\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{$1<c<\sqrt2$}
\define\qq#1{\bar q_{#1}(x)}
\define\ct{$\sqrt2<c<2$}
\define\fh#1{f^h_{{#1},N}(x)}
\define\Qb#1{\bar Q_{#1}(x)}
 
\leftheadtext\nofrills{{\rightline{P. M. Bleher and P. Major}}}
\rightheadtext\nofrills{\leftline{Dyson's vector-valued hierarchical
model}}
 
\topmatter
\title RENORMALIZATION OF DYSON'S VECTOR-VALUED \\HIERARCHICAL MODEL
AT LOW TEMPERATURES\endtitle
\author P. M. Bleher$^{(1)}$ {\rm and} P. Major$^{(2)}$\endauthor
\endtopmatter
\noindent
$^{(1)}$Keldysh Institute of Applied Mathematics of the Soviet Academy
of Sciences\newline
Moscow \newline
$^{(2)}$Mathematical Institute of the Hungarian Academy of
Sciences\newline
Budapest
\vskip0.7cm
 
In this paper we discuss Dyson's vector-valued model and explain how
its so-called large-scale limit can be calculated. Formally, this is a
limit theorem for the distribution of sums of random variables.
Hence one would expect that the classical methods of probability
theory, worked out for the investigation of such problems,  can be
applied  in this case too. However, this model has a much more
complex behaviour than the analogous models in classical probability,
and the usual methods of probability theory are not appropriate for
its investigation. The aim of the present paper is to
explain the main motivations (statistical physical and
probabilistic) for studying Dyson's model and the method of
of investigation. Here we restrict ourselves to  vector-valued models.
A detailed discussion of scalar-valued models is given in [2]. On the
other hand, vector-valued models have a peculiar behaviour which
cannot be guessed from the behaviour of scalar models.
The behaviour of these models strongly depend on a physical parameter,
on the temperature $T$. In scalar-valued case there is a special,
so-called critical value of the parameter, where the large-scale limit
exists with a different scaling. This means a non-classical limit
theorem with an unusual normalization.
 
In vector-valued  case a  similar phenomenon appears not only at the
critical, but  at all low temperatures. On low temperatures there is a
phase transition, hence first we have to construct the random field,
called pure state in the literature, which we want to renormalize. This
field has a spontaneous magnetization, and a different normalization is
needed in the direction of the spontaneous magnetization and in the
direction orthogonal to it. In the direction orthogonal to the
spontanous magnetization one has to normalize similarly to the
normalization at the critical temperature.  In the direction of the
magnetization the situation is even more complex. Here the right
normalization and the large-scale limit depends on another parameter
of the model which measures the strength of the interaction.  Let us
emphasize again that the above statements hold not only for a singular
critical value of the parameter $T$, but for all sufficiently low
temperatures $T$. A detailed description of Dyson's vector-valued
hierarchical model together with a complete proof is contained in the
works [1], [4] and [5]. Since these papers are burdened with many
technical details we have found it useful to discuss separately the
most important analytical problems one has to study during these
investigations.
 
The peculiar behaviour of vector-valued models is related to their
invariance  with respect to rotations. This is a symmetry with
respect to a  continuous group. Actually, our main motivation for
studying this problem is to understand the role of continuous
symmetries.
 
The most important part of the problem we are dealing with
can be translated to purely analytical questions. One has to
investigate the action of large powers of a certain integral
operator to a starting function. This operator  is very similar
to the convolution operator (i.e.\ to the convolution of a
function with itself), which is a well-known object in classical
probability. There is however one essential difference between
these operators. The operator appearing in our problem has, unlike
the convolution operator, the following  instability property:
When a large power of this operator is applied to a starting function
then the result strongly depends on the starting function. This
instability property is the main reason for  the peculiar
behaviour of models in statistical physics. ( See paper [3]
for a heuristic explanation of this question.) Actually this
instability is behind the different behaviour of
scalar and vector-valued models too.
 
Let us discuss our problem in some more detail. We are
interested in the behaviour of equilibrium states.
Equilibrium states are appropriately defined probability measures
on the space of all configurations $\s=\{\sigma(j),\;j\in \zb\}$.
The set $\zb$ is chosen generally in translation invariant
models as the integer lattice of the $d$-dimensional
Euclidean space $R^d$ with some $d\ge 1$. In Dyson's model we
choose $\zb=\{1,2,\dots\}$. The variables $\s(j)$, $j\in
\zb$, which are called in the literature  spin variables,
take values in the Euclidean space  $R^p$ with some $p\ge1$. We speak
of scalar-valued models if $p=1$ and vector-valued models if $p\ge
2$. The probability measure that we call equilibrium space is a
Gibbs measure which is a well-known object in statistical physics. It
depends on the following three quantities: A Hamiltonian function
$\ch(\s)$ defined on  the configurations $\s$, a  probability
measure $\nu$, called the  free measure, on $R^p$, a physical
parameter $T$, called the temperature. Formally it is defined as the
probability measure $\mu$ given by the formula
$$
\mu(d\s)=const.\exp\biggl\{-\frac1T\ch(\s)\biggr\}
\prod_{j\in\zb}\nu(d\s(j))\,.\tag 1
$$
Formula (1) would require a more detailed explanation. We
omit it in this discussion, because it is needed only to
translate the problems we are interested in into analytical
questions. A precise definition of  equilibrium states
can be found also in Appendix D of ~[4].
 
Let $\s=\{\s(j),\;j\in \zb\}$ be a $\mu$ distributed random
field. We are interested in its large-scale limit. Let us
explain the meaning of this problem. Since we restrict
ourselves to Dyson's model we shall assume that
$\zb=\{1,2,\dots\}$.  For all $n=1$,~2, $\dots$ define the
fields
$$
Y_{n}(j)=\frac1A_{n}\sum_{k=(j-1)2^n+1}^{j2^n}\s(k)-E\s(k)\,,
\quad j\in\zb\,.\tag2
$$
Let us choose the norming constant $A_n$  in formula (2) in
such a way that the finite dimensional distributions of the
fields $Y_n(j)$ have a nontrivial limit as $n\to\infty$. We
are interested in the right choice of the norming factor
$A_n$ and the distribution of the limit field which is
called the large-scale limit of the original field $\s(j)$ in
the literature. In case of vector-valued models, i.e.\ when
$\s(j)=(\s^{(1)}(j),\dots,\s^{(p)}(j))$ with some $p\ge2$ we
normalize in each coordinate independently. This means that
we define
$$
Y_{n}^{(l)}(j)=\frac1{A_{n}^{(l)}}
\sum_{k=(j-1)2^n+1}^{j2^n}\s^{(l)}(k)-E\s^{(l)}(k)\,,
\quad j\in\zb,\;l=1,\dots,p\,.\tag2$^{\prime}$
$$
 
In formulas (2) and (2$^{\prime}$)  we have  formulated a
problem about the limit distribution of partial
sums of random variables, and this  is a very natural problem
of classical probability theory. If the variables  $\s(j)$ are
independent or very weakly dependent then $A_n=2^{n/2}$ is the
right choice, and the large-scale limit consists of independent
Gaussian random variables. But  for certain choice of  the
Hamiltonian function $\ch(\s)$ and  the temperature $T$ we
have to apply a different normalization, and also the large-scale
limit has a different structure. Our main goal is to obtain a
possibly complete picture of  different possibilities.
 
It is quite natural that the large-scale limit strongly
depends on the Hamiltonian function of the equilibrium state.
The surprising fact, which requires an explanation, is the
dependence of the large-scale limit and of the norming
constant on the temperature $T$. In interesting models the
following picture holds true: There is a so-called critical
temperature~$T_{cr}$, where the model has a peculiar
behaviour. For all $T\ne T_{cr}$ the large-scale limit is a
field of independent Gaussian variables, and the norming
constant is the classical one, i.e.\ one has to divide by
the square-root of the number of summands in a bloc. The only
exceptional case is when $T=T_{cr}$. In this case one has to
divide by a different power, and the large-scale limit may be
non-Gaussian. Moreover, the behaviour of translation
invariant models at the critical temperature strongly depends
on the dimension of the lattice where they are defined.
 
It  is an outstanding problem of the
statistical physics to find a satisfactory explanation for
the above mentioned phenomena. Our paper [2] was devoted
mainly to
this problem. We have also understood that in vector-valued
models some even more complex phenomena appear. Such
models can show some phenomena similar to the critical
behaviour for {\it all}\/ low temperatures. The discussion of
this question is the main topic of the present paper.
Let us remark that since here a model living on the integer
lattice is investigated, i.e.\ a model whose dimension is
always one, it is not possible to investigate  the
dependence of the behaviour of different models on their
dimension directly with the help of this model. Nevertheless,
this model can give some useful information also
about this problem.
 
Our model has the  Hamiltonian function
$$
\ch(\s)=-\sum_{i\in\zb}\sum\Sb j\in \zb\\j>i\endSb
d(i,j)^{-a}\s(i)\s(j)\,,   \tag 3
$$
where $1<a<2$, \ $d(i,j)=2^{n(i,j)-1}$, and
$$
n(i,j)=\min n,\;\exists \;\text{some $k$ such that
}(k-1)2^n<i,j\le k2^n\,.
$$
(In vector-valued models $\s(i)\s(j)$ denotes scalar product
in formula (3).) For the sake of convenience we work with
the number $c=2^{2-a}$ instead
of the parameter $a$. Here the function $d(i,j)$, called
hierarchical distance, is a modification of the usual
distance $|i-j|$. It behaves very similarly, and we have
introduced it, because models defined with the help of such a
distance are simpler to handle because of their symmetry.
Let us observe that in formula~(3) there is an interaction
$d^{-a}(i,j)$ between all pairs $(i,j)$. This interaction
is power-like decreasing with respect to the
hierarchical distance, and the exponent $a$ of this power
is a most important parameter of the model. It plays a role
similar to the dimension in translation invariant models.
 
 The description of the large-scale limit of
an equilibrium state is a very natural
probabilistic problem, but it needs some justification
from  the point of view of statistical physics. It seems very
unlikely that
some physical effect can be thought out which depends on what
kind of limit distribution appears in the large-scale limit
of an equilibrium state. In statistical physics one would be
satisfied with the answer to such at the  first sight  much
simpler questions like the order of decrease of the
correlation
function $E\s(i)\s(j)-E\s(i)E\s(j)$ as $|i-j|\to\infty$. But,
disregarding some special solvable models, there seems to be
no way
to determine the decrease of the covariance function without
solving the more complex problem about the large-scale limit.
This is a very important peculiarity of the problem we are
dealing with, and it also indicates the special character of
the probabilistic problem one has to solve when investigating
the large-scale limit of an equilibrium state. In formula~
(2) the norming constant $A_n$ has to be chosen in such a way
that the variance  of $Y_n(j)$ be separated both from zero
and infinity. If one knows the right choice of $A_n$ one
can also determine the decrease of the correlation function.
But we do not know the behaviour of the correlation
function at the start. Moreover,  the correlation function
behaves essentially differently for  $T=T_{cr}$ and $T\ne
T_{cr}$, and  no  method is known to give an analytic
formula for the critical temperature. Probably, it is
principally impossible to give such an explicit formula. What
we are
able to do is to carry out a procedure which enables us to
approximate the critical temperature at each step better
and
better. We can get the information we are interested in with
the help of such a procedure. This fact also indicates an
essential difference between the determination of the
large-scale limit of equilibrium states and the usual limit
theorems in classical probability. In problems of probability
theory  one generally  knows at the start whether one has to
deal with strongly or weakly dependent random variables. Here
we learn it only after solving the limit problem.
 
To give a more detailed discussion  we have
to
translate our problem to purely analytical questions. To do
it first we have to define our model completely.
We consider a model  with the Hamiltonian function $\ch(\s)$
defined in (3) and the free measure  $\nu$ defined by the
formula
$$
p_0(x)=\frac {d\nu}{dx}(x) =C(t)\exp\left\{-
\frac{x^2}2-\frac t4|x|^4\right\}\,,  \tag 5
$$
where  $t>0$ 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 $1<c<\sqrt
2$ and $\sqrt 2<c<2$. For $\sqrt 2<c<2$ there is a sequence
$M_n$ such that $2^{-n/2}P_n(2^{-n/2}z+M_n)$ tends to a Gaussian
density function with expectation zero and positive variance,
i.e.\ the situation is similar to the scalar-valued case. On
the other hand, for $1<c<\sqrt2$ we have to normalize
otherwise. In this case $c^{-n}P(c^{-n}z+M)$ has a
non-trivial limit with some $M>0$ 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 $1<c<\sqrt2$ and $\sqrt2<c<2$.)
 
The main goal of our investigations is to give an explanation
for the above discussed phenomena and some related questions.
For this aim let us rewrite Problem $1'$ in a form more
appropriate for us.
Let us introduce the functions
$$
\qb_n(x) =\qb_n(x,T)=B_n
\exp\biggl\{\frac{a_0}{2a_1}c^nx^2\biggr\}p_n\biggl(
\sqrt{\frac T{a_1}}x\biggr)
$$
and
$$
q_n(x)=c^{-n/2}\qb_n\left(c^{-n/2}x\right)\,
 $$
with $a_0=\frac2{2-c}$ and $a_1=a_0+1$, where the function
$p_n(x)$ is the same as in Problems 1 and 1$^{\prime}$.
Simple calculation shows that
$$
\qb_{n+1}(x)=\int
\exp\{-c^nu^2\}\qb_n(x-u)\qb_n(x+u)\,du\tag7 $$
and
$$q_{n+1}(x)= \rb q_n(x)
$$
with
$$
\rb q(x)=\int
\exp\{-u^2\}q\left(\frac x{\sqrt c}-u\right)
q\left(\frac x{\sqrt c}+u\right)\,du\tag7$^{\prime}$ $$
with some $B_n$. (We are not really interested in the value
of $B_n$, because it influences only the norming factor when
$p_n(x)$ is expressed through $q_n(x)$. On the other hand,
the norming constant in $p_n(x)$ is determined by the fact
that $p_n(x)$ is a density function.) We also
have $$ \qb_0(x)=q_0(x)=B_0\exp\left\{\frac{a_0-T}{2a_1}x^2
-\frac{tT^2}{4a_1 ^2}|x|^4\right\} .
$$
Since the function $\p n$ can be simply expressed by
$\q n$ or $\qq n$, their investigations are equivalent
problems. It is more convenient to work with the function $\q
n$ or $\qq n$ than directly with $\p n$. Let us emphasize
that the formulas expressing
$\q{n+1}$ and $\qq{n+1}$ through $\q n$ and $\qq n$ do not
depend on the parameter $T$. Moreover, in the case of the
function $\q n$ it depends neither on $n$. (This was our main
reason for introducing the function $\q n$.) All dependence
on $T$ is contained in the starting function $\q 0$. But this
dependence is very essential, because the operator $\rb$,
unlike its classical  probability counterpart the
convolution operator, ``remembers'' of the starting function.
The limit of $\rb ^n\q 0$, as $n\to\infty$, is a small
perturbation of the starting function. In our paper [3] we
discussed this property and pointed out  that this is the
final cause of phase transitions and critical phenomena in
statistical physics. Here we show that it has other
far-reaching consequences. It also implies that vector-valued
models with continuous symmetries have some properties which
have no scalar-valued counterpart. Let us explain this in
more detail.
 
The function $\qq n$ depends on $x$  only through its
absolute value $|x|$, i.e.\ it remains invariant if we rotate
$x$. This property, called $\bold O(p)$ symmetry in the
literature, is a continuous symmetry, and it has far-reaching
consequences. For the sake of simpler notations let us assume
that $p=2$. Because of this property it is natural to
introduce the function $Q_n(x)$ defined by the relation
$Q_n(x)=\bar q_n((x,0))$ and to work with this function
instead of the original function $\bar q_n(x)$. Let us
rewrite formula  (7) for the functions $Q_n(x)$. We get that
 $$
Q_{n+1}(x)= C_n\int e^{-c^n(u^2+v^2)}
Q_n\left(\sqrt{(x+u)^2+v^2}\right)
Q_n\left(\sqrt{(x-u)^2+v^2}\right) \,du\,dv  \tag8
$$
The argument of the function $Q_n$ in formula (8) is rather
complicated. Hence it is natural to substitute it by a
simpler expression and to control the error caused by this
substitution. This  can  be done in both cases $1<c<\sqrt2$
and $\sqrt2<c<2$, but the right substitutions are different
in the two cases. This is the reason of their  different
behaviour. We give a short informal explanation for this.
 
In formula (8) we have to integrate in the whole space $R^2$,
but because of the kernel $\exp\{-c^n(u^2+v^2)\}$ in the
integral an essential contribution for the integral is
supplied only if $u$ and $v$ are small. (Actually this is the
cause why the operator ~$\rb $ ``remembers'' to the starting
function.) The function ~$Q_n(x)$ has a maximum ~$M=M_n$
which
depends very weakly on ~$n$, hence at this heuristical level
we can simply disregard this dependence. Actually it is
enough to give a good asymptotic formula for $Q_n(x)$ only in
a small
neighbourhood of ~$M$. When $x\sim M$ and $u$ and $v$ are
small it is natural to try to  make one of  the following
approximations in the argument of ~$Q_n$ in formula ~(8):
$$
\align
\sqrt{(x\pm u)^2+v^2}&\sim x\pm u+\frac {v^2}{2(x\pm u)}
\sim x\pm u+\frac {v^2}{2M} \,,\\
\intertext{or}
\sqrt{(x\pm u)^2+v^2}&\sim x\pm u \,.
\endalign
$$
The first approximation would suggest  the formula
$$
\align
& Q_{n+1}(x)\\&\qquad={C}_n\int
\exp\left\{-c^n(u^2+v^2)\right\}
Q_n\left(x+u+\frac{v^2}{2M} \right)
Q_n\left(x-u+\frac{v^2}{2M} \right)  \,du\,dv+\e^1_n(x)
\\
&\qquad=\int\exp\left\{-c^n u^2\right\}
Q_n\left(x+u+\frac{v^2}{2M} \right)
Q_n\left(x-u+\frac{v^2}{2M} \right)\,du\,dv+\e_n(x)\,,\tag9
\endalign
$$
and the second one
$$
\align
Q_{n+1}(x)&={C}_n\int \exp\left\{-c^n(u^2+v^2)\right\}
Q_n(x+u)
Q_n(x-u)  \,du\,dv+\bar\e^1_n(x)          \\
&=\int Q_n(x+u)Q_n(x-u)\,du+\bar\e_n(x)  \,.\tag9$'$
\endalign
$$
(If the  approximation in the first row is allowed in the
latter
formulas then the approximation in the second row  is also
permitted.) Whether the first approximation ~(9) has to be
applied or the second rougher approximation ~(9$'$) is also
permitted
that depends on whether the error terms ~$\e_n(x)$ and
$\bar\e_n(x)$
are negligible or not. A detailed analysis shows that in the
case $1<c<\sqrt2 $ the first approximation (9) is the right
choice and in the case $\sqrt2<c<2$ the second one (9$'$).
 
For \co{} let  us introduce the function
$\Qb{n}=c^{-3/2n}Q_n(c^{-n}x+M)$. Formula ~(9) can be
rewritten as
$$
\Qb{n+1}=\bt \Qb{n}+\bar\e_n(x) \tag 10
$$
with
$$
\bt f(x)=\bt_c f(x)=\int \exp(-v^2)
f\left(\frac xc-u+\frac
{v^2}{2M}\right)
f\left(\frac xc+u+\frac
{v^2}{2M}\right)\,du\,dv \,.
$$
In this case ~$\Qb{n}$ tends to the unique solution of the
fixed point equation $Q=\bt Q$, and this fact supplies the
answer to Problem ~$1'$. To prove it we have to show that the
solution ~$Q(x)$ of the fixed point equation is sufficiently
stable for our purposes, i.e.\ the relation
$\lim_{n\to\infty}\bt^n\bigl(Q(x)+\e(x)\bigr)=Q(x)$ holds for
a small
perturbation of the fixed point $Q(x)$, and this convergence
is sufficiently fast. This stability property holds only in
the case \co{}, and this is the reason why the solution of
Problem ~$1'$ is different in the cases \co{} and \ct{}. To give
a complete proof of the convergence $\Qb{n}\to Q(x)$ we have to
overcome several technical difficulties. The  most important
one among them is to show that the error term $\bar\e_n(x)$
is really negligibly small. This question is discussed in
the Preface and in the first three Sections of Part ~I of
paper [4] in more detail.
 
In the case \ct{} it is more natural to work with the function
$\tilde Q_n(x)=\allowmathbreak 2^{-n/2}Q_n(2^{-n/2}x+M)$.
Relation ~($9'$) implies that
$$
\tilde Q_{n+1}(x)=\bold U\tilde  Q_n(x)+\bar \e'_n(x)
\tag$10'$
$$
with
$$
\bold Uf(x)=\int
f\left(\frac x{\sqrt 2}-u\right)
f\left(\frac x{\sqrt 2}+u\right)\,du\,.
$$
The fixed point of the operator ~$\bold U$, i.e.\ of the
convolution operator, are normal density functions with
expectation zero. In the case \ct{} the error term ~$\bar
\e_n(x)$ is negligibly small for \ct{}. This implies that in
this case \ct{} $\tilde Q_n(x)$ tends to a normal  density
function, and this yields the solution of Problem ~$1'$
in this case too.
 
We discuss the investigation of Problem ~2 more briefly. We
have introduced the dependence of the measure ~$\mu^h_N$ on
$h$ in order to investigate the influence of an external
magnetic field to the model. We need  such an approach in the
investigation of the model at low temperatures. At low
temperatures there is a phase transition, i.e.\ several
different equilibrium states exist with the same Hamiltonian
function, free measure and temperature. Hence first we must
clarify which equilibrium state we are working with. We
select out a pure state, i.e.\ an equilibrium state which
cannot be written as the mixture of different equilibrium
states. A natural way to construct pure states, and we
choose this approach,  is to introduce an external field
~$h_N$ in the volume ~$V_N$ and take the limit of the
measures
~$\mu^{h_N}_N$ in ~$V_N$ as $N\to\infty$ and $h_N\to 0$. To
carry
out this program we have to solve Problem ~2. The pure state
we construct in this way has a spontaneous magnetization in
the direction $e^{(1)}=(1,0)$, i.e.\ $E\s^{(1)}(j)=M>0$ 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