\magnification=\magstep 1
\input amstex
\documentstyle{amsppt}
\hsize=16truecm
\TagsOnRight
\parskip=3 pt plus 1pt
\define\zb{\bold Z}
\define\s{\sigma}
\define\e{\varepsilon}
\define\ch{\Cal H}
\define\qb{\bar q}
\define\bt{\bold T}
\define\p#1{p_{#1}(x)}
\define\q#1{q_{#1}(x)}
\define\rb{\bold R}
\define\co{$1i\endSb
d(i,j)^{-a}\s(i)\s(j)\,, \tag 3
$$
where $10$ is another parameter of the model, and $C(t) $
is an appropriate norming constant. We shall assume that
$t>0$ is sufficiently small. What is important for us is
that $p_0(x)$ is a small perturbation of the normal density
function, and it tends to zero very fast.
Our first problem is to construct an equilibrium state with
the above defined Hamiltonian function and free measure at
some temperature $T$ and then to describe its large-scale
limit. Both problems can be solved with the help of some
limiting procedure if one solves first the following two
problems:
\demo{Problem 1} Put $V_n=\{1,2,\dots,2^n\}$, and
$$
\ch_{V_n}(x_1,\dots,x_{2^n})=-\sum_{i\in V_n}\sum\Sb j\in
V_n\\j>i\endSb d(i,j)^{-a}x_ix_j.
$$
Define the probability measure $\mu_n= \mu_{n,T}$ on
$R^{V_n}$ (on $(R^p)^{V_n}$ if we have a model with
$p$-dimensional spins) with the density function
$p_n(x_1,\dots,x_{2^n})$ by the following formula:
$$
p_n(x_1,\dots,x_{2^n})= \frac{d\mu_n(x_1,\dots,x_{2^n})}
{dx_1\dots dx_{2^n} }
=C_n\exp\left\{-\frac1T\ch_{V_n}(x_1,\dots,x_{2^n})\right\}\,.
$$
Let $\left(\s(1), \s(2), \dots,\s(2^n)\right)$ be a
$\mu$~distributed
random vector, and let $p_n(x)$ denote the density function
of the average $2^{-n}\sum_{i=1}^{2^n}\s(i)$. Give a good
asymptotic formula for $p_n(x)$.
\enddemo
\demo{Problem 2} Let $N\ge n$, and $h>0$. Define the
probability measure $\mu_N^h $ on $R^{V_N}$ (on
${(R^p)}^{V_N}$ in $p$-dimensional models) with
density function $p_N^h(x_1,\dots,x_{2^N})$ by the formula
$$
p_N^h(x_1,\dots,x_{2^N})
=C_{N}^h\exp\left\{-\frac1T
\ch_{V_N}^h(x_1,\dots,x_{2^N})\right\} \,,
$$
where
$$
\ch_{V_N}^h(x_1,\dots,x_{2^N})
=\ch_{V_N} (x_1,\dots,x_{2^N})-h\sum_{i=1}^{2^N} x_i\,,
$$
and $C^h_N$ is an appropriate norming constant. (In the
vector-valued case we consider $h$ as the vector
$(h,0,\dots,0)$ with some $h>0$, and product means scalar
product in the last
formula.) Let $\mu_{n,N}^h$ denote the restriction of the
above defined measure $\mu_N^h$ to the volume $V_n$, and
let us consider the Radon--Nikodym derivative
$$
f_{n,N}^h(x_1,\dots,x_{2^n})=\frac{d\mu_{n,N}^h}{d\mu_n}
(x_1,\dots,x_{2^n}) \,.
$$
Give a good asymptotic formula on the function
$f_{n,N}^h(x_1,\dots,x_{2^n})$.
\enddemo
Both problems can be translated to purely analytical
questions. It can be seen e.g.\ with the help of Appendix A
of [4] that Problem 1 is equivalent to the following
\demo{Problem $1^{\prime}$} Define the sequence of
density functions $p_n(x)=p_n(x,T)$ by the recursive
formula
$$
p_{n+1}(x)=C_n(T)\int
\exp\left\{\frac{c^n}T
(x^2-u^2)\right\}p_n(x-u)p_n(x+u)\,du\,,
$$
and let $p_0(x)$ be defined by formula (5). Give a good
asymptotic formula on $p_n(x)$.\enddemo
Problem 2 can be translated with the help of the
result in Appendix C of [4] to the following
\demo{Problem 2$^{\prime}$}
Define the functions
$f^{h}_{n,N}(x)$, \ $N\ge n$, by the relations
$$
\align
f^{h}_{N,N}(x)&=K(N,h)\exp
\left(\frac{2^Nhx^{(1)}}T\right) \tag 6\\
f^{h}_{n,N}(x)&=K(n,N,h)\bold S_nf^{h}_{n+1,N}(x) \tag 6$^{\prime}$
\endalign
$$
with
$$
\bold S_nf(x)=\int\exp\biggl(\frac{c^n}Txy\biggr)f\biggl(\frac{x+y}2
\biggr)p_n(y)\,dy\,, \tag6${}^{\prime\prime}$
$$
where $p_n(x)$ is the same as in Problems 1 and $1'$, and
$K(n,N,h)$ is an appropriate normig constant. Find
a good asymptotic formula for the above defined functions
$f^{h}_{n,N}(x)$. \enddemo
It is proved in Appendix C of [4] that
$$
\frac{d\mu^{h}_{n,N}}{d\mu_n}\left(x_1,\dots,x_{2^n}\right)=
f^{h}_{n,N}\biggl(2^{-n}\sum^{2^n}_{j=1}x_j\biggr),\quad n \le N\,,
$$
hence Problems 2 and 2$^{\prime}$ are equivalent.
The main part of our investigation consists of solving
Problems $1'$ and $2'$. In this paper the
vector-valued case (i.e.\ the case when $p\ge2$) is
considered.
Formally the problems change very little
when scalar-valued models are replaced with vector-valued
ones. Thus in Problem $1'$ the only change is that $|x|$
means the absolute value of a vector, and $xy$ denotes scalar
product. Nevertheless, and this is the most striking feature
of the problem we are investigating, these seemingly
unessential modifications radically change the behaviour of
the model. Thus the functions $\p{n}$ defined in Problem
$1'$ have the following behaviour for small $T$ in the
scalar-valued case: Since $\p{n}=p_n(-x)$, it is enough to
consider $\p n$ for $x\ge0$. There is some sequence $M_n=
M_n(T)$, \ $M_n>0 $, \ $M_n\to M$ with some $M=M(T)>0$ such
that $2^{-n/2}p(2^{-n/2}x+M_n)$ tends to a normal density
function with expectation zero and some positive variance.
(The number $M_n$ is called the spontaneous magnetization in
the literature.) This
means a central limit theorem with the usual normalization.
The behaviour of the model in the vector-valued case is
more complex. It is not difficult to see that $\p n$ depends
on $x$ only through its absolute value $|x|$,
i.e.\ there is a function $P_n(z)=P_n(z,T)$, \ $z\in R^1$ and
$z\ge0$ such that $\p n=P_n(|x|)$ for all $x\in R^p$.
Hence, it is natural to investigate the function $P_n(z)$
instead of $\p n$. For small $T$ the behaviour of the
function $P_n(z, T)$ is essentially different for $10$ which can be described as
the solution of an integral equation. This means a
non-central limit theorem with an unusual normalization.
(In scalar-valued cases there is only at $T=T_{cr}$ such a
big
difference between the cases $10$ and
$E\s^{(2)}(j)=0$ for a random field $\s=\{\s(j),\,j\in \bold
Z\}$ with the distribution of the pure
state we have constructed. In the large-scale limit defined
by formula ~($2'$) we have to normalize differently in the
direction of the magnetization and in the
direction orthogonal to it. Our main interest in
this work is the description
of the large-scale limit in both directions.
The operator $\bold S_n$ defined in formula ~($6''$) depends on the
function $p_n(x)$ appearing in Problem ~1. This implies that
the recursive formula expressing $\fh{n}$ through
$\fh{n+1}$ has an essentially different form in the cases
\co{} and \ct{}. Nevertheless, the asymptotic behaviour of the
function $\fh n$ is the same in the two cases.
More
precisely, we need a good asymptotic formula for the function
$\fh n$ only in a typical region, and outside this region it
is enough to give some upper bound on it. Actually we are
interested in the product $p_n(x)\fh n$, and not the function
$\fh n$ itself. Hence the typical region, where we need a
good approximaton, is a small neighbourhood of the maximum of
the above mentioned product. In this domain the
Radon--Nikodym derivative has the following form:
$$
\fh n=C_n\exp\left\{g_nx^{(1)}+A_nx^{(2)2}+\e_n(x)\right\}\,,
\tag 11
$$
where $\e_n(x)$ is a small error term, and the constants
$g_n$ and $A_n$ are defined by a recursive formula.
This formula holds both for \co{} and \ct{}, and even the
recursive formulas on $g_n$ and $A_n$ have the same
structure in the two cases. This means that in Problem ~$2'$
there is no essential change between models with different
parameters ~$c$. A heuristic explanation of this fact is
contained in the last two formulas at page 466 of [2], and
they are the basis for our investigations of Problem 2. When giving
an upper bound outside the typical region we have to work
differently if the absolute value ~$|x|$ is not typical and
if it is typical, but the vector ~$x$ is not in the typical
region because of its direction. This question is discussed
in Section 2 of Part II of [4] in more detail.
The solution of Problem ~2 in the case \ct{} is given in our
paper [1]. Actually the greatest part of that work deals with
this question. The solution of this problem (which also
contains the investigation of the asymptotic behaviour of the
sequences $g_n$ and $A_n$) is the main ingredient in the
description of the large-scale limit of the equilibrium
state. For \ct{} one has to normalize with
$A_n^{(2)}=2^nc^{-n/2}$ in the direction orthogonal to the
direction of the spontaneous magnetization, and the limit is
a field of dependent Gaussian random variables whose
distribution we can describe explicitly. In the direction of
the magnetization the classical norming $A^{(1)}_n=
2^{n/2}$ has to be applied, and the limit is a field of
independent Gaussian random variables. (The bounary case
$c=\sqrt2$ is similar to the above case \ct. (See [5]). The only
difference is that in this case the normalization
$A^{(1)}_n=2^{n/2}\sqrt n$ has to be applied in the direction of
the spontaneous magnetization.) This means that in the
direction orthogonal to the spontaneous magnetization a
``critical'' normalization has to be applied for all low
temperatures, i.e.\ the same normalization as at the critical
temperature. This result has no equivalent in scalar-valued
models.
The large-scale limit of Dyson's model in the case \co{}
is similar to the case \ct{} in the direction orthogonal to the
spontaneous magnetization, and it is different in the
direction of the magnetization. The reason for it lies
in the fact that the solution of Problem~2 is similar in the
two cases, and the solution of Problem~1 is different. In the
direction orthogonal to the spontaneous magnetization one has
to divide again by $A_n^{(2)}=2^nc^{-n/2}$, and the limit is
a field of dependent Gaussian random variables. In the
direction of the spontaneous magnetization one has to divide
by $A^{(1)}_n=2^nc^{-n}$, and the limit field is
non-Gaussian. The explicit form of the large-scale limit is
given in Theorem 2 of Part II in paper [4]. The proof of this
result consists of a limit procedure which can be carried out if
Problems~1 and 2 are already solved.
\bigskip
\noindent \bf References: \rm
\parindent=20pt
\item{[1]} Bleher, P. M., Major, P.,:
Renormalization of Dyson's
hierarchical vector valued
$\varphi^4$~ model at low temperatures. {\it Comm. Math. Physics\/}
{\bf 95} (1984), 487--532.
\item{[2]} Bleher, P. M., Major, P.,:
Critical phenomena and universal exponents in statistical
physics. On Dyson's hierarchical model. {\it Annals of Probability\/}
{\bf15} 1987, 431--477.
\item{[3]} Bleher, P. M., Major, P.,:
Limit theorems in statistical Physics; on Dyson's
hierarchical model. In the conference volume of the first world
conference on probability and statistics of the Bernoulli Society.
\item{[4]} Bleher, P. M., Major, P.,:
The large-scale limit of Dyson's hierarchical vector-valued
model at low temperatures. The non-Gaussian case. {\it Annales de
l'Institut Henri Poincar\'e,} S\'erie Physique, Volume {\bf49}
fascicule 1 (1988),
\item{[5]} Bleher, P. M., Major, P.,:
The large-scale limit of Dyson's hierarchical
vector-valued model at low temperatures. The marginal case $c=\sqrt2$.
{\it Comm. Math. Physics\/}
\bye