immediately follows from (1.4) in Part I with $n=0$.  To
prove (2.1) let us first observe that the recursive relation
$$\align
\Cal H_{n+1}\left(x_1,\dots, x_{2^{n+1}}\right)&=
\Cal H_n\left(x_1,\dots, x_{2^n}\right)+
\Cal H_n\left(x_{2^n+1},\dots, x_{2^{n+1}}\right)\tag
A1\\&\qquad\qquad
-c^n  \biggl(2^{-n}\sum_{i=1}^{2^n}x_i \biggr)
     \biggl(2^{-n}\sum_{j=2^n+1}^{2^{n+1}}x_j
\biggr)\endalign $$
holds for $n\ge0$, where
$$
\Cal H_n(x_1,\dots,
x_{2^n})=-\sum_{i=1}^{2^n}\sum_{j=i+1}^{2^n} U(i,j)x_i x_j\;,
$$
and $U(i,j)$ is defined by (1.1) and (1.2${}^{\prime}$) in Part I. By
relation (1.4) in Part I
$$
\align
p_{n+1}(x,T)=\frac1{Z_{n+1}(T,t)} \int\exp
\biggl\{-\frac1T&\Cal H_n(x_1,\dots,x_{2^{n+1}})\biggr\}\tag
A2\\ &\delta\biggl(
2^{-(n+1)}\sum_{i=1}^{2^{n+1}}x_i-x\biggr)
\prod_{i=1}^{2^{n+1}}p(x_i)\,dx_i\;, \endalign
$$
where $ Z_{n+1}(T,t) $ is an appropriate norming constant,
and $\delta\bigl( 2^{-(n+1)}\sum_{i=1}^{2^{n+1}}x_i-x\bigr)$
means that integration in (A2) is taken on the hyperplane
$2^{-(n+1)}\sum_{i=1}^{2^{n+1}}x_i=x$ 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
$2^{-n} \sum_{i=1}^{2^n}x_i=x+u$  and
$2^{-n} \sum_{i=2^n+1}^{2^{n+1}}x_i=x-u$ and then by
integrating by $u$. We get with the help of  relations
(A1) and (A2) that
$$
\align
p_n(x,T)&=\frac1{Z_{n+1}(T,t)}\int \exp\left\{\frac{c^n}T
(x+u)(x-u)\right\}\\
&\left[\int\exp \left\{-\frac1T\Cal
H_n(x_1,\dots,x_{2^n})\right\}\delta\left(2^{-n}\sum_{i=1}^{2^n}
x_i-(x+u)\right)\prod_{i=1}^{2^n}p(x_i)\,dx_i\right]   \\
&\biggl[\int\exp \left\{-\frac1T\Cal
H_n(x_{2^n+1},\dots,x_{2^{n+1}})\right\}
\delta\biggl(2^{-n}\sum_{i=2^n+1}^{2^{n+1}}
x_i-(x-u)\biggr)\\
&\qquad\prod_{i=2^n+1}^{2^{n+1}}\!\!p(x_i)\,dx_i\biggr]\,du \\
&=C_n\int\exp \left\{\frac{c^n}T(x^2-u^2)\right\}
p_n(x+u)p_n(x-u)\,du\;,
\endalign
$$
as we have claimed.
 
 \subheading{Appendix B. The proof of Theorem C}
 Since the measure $\mu_{n+k}$ has the density function
$$
\frac1Z_{n+k}\exp \left\{-\frac1T\Cal
H_{n+k}(z_1,\dots,z_{2^{n+k}})\right\}
\prod_{i=1}^{2^{n+k}}p(z_i)     \,           ,
$$
the density function of the measure $Q_n  \mu_{n+k}$, the
function $h_k(x_1,\dots,x_{2^k})$ equals to
$$
  \align
h_k(x_1,\dots,x_{2^k})=\frac1Z_{n+k}\int
\exp &\left\{-\frac1T\Cal
H_{n+k}(z_1,\dots,z_{2^{n+k}})\right\}\tag B1\\
&\prod_{l=1}^{2^k}\delta\biggl(2^{-n}\! \!
\sum_{j=(l-1)2^n+1}^{l2^n} \!\! z_j-x_l\biggr)
\prod _{i=1}^{2^{n+k}} p(z_i)\,dz_i \,,
\endalign
$$
where
 $\prod_{l=1}^{2^k}\delta(2^{-n}\sum_{j=(l-1)2^n+1}^{l2^n}
 z_j-x_l) $ in the integral (B1) means that integration is
taken on the hyperplane defined by the relations
$2^{-n}\sum_{j=(l-1)2^n+1}^{l2^n}z_j=x_l$, \ $l=1,\dots,2^k$,
with respect to the Lebesgue measure. The special structure
of the hierarchical distance implies that
 $$
\Cal H_{n+k}(z_1,\dots,z_{2^{n+k}}) =
\sum_{l=1}^{2^k}\Cal H_n(z_{(l-1)2^n+1},\dots,z_{l2^n})-
\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^k} c^nU(i,j)\bar z_i\bar
z_j\,, $$
with
$\bar z_i=2^{-n}\sum_{p=(i-1)2^n+1}^{i2^n} z_p$,  \
$i=1\dots,2^k$.
 
Hence relation (B1) can be rewritten as
$$
\align
h_k(x_1,\dots,x_{2^k})&=\frac1Z_{n+k}\exp \left\{ \frac1T
\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^k} c^nU(i,j)x_ix_j \right\}\\
&\qquad\cdot\prod_{l=1}^{2^k}\int \exp \left\{-\frac1T
\Cal H_n(z_{(l-1)2^n+1},\dots,z_{l2^n})\right\}\\
&\qquad\qquad \delta\biggl(2^{-n}\sum_{j=(l-1)2^n+1}^{l2^n}
z_j-x_l\biggr) \prod_{j=(l-1)2^n+1}^{l2^n}p(z_j)\,dz_j \\
&=C_{k,n}\exp \left\{-\frac1T\Cal H_k(c^{n/2}x_1,\dots,
c^{n/2}x_{2^k})\right\}\prod_{l=1}^{2^k}p_n(x_l,T)\,,
\endalign
$$
as we have claimed.
 
\subheading{Appendix C. The calculation of the
Radon--Nikodym derivatives. The proof of formulas
(2.1)---(2.3${}^{\prime}$) 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,
$$
\align
P_{n+1}(x_1,\dots,x_{2^{n+1}})&=C_n
P_n(x_1,\dots,x_{2^n}) P_n(x_{2^n+1},\dots,x_{2^{n+1}})\\
&\qquad  \exp\left\{\frac{c^n}T  \biggl(2^{-n}\sum_{j=1}^{2^n}
x_j\biggr) \biggl(2^{-n}\!\sum_{j=2^n+1}^{2^{n+1}}\!
x_j\biggr) \right\}
\endalign
$$
with some norming  constant $C_n$.  Given some  measurable
set $A\subset (R^p)^{2^n}$ define the cylindrical set
$\tilde A \subset (R^p)^{2^{n+1}}$ as $\tilde A=A\times (R^p)^{2^n}$.
By our inductive hypothesis for $n+1$
$$
\align
\mu_{n,N}^{h_N}(A)&=\int_{\tilde A} f_{n+1,N}^{h_N}
\biggl(2^{-(n+1)}\sum_{j=1}^{2^{n+1}} x_j  \biggr)
P_{n+1}(x_1,\dots,x_{2^{n+1}})\,dx_1\dots dx_{2^{n+1}}\\
&=C_n  \int_{\tilde   A}   f_{n+1,N}^{h_N}
\biggl(2^{-n}\biggl(\sum_{j=1}^{2^n}\frac{ x_j}2 +
\sum_{j=2^n+1}^{2^{n+1}}\!\frac {x_j}2
\biggr) \biggr)\\
&\qquad\qquad P_n(x_1,\dots,x_{2^n})
P_n(x_{2^n+1},\dots,x_{2^{n+1}}) \\
&\qquad\qquad\exp\left\{\frac{c^n}T
\biggl(2^{-n}\sum_{j=1}^{2^n} x_j \biggr)
\biggl(2^{-n}\sum_{j=2^n+1}^{2^{n+1}}
x_j\biggr)\right\}
\,dx_1\dots dx_{2^{n+1}} \;.
\endalign
$$
Let us  calculate the last integral by first integrating on the
hyperplanes where  $x_1,\dots,x_{2^n}$  and  $y=
\frac1{2^n}\sum_{j=2^n+1}^{2^{n+1}} x_j$  are fixed.  Since
$P_n(x_{2^n+1},\dots,x_{2^{n+1}})$
 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
$$
\align
\mu_{n,N}^{h_N}(A)&=C_n\int_{A\times R^p}
f_{n+1,N}^{h_N}
\biggl(2^{-n}\Bigl(\sum_{j=1}^{2^n} \frac{x_j}2 +\frac
y2\Bigr) P_n(x_1,\dots,x_{2^n})\biggr)p_n(y) \\
&\qquad\exp\biggl\{\frac{c^n}T
2^{-n}(\sum_{j=1}^{2^n}
x_j)y\biggr\}\,dx_1\dots\,dx_{2^n}dy\;.
\endalign
$$
 
Hence we get, by integrating first by the variable $y$ that
$$
\mu_{n,N}^{h_N}(A)  =C_n^{\prime}\int_AS_n
f_{n+1,N}^{h_N}
\biggl(2^{-n}\sum_{j=1}^{2^n} x_j  \biggr)
P_n(x_1,\dots,x_{2^n})   \,dx_1\dots\, dx_{2^n}\;.
$$
Since this relation holds for all measurable sets
$A\subset (R^p)^{2^n}$, it implies our inductive hypothesis
for $n$.
 
\subheading{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 $\bold Z\subset \bold
Z^d$ of the $d$-dimensional integer lattice and a closed set
$K\subset R^p$ in the $p$-dimensional Euclidedan space. We
consider a Hamiltonian function $\Cal H(\sigma)$ of the form
$$
\Cal H(\sigma)=-\sum_{i,j\in\bold Z}
U(i,j)\sigma(i)\sigma(j),\quad
\sigma=\{\sigma(j),\;\sigma(j)\in K,\;j\in \bold Z\},
$$
where $U(\cdot,\cdot)$ is a given function,
$ U:\;\bold Z\times\bold Z\to R^1$, and
$\sigma(i)\sigma(j)$ 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  $V\subset \bold Z$, we define the energy function
$\Cal H_V(\sigma)$
as
$$
\Cal H_V(\sigma)=-\sum_{i,j\in V}
U(i,j)\sigma(i)\sigma(j)
\,,\qquad \sigma=\{\sigma(i),\;i\in V\},  \tag D1
$$
and the conditional energy in $V$ with respect to a
configuration $\bar\sigma$ in $\bold Z-V$ as
$$
\align
\Cal H_V(\sigma|\bar \sigma)&=\Cal H_V(\sigma)-
\sum_{i\in V}\sum_{j\in \bold Z-V}
U(i,j)\sigma(i)\bar\sigma(j)  ,\tag D2 \\
&\qquad\sigma=\{\sigma(i),\;i\in V\},\;
\bar\sigma=\{\bar\sigma(i),\;i\in \bold Z-V\},
\endalign
$$
provided that the last sum is convergent. Given some $h\in
R^p$, we also introduce the energy of a configuration
$\sigma$ in the volume $V$ with respect to the external field
$h$  as
$$
\Cal H_V^h(\sigma)=\Cal H_V(\sigma)-h\sum_{i\in V}\sigma(i).\tag D3
$$
 
Given some finite set $V\subset\zz$, a configuration
$\ssb=\{\ssb(j),j\in\zz-V\}$ outside $V$, a Hamiltonian
function $\Cal H(\sigma))$ and a free measure
$P(\,dx)$ on $K$, we define the Gibbs measure in volume $V$
with respect to the external field $\ssb$ at temperature $T$
as the probability measure
$\mu_{V,T}(\,\cdot\, |\ssb)$
on $K^V$  given by the formula
$$
\mu_{V,T}(\sigma\in A|\ssb)=\frac1{Z(V,T,\ssb)}\int_A\exp
\biggl\{- \frac1T\Cal H_V(\sigma|\ssb)\biggr\}
\prod_{j\in V} P(\,d\sigma(j)), \tag D4
$$
where $A\in K^V$ is an arbitrary measurable set,
$\sigma=\{\sigma(j),j\in V\}$
and
$Z(V,T,\ssb)$ is an appropriate norming constant, provided
that the above expression is meaningful.
Now we formulate the following
 
 \proclaim{Definition of Gibbs
states}\it A probability measure $\mb$ is a Gibbs state with
Hamiltonian function $\Cal H$  and free measure $P$ at
temperature $T$ if a $\mb$ distributed random field  $\sigma
(j),\;j\in\zz$ satisfies the following relation: For any
finite set $V$ and measurable set $A\subset K^V$ the
conditional probability of the event $\sigma\in A$, \
$\sigma=\{\sigma(j),
j\in V\}$ with respect to the condition $ \{\sigma(j)=\ssb,\;
j\in
\zz-V\}$ with a configuration $\ssb=\{\ssb(j),\,j\in \zz-V\}$
equals to
$$
\mb (\sigma\in A\,|\,\{\sigma(j),\,j\in \zz-V\}=\ssb)=
\mu_{V,T}(A\,|\,\ssb)
$$
with $\mb$ probability one, where $\mu_{V,T}$ is defined in
(D4).\endproclaim
 
The question arises whether Gibbs states on $K^{\zz}$ 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 $V_n\subset\zz$, \ $\cup V_n=\zz$,
fix a configuration $\ssb=\ssb^{(n)}=
\{\ssb(j),\,j\in \zz-V_n\}$ for each $V_n$, and consider the
measures $\mu_{V_n,T}(\,\cdot\,|\ssb)$ 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 $\ssb^{(n)}$, different limits can appear in
the above construction. A slightly different, and often useful
approach is to choose a sequence $h_n\in R^n$, $h_n\to 0$,
and try to construct Gibbs states as the limit of a sequence
of measures of the form  $\mu_{V_n,T}^{h_n}$, where we define
the probability measure  $\mu_{V,T}^h$ as
$$
\mu_{V,T}^h(A)=
\frac1{Z(V,T,h)}  \int_A\exp
\biggl\{ -\frac1T\biggl(\sum_{i\in V}\sum_{j\in
V}U(i,j)x_ix_j-h\sum_{i\in V}x_i\biggr)\biggr\}
\prod_{i\in V}P(dx_i)\,.\tag D5
$$
 
If $K$ is a compact subset of $R^p$ then standard results in
probability theory imply the compactness of the measures
$\mu_{V_n,T}(\,\cdot\,|\ssb)$ or of $\mu_{V_n,T}^{h_n}$ 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
 $\mu_{V_n,T}(\,\cdot\,|\ssb)$ (or $\mu_{V_n,T}^{h_n}$) is
really a Gibbs state it is worth while to rewrite the definition
of Gibbs states in an equivalent integral form. Let
$f=f(x_{j_1},\dots,x_{j_k})$ and $g=g(x_{l_1},\dots,x_{l_k})$,
$x_{j_i}\in R^p$, $x_{l_i}\in R^p$, be two bounded and continuous
functions with finitely many arguments, $V=\{j_1,\dots,j_k\}$,
$W=\{l_1,\dots,l_k\}$, $V\subset \zz$, $W\subset \zz$  such that
$V\cap W=\emptyset$.
 
The measure $\mb$ on $K^{\zz}$ is a Gibbs state if and only if
$$
\int_{K^{\zz}}fg\,d\mb=\int_{K^{\zz-V}}\mu (f)g \,d\mb\tag D6
$$
for all functions $f$ and $g$ with the above properties,
where
$$
\align
\mu(f)=\mu(f)(\ssb)=\int_{K^V}
& f(\sigma(j_1),\dots,\sigma(j_k))\
 \mu_{V,T}(\,d\sigma |\ssb) \tag D7\\
&\qquad \sigma=\{\sigma(j),j\in V\},\quad   \ssb=\{\sigma(j),j\in
\zz-V\} ,
\endalign
$$
and $\mu_{V,T}$ is defined in (D4).
 
Let us consider an arbitrary sequence of sets $V_n\subset
\zz$, \ $\cup V_n=\zz$, and numbers $h_n\in R^p$, \ $h_n\to0$.
Some calculation shows that for sufficiently large $n$ (if
$V\subset V_n $, $W\subset V_n$)
$$
\int_{K^{V_n}}fg\,d\mu_{V_n,T}^{h_n}=\int_{K^{V_n-V}}
\mu^{h_n} (f)g \,d\mu_{V_n,T}^{h_n}    \tag D8
$$
with
$$
\align
&\mu^{h_n}(f)=\mu_{V,V_n,T}^{h_n} (f;\sigma(j),j\in  V_n-V) \\
&=\frac{ \int f(\sigma(j_1),\dots,\sigma(j_k))  \exp
\bigl\{- \frac1T(\Cal H_{V,V_n}(\sigma)
-h_n\sum\limits_{i\in V}\sigma(i))
\bigr\} \prod\limits_{i\in V}  P(\,d\sigma(i))}
{\int \exp\bigl\{- \frac1T(\Cal H_{V,V_n}(\sigma)
-h_n\sum\limits_{i\in V}\sigma(i))
\bigr\} \prod\limits_{i\in V}  P(\,d\sigma(i))  } \,,
\endalign
$$
where
$$
\Cal H_{V,V_n}(\sigma)=\sum_{i\in V}\sum_{j\in V_n}\sigma(i)\sigma(j)\;.
$$
 
If the sequence $\mu_{V_n,T}^{h_n}$ tends weakly to the
measure $\mb$ then the left-hand side of (D8) converges to
that of (D6). Hence to prove that the limit measure $\mb$ 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 $|i-j|\ge r$ then it is
not difficult to see that
$\mu^{h_n}(f)(\ssb)\to\mu(f)(\ssb)$, 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 $\mu_N^{h_n}$ to  $\mb$. (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.
 
\subheading{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 $V=\{1,2,\dots,2^k\}$,
$W= \{2^k+1,2,\dots,2^m\}$ with some $0\le  k< m$, i.e.
$f=f(x_1,\dots,x_{2^k})$, $g=g(x_{2^k+1},\dots,x_{2^m})$ and
$V_N=\{1,2,\dots,2^N\}$. (We apply the notation of Appendix D.)
Introduce the functions
$$
\align
&p^h_{k,N}(x_1,\dots,x_{2^k}|  x_{2^k+1},\dots,x_{2^N})\tag
E1   \\            &\qquad=\frac
{ \exp\bigl\{- \frac1T(\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^N}
U(i,j)x_ix_j- h\sum_{i=1}^{2^k}x_i) \bigr\}  }
{   \int    \exp
\bigl\{- \frac1T(\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^N}
 U(i,j)x_ix_j- h\sum_{i=1}^{2^k}x_i) \bigr\}
\prod_{i=1}^{2^k} p(x_i)\,dx_i       }\,, \\
\vspace{1\jot}
  &\qquad\qquad\qquad x_j\in
R^p,\;j=1,\dots,2^N \endalign
$$
and
$$
\align
&p_{k}(x_1,\dots,x_{2^k}|  x_{2^k+1},\dots)\tag
E1${}^{\prime}$  \\            &\qquad=\frac
{ \exp\bigl\{- \frac1T\sum_{i=1}^{2^k}\sum_{j=i+1}^{\infty}
 U(i,j)x_ix_j\bigr\}  }
{   \int    \exp
\bigl\{- \frac1T\sum_{i=1}^{2^k}\sum_{j=i+1}^{\infty}
U(i,j)x_ix_j \bigr\}
\prod_{i=1}^{2^k} p(x_i)\,dx_i       }  \,,  \\
\vspace{1\jot}
&\qquad\qquad\qquad  x_j\in
R^p,\;j=1,\dots, \endalign
$$
where the function $U(\cdot,\cdot)$ is defined in (1.2) and
$p(x)$ in (1.3) of Part I. Put
$$
\align
\mu^h_{k,N}(f)(   x_{2^k+1},\dots,x_{2^N})  &=\int
f(x_1,\dots,x_{2^k})  \tag E2  \\
&\qquad
p^h_{k,N}(x_1,\dots,x_{2^k}|  x_{2^k+1},\dots,x_{2^N})
      \prod_{i=1}^{2^k} p(x_i)\,dx_i
\endalign $$
and
$$
\mu_k(f)(   x_{2^k+1},\dots) = \int
f(x_1,\dots,x_{2^k})
p_k(x_1,\dots,x_{2^k}|  x_{2^k+1},\dots)
      \prod_{i=1}^{2^k} p(x_i)\,dx_i   .
 \tag E2${}^{\prime}$
 $$
 
The convergence of the right hand side of (D8) to that of
(D6) is equivalent to the relation
$$
\lim_{N\to\infty}\int g\mu_{k,n}^{h_N}(f)\,d\mu_N^{h_N}=
\int g\mu_k(f)\,d\mb  \tag E3
$$
in our case, where $ \mu_N^{h_N} $ and $\mb$ are the same
probability measures on $(R^p)^{2^N}$ and $(R^p)^{\zz}$ 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 $K\in R^1$, \ $K>0$, \ $k, n, N\in
\zz$ and  $k<n<N$, defined by the formulas
$$
\align
& A(K,k,n,N)=\bigl\{
( x_{2^k+1},\dots,x_{2^N}),\;x_j\in R^p ,\;j=2^k+1,\dots
2^N, \;|x_j|<K  \\
&\qquad\qquad \text{if }
2^k<j\le 2^n,\;\text{and }\; |x_j|<2^{l\alpha}
\text{ if }2^l<j\leq 2^{l+1},\;l=n,\dots,N-1\bigr\}
\endalign
$$
and
$$
\align
  A(K,k,n)&=\bigl\{
( x_{2^k+1},\dots),\;x_j\in R^p ,\;j=2^k+1,\dots
, \;|x_j|<K  \\
&\qquad \text{if }
2^k<j\le 2^n,\;\text{and }\; |x_j|<2^{l\alpha}
\text{ if }2^l<j\le 2^{l+1},\;l=n,n+1,\dots \bigr\}  ,
\endalign
$$
where $\alpha=\frac 34-\frac 12 \frac{\log c}{\log2}$.
 
We claim that for all $\e>0$ some $n=n(\e)$ and $K=K(\e,n)
$ can be chosen in such a way that
 $$
\mu^{h_N}_N \bigl  ((x_{2^k+1},\dots,x_{2^N})
\notin A(K,k,n,N)\bigr)<\e \tag E4
$$
and
$$
\mb     \bigl (( x_{2^k+1},\dots,)
\notin A(K,k,n,)\bigr)<\e\,. \tag E4${}^{\prime}$
$$
To prove relations (E4) and (E4${}^{\prime}$) let us first observe
that there is a universal constant $C$, $C>0$, such that if
$\s(j)$  is the
$j$-th coordinate   of a $\mu_N^{h_N}$ or $\mb$ distributed
random variable then the inequality $E\sigma^2(j)<C$ holds for
all $j\in \zz$ and measures   $\mu_N^{h_N}$ and $\mb$.
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
$\sigma(j)$ of a   $\mu_N^{h_N}$  distributed random vector $\sigma$
converge to the corresponding moment of the $j$-th coordinate
of a $\mb $ distributed random vector as $N\to\infty$. Then
we get,by exploiting that $1-2\alpha>0$ that
 $$
\mu^{h_N}_N \left\{(x_{2^k+1},\dots,x_{2^N})
\notin A(K,k,n,N)\right\}\le C\biggl(\frac{2^n-2^k}{K^2}+
\sum_{j=n}^N 2^{j(1-2\alpha)}\biggr)<\e
$$
if first $n$ and then $K$ is chosen sufficiently large. The
proof of (E4${}^{\prime}$) is the same. (We remark that relations (E4)
and (E4${}^{\prime}$) hold with arbitrary $\alpha>0$ in the definition of
the sets $A(\cdot,\cdot,\cdot)$. To prove it we have to
apply the stronger statement $E\sigma^{2k}<C_k$ for all $k\ge1$.
This observation is needed if we want to prove Theorem B in
the case $\sqrt2<c<2$ too.)
 
We claim that for all $\e>0$ there is some $N_0=N_0(\e,K,n)$
such that for $N>N_0$
$$
\align \biggl |
p_{k,N}^{h_n}(x_1,\dots,x_{2^k}|x_{2^k+1}\dots,x_{2^N})&-
p_{k,N_0}^{h_n}(x_1,\dots,x_{2^k}|x_{2^k+1}\dots,x_{2^{N_0}})\biggr|
\tag E5\\
&\quad<\e\biggl(\sum_{i=1}^{2^k}|x_i|\biggr)\exp\biggl\{(2Kn+1)
 \sum_{i=1}^{2^k}|x_i|\biggr\}
 \endalign
$$
and
$$
\align \biggl     |
p_k(x_1,\dots,x_{2^k}|x_{2^k+1}\dots)&-
p_{k,N_0}^0(x_1,\dots,x_{2^k}|x_{2^k+1}\dots,x_{2^{N_0}})\biggr
| \tag E5${}^{\prime}$\\
&\quad<\e\biggl(\sum_{i=1}^{2^k}|x_i|\biggr)\exp\biggl\{(2Kn+1)
 \sum_{i=1}^{2^k}|x_i|\biggr\}
 \endalign
$$
if
$(x_{2^k+1},\dots,x_{2^N})\in A(K,k,n,N)$ and
$(x_{2^k+1},\dots)\in  A(K,k,n)$. (The constants $K$ and $n$
in formulas (E5) and (E5${}^{\prime}$) are the same as in the
 definition
of the sets $A(K,k,n,N)$.)
 First we show that (E5) and (E5${}^{\prime}$) together with (E4)
 and
(E4${}^{\prime}$) imply (E3), hence also Theorem B. Indeed, since
$p_n(x)$ decreases at infinity faster than $\exp(-x^2/2)$
hence (E5) and (E5${}^{\prime}$) imply that
$$
\left|\mu_{k,N}^{h_N}(f)(x_{2^k+1},\dots,x_{2^{N}})-
\mu_{k,N_0}^0(f)(x_{2^k+1},\dots,x_{2^{N_0}}) \right|
<const.\e
$$
and
$$
\left|\mu_k(f)(x_{2^k+1},\dots)-
\mu_{k,N_0}^0(f)(x_{2^k+1},\dots,x_{2^{N_0}}) \right|<\text{const.}\e\;.
$$
if $(x_{2^k+1},\dots)\in A(K,k,n)$. This implies that
$(x_{2^k+1},\dots,x_{2^N})\in A(K,k,n,N)$.
Since this relation holds on a set of $1-\e$ $\mu_N^{h_N}$
resp. $\mb$ probability by (E4) and (E4${}^{\prime}$), the
functions $f$, $g$, $\mu_{k,N}^{h_N}(f)$ and $\mu(f)$ are bounded,
hence an error less than $const.\e$ is committed if $\mu_{k,N}^{h_N}$
and $\mu_k(f)$ is replaced by $\mu_{k,N_0}^0$ in formula
(E3). After this replacement relation (E3) holds, because the
projections of the measures $\mu_N^{h_N}$ to $(R^p)^{2^N}$
converge to the projection of $\mb $ to the same subspace.
Since $\e>0$ can be chosen arbitrary small, relation (E3)
holds in its original form.
 
We prove only (E5) the proof of (E5${}^{\prime}$) being the same. Let us
first observe that for any $\eta>0$ there is some $N_0=N_0
(K,n,\eta)$ such that for $N\ge N_0$ and
$(x_{2^k+1},\dots,x_{2^N})\in A(K,k,n,N)$
$$                  \align  &  \tag E6 \\
&\left| \exp\biggl\{-
\frac1T(\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^N}
   U(i,j)x_ix_j- h_N\sum_{i=1}^{2^k}x_i) \biggr\}  -
 \exp\biggl\{- \frac1T\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^{N_0}}
   U(i,j)x_ix_j\biggr\}\right|   \\
&=\exp\biggl\{- \frac1T\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^N}
   U(i,j)x_ix_j\biggr\}\\
&\qquad\left| \exp\left\{-
\frac1T\biggl(\sum_{i=1}^{2^k}\sum_{j=2^{N_0}+1}^{2^N}
   U(i,j)x_ix_j- h_N\sum_{i=1}^{2^k}x_i\biggr) \right\}  -
1\right|\\
  &\le\eta \frac{\sum_{i=1}^{2^k}|x_i|}T\exp
\left\{\frac{2Kn+\eta}T
\sum_{i=1}^{2^k}|x_i|\right\}
  .    \endalign
$$
 In the last relation we have applied the inequality $|e^x-1|
\le |x|e^{|x|}$ together with the relations $h_N<\eta/2$, \ $
|
\sum_{j=2^{N_0}+1}^{2^N}U(i,j)x_j|\le |\sum_{j=N_0}^{\infty}
2^{j(\alpha-1)}\bigl(\frac c4\bigr)^j|<\eta/2$ and
$|\sum_{j=i+1}^{2^N}U(i,j)x_j|\le 2Kn$ if $N>N_0$, \ $j>2^k$,
\ $(x_{2^k+1},\dots,x_{2^N})\in A(K,k,n,N)$ and $N_0$ is
sufficently large. Integrating inequality (E6) with respect
to the measure $\prod_{i=1}^{2^k}p(x_i)\,dx_i$ we get that
$$
\align
&\biggl|\int \exp\biggl\{-
\frac1T\biggl(\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^N}
   U(i,j)x_ix_j- h_N\sum_{i=1}^{2^k}x_i\biggr)
\biggr\}\prod_{i=1}^{2^k}p(x_i)\,dx_i  \tag E7\\ &\qquad-
\int \exp\biggl\{-
\frac1T\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^{N_0}}
   U(i,j)x_ix_j\biggr\}
\prod_{i=1}^{2^k}p(x_i)\,dx_i   \biggr|
<const.\eta\;,
\endalign
$$
 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 $\eta$ 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 $\eta>0$ can be
taken arbitrary small if $N_0=N_0(K,n,\eta) $ is sufficiently
large.  Hence we prove (E5) by showing that
$$
\int \exp\bigl\{-
\frac1T\sum_{i=1}^{2^k}\sum_{j=i+1}^{2^{N_0}}
   U(i,j)x_ix_j\bigr\}
\prod_{i=1}^{2^k}p(x_i)\,dx_i   >D \tag E8
$$
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 $|x_j|<1$, $j=1,2,\dots,2^k$ and $(x_{2^k+1},\dots,
x_{2^N})\in A(K,k,n,N_0)$ then the integrand in
(E8) is separated from zero.
 
 
 
\heading Table of Contents\endheading
 
\vskip 0.5cm
 
 
\subheading{Preface}
 
\bigpagebreak
\subheading{Part I} Limit theorems for the average spin.
 
\item{1.}Introduction
\item{2.}On the content of Theorem 1. Convergence to the
solution of the fixed point equation.
\item{3.}On the strategy of the proof. The inductive
procedure.
\item{4.}The proof of Proposition 1 and its Corollary. The
first step of the inductive procedure.
\item{5.}The proof of Proposition 2. The second step of the
inductive procedure.
\item{6.}The proof of Proposition 3 and some of its
consequences.
\item{7.}On the fixed point equation $T_M g=g$.
\item{8.}The proof of Theorems 1 and 1$^{\prime}$.
\item{9.}The proof of Theorem 2. The behaviour of the density
of the average spin at infinity.
\item{10.}The  proof of Lemmas 17 and 18.
 
\subheading{Part II}Description of the large-scale limit.
 
\item{1.}Introduction.
\item{2.}On the basic estimates needed during the proof.
Reduction to integral equations.
\item{3.}The proof of Lemma 1.
\item{4.}Some preparatory remarks to the proof of Proposition
1$^{\prime}$.
\item{5.}The proof of Proposition 1$^{\prime}$.
\item{6.}The proof of Theorem 1 and Proposition 2. Existence
of the thermodynamical limit.
\item{7.}The proof of Theorem 2. Existence of the large-scale
limit.
\item{8.}Some open problems and conjectures.
 
 
\subheading{Appendix}
\item{A.}The proof of the basic recursive relations (2.1) and
(2.1$^{\prime}$) in Part I.
\item{B.}The proof of Theorem C.
\item{C.}The calculation of the Radon--Nikodym derivatives.
The proof of formulas (2.1)---(2.3${}^{\prime}$) in Part
II. \item{D.}On limit Gibbs states.
\item{E.}The proof of Theorem B.
 
\bye
 
