\input amstex \documentstyle{amsppt} \magnification=\magstep1 \hsize=16truecm \TagsOnRight \define\di{\displaystyle} \define\({\left(} \define\){\right)} \define\[{\left[} \define\]{\right]} \define\a{\alpha} \define\e{\varepsilon} \define\g{\gamma} \define\la{\lambda} \define\k{\kappa} \define\const{\text{\rm const.}\,} \define\nn{\tilde n(T)} \define\bbeta#1 { \dfrac{\beta_{#1}(T)}{c^{{(#1)}}}} \define \N{\Bbb N} \define \R{\Bbb R} \topmatter \title On a conjecture of Dyson \endtitle \author P. M. Bleher$^{(1)}$ and P. Major$^{(2)}$ \endauthor \affil $^{(1)}$ Department of Mathematical Sciences, \\ Indiana University -- Purdue University at Indianapolis, \\ 402 N. Blackford Street, Indianapolis, Indiana 46202, USA \\ E-mail: bleher\@math.iupui.edu \\ $^{(2)}$ Mathematical Institute of the Hungarian Academy of Sciences and \\ Bolyai College of E\"otv\"os Lor\'and University, \\ Budapest, HUNGARY \\ E-mail: major\@math-inst.hu \endaffil \dedicatory To the memory of Roland L'vovich Dobrushin \enddedicatory \abstract In this paper we study Dyson's classical $r$-component hierarchical model with a Hamiltonian function which has a continuous $O(r)$-symmetry, $r\ge 2$.This is a one-dimensional ferromagnetic model with a long range interaction potential $U(i,j)=\allowmathbreak -l(d(i,j))\allowmathbreak d^{-2}(i,j)$, where $d(i,j)$ denotes the hierarchical distance. We are interested in the case when $l(t)$ is a slowly increasing positive function. For a class of free measures, we prove a conjecture of Dyson. This conjecture states that the convergence of the series $l_1+l_2+\dots$, where $l_n=l(2^n)$, is a necessary and sufficient condition of the existence of phase transition in the model under consideration, and the spontaneous magnetization vanishes at the critical point, i.e. there is no Thouless' effect. We find however that the distribution of the normalized average spin at the critical temperature $T_c$ tends to the uniform distribution on the unit sphere in $\R^r$ as the volume tends to infinity, a phenomenon which resembles the Thouless effect. We prove that the limit distribution of the average spin is Gaussian for $T>T_c$, and it is non-Gaussian for $T0. $$ Let $\nu(dx)$ be a free probability measure on $\R^r$. Then the Gibbs measure in $V_n$ at a temperature $T>0$ with free boundary conditions is defined as $$ \mu_n(dx;T)=Z^{-1}_n(T)\exp\left\{-\beta \Cal H_n(x)\right\}\prod_{j=1}^{2^n}\nu(dx_j), \quad \beta=T^{-1}. $$ We will assume that the free measure $\nu(dx)$ is invariant with respect to the group $O(r)$ of orthogonal transformations, i.e., $\nu(UA)=\nu(A)$ for all $U\in O(r)$ and all Borel sets $A\in B(\R^r)$. Then the Gibbs measure $\mu_n(dx;T)$ is $O(r)$-invariant as well, $$ \align \mu_n(UA_1,\dots, UA_{2^n};T)=\mu_n(A_1,\dots,A_{2^n};T), \quad &\text{for all } U\in O(r),\\ &\quad A_j\in B(\R^r),\; j=1,\dots, 2^n. \endalign $$ In [Dys2], Dyson proved the following theorem (see also [Dys3]). Define $$ l_n=l(2^n).\tag1.2 $$ Assume that $r=3$ and $\nu(dx)$ is a uniform measure on the unit sphere in $\R^3$. This is the classical Heisenberg hierarchical model. \proclaim{Theorem 1.1}{\rm (see [Dys2]).} {\it The classical Heisenberg hierarchical model has a phase transition if $$ B=\sum_{n=1}^\infty l_n^{-1}<\infty.\tag1.3 $$ It has a long-range order so long as $$ \beta>B. $$} \endproclaim Dyson also formulated the following conjecture (see [Dys2]): ``It also seems likely that for sequences $l_n$ which are positive and increasing with $n$ the condition (1.3) is necessary for a phase transition in Heisenberg hierarchical models.'' The goal of this paper is to {\it prove} Dyson's conjecture for a class of hierarchical models and to study the {\it limit distribution of the average spin} both below and above the critical temperature if condition (1.3) holds. Dyson's proof is a clever application of {\it correlation inequalities}. Our approach is based on an analytical study of the {\it renormalization transformation} for the hierarchical model. We apply a perturbation technique which only works if the free measure $\nu(dx)$ is a {\it small perturbation} of the Gaussian measure. Hence, we cannot treat the case when $\nu(dx)$ is a uniform measure on the unit sphere. On the other hand, we will consider {\it arbitrary} spin dimension $r\ge 2$. We will focus on free measures $\nu(dx)$ which have a density function $p(x)$ on $\R^r$ such that $p(x)$ is close, in an appropriate sense, to the density function $$ p_0(x) =C(\k)\exp\left\{-\frac{|x|^2}2-\k\, \frac{|x|^4}4\right\} \tag1.4 $$ with a sufficiently small parameter $\k>0$. Precise conditions on $p(x)$ are given below. We also will assume some regularity conditions about the sequence $l_n=l(2^n)$ (see below). We are investigating the following question. Let $p_n(x,T)$ denote the density function of the average spin $2^{-n}\sum\limits_{j=1}^{2^n}\sigma(j)$, where $(\sigma(1),\dots,\sigma(2^n))$ is a $\mu_n(T)$-distributed random vector. Because of the rotational invariance of the model, the function $p_n(x,T)$ is a function of $|x|$. We are interested in the limit behaviour of the function $p_n(x,T)$ as $n\to\infty$, with an appropriate normalization. In our papers [BM1,3,4] this problem was considered for the polynomial function $l(t)=t^\a$ with $0<\a<1$, when the potential function $l(d(j,k))d^{-2}(j,k)$ in (1.1) is $d^{-2+\a}(j,k)$. We have distinguished three cases for~$\a$: \medskip (i) $(1/2)<\a<1$, \ (ii) $\a=1/2$, and (iii) $0<\a<(1/2)$. \smallskip The difference between these cases appears in the asymptotic behavior of $p_n(x,T)$ at small $T$. When $T$ is small the spontaneous magnetization $M(T)$ is positive, and the function $p_n(x,T)$ is concentrated in a narrow spherical shell near the sphere $|x|=M(T)$. The question is what the width of this shell is and what the limiting shape of $p_n(x,T)$ is like along the radius after an appropriate rescaling. In case (i), the width is of the order of $2^{-n/2}$ and the limit shape of $p_n(x,T)$ is Gaussian (see [BM1]). In case (ii), there is a logarithmic correction in the asymptotics of the width, but the limit shape is still Gaussian (see [BM4]). In case (iii), the width of the shell has a nonstandard asymptotics of the order of $2^{-n\a }$, and the limit shape of $p_n(x,T)$ along the radius (after a rescaling) is a non-Gaussian function which is the solution of a nonlinear integral equation (see [BM3] and the review [BM2]). In the present paper we are interested in the marginal case when $l(t)$ has a sub-polynomial growth. Before formulating the main results we would like to discuss the importance of Dyson's condition (1.3). In the case of the Ising hierarchical model ($r=1$), Dyson proved in [Dys2] that there exists the ``weakest'' interaction function $l(t)$ for which the hierarchical model (1.1) has a phase transition. This function is $l(t)=\log\log t$, which corresponds to $l_n=\log n$. Dyson has proved that if $$ \lim_{n\to\infty} {l_n\over \log n}=0, \tag1.5 $$ then the spontaneous magnetization is equal to zero for all temperatures $T>0$. On the other hand, if $$ {l_n\over \log n}>\varepsilon \quad\text{for all } n>0\text { with some }\e>0, $$ then the spontaneous magnetization is positive at sufficiently low temperatures $T>0$. In the borderline model, when $$ l_n=J\log n,\qquad J>0, \tag1.6 $$ Dyson proved that the spontaneous magnetization $M(T)$ has a jump at the critical temperature $T_c$. The existence of the jump for the 1D Ising model with long-range interaction was first predicted by Thouless (see [Tho], and also the work [AYH] of Anderson, Yuval, and Hamann and references therein) for the translationally invariant Ising model with the interaction $$ H(\sigma)=-\sum_{j,k} {\sigma(j)\sigma(k)\over (j-k)^2}. \tag1.7 $$ This phenomenon (the jump of $M(T)$ at $T=T_c$) is called the Thouless effect. A rigorous proof of the existence of the Thouless effect in the Ising model with the inverse square interaction (1.7) was given by Aizenman, J. Chayes, L. Chayes, and Newman [ACCN]. Simon proved in [Sim] the absence of continuous symmetry breaking in the one-dimensional $r$-component model with the interaction (1.7), in the case when $r\ge 2$. Dyson formulated a general heuristic principle in [Dys2] which tells us when one should expect the Thouless effect in a 1D long-range ferromagnetic model: It should occur for the ``weakest'' interaction (if it exists) for which a phase transition appears. Dyson wrote that in the hierarchical model ``in the Ising case, there exists a borderline model $l_n=\log n$ which is the `weakest' ferromagnet for which a transition occurs, and this borderline model shows a Thouless effect. In the Heisenberg case there exists no borderline model, since there is no `most slowly converging' series (1.3). Thus we do not expect to find a Thouless effect in any one-dimensional Heisenberg hierarchical ferromagnet.'' This conjecture of Dyson, about the absence of a Thouless effect in the Heisenberg case, plays a very essential role in our investigation. We show that in the class of the $r$-component hierarchical models under consideration, the spontaneous magnetization $M(T)$ approaches zero as $T$ approaches the critical temperature, i.e., there is no Thouless effect. On the other hand, we observe a phenomenon which resembles the Thouless effect: at $T=T_c$ the rescaled distribution $$ \bar M_n^r(T_c) p_n(\bar M_n(T_c) x,T_c)\,dx,\qquad \bar M_n(T)=\(\int_{\R^r} x^2 p_n(x,T)\,dx\)^{1/2}, \tag1.8 $$ approaches, as $n\to\infty$, a uniform measure on the unit sphere in $\R^r,\;r\ge 2$. Thus, although the spontaneous magnetization $M(T_c)=\lim\limits_{n\to\infty} \bar M_n(T_c)$ is equal to zero at the critical point, the distribution of the normalized average spin converges to a uniform measure on the unit sphere. This is a ``remnant'' of the spontaneous magnetization at the critical temperature $T_c$. To formulate our results we will need some conditions on the sequence $l_n=l(2^n)$. We need different conditions on $l_n$ in different theorems. We formulate the conditions we shall later apply. \medskip\noindent {\it Conditions on the sequence $l_n$, $n=0,1,2,\dots$}: Let us introduce the notation $$ c_n={l_n\over l_{n-1}}, \quad n=0,1,\dots,\text{ with}\quad l_{-1}=1. \tag1.9 $$ \proclaim {Condition 1} $$ l_0=1;\quad 1\le c_n\le 1.01,\quad\text{for all } n; \qquad \lim_{n\to\infty} c_n=1. \tag$1.10$ $$ \endproclaim {\it Remark.} The relation $l_0=1$ is not a real condition, it can be reached by a rescaling of the temperature. We use it just for a normalization. \proclaim {Condition 2} {\it $$ \lim_{n\to\infty}l_n\sum_{j=n}^\infty l_j^{-1}=\infty. \tag1.11 $$ Moreover, the above relation is uniform in the following sense: For all $\e>0$ there are some numbers $K=K(\e)>0$ and $L=L(\e)>0$ such that $$ l_n\sum_{j=n}^{n+K}l_j^{-1}\ge \e^{-1} \tag1.12 $$ for all $n>L$.} \endproclaim \proclaim {Condition 3} {\it $$ \sup_{1 400\,\k^{-1}. \tag1.14 $$} \endproclaim \proclaim {Condition 5} {\it $$ {l_n\over l_{n+k}}>\bar\eta\quad\text{for all }\;n=0,1,2,\dots, \quad \text{and all } k=1,\dots, L. \tag1.15 $$} \endproclaim The numbers $\k,\bar\eta>0$, and $L\in\N$ in these conditions will be chosen later. An example of sequences $l_n$ satisfying Conditions 1--5 is given in the following proposition. \proclaim {Proposition 1.2} The sequence $$ l_n=(1+an)^\lambda,\qquad a>0,\;\lambda>1, \tag1.16 $$ satisfies Conditions 2 and 3 for all $a>0$ and $\lambda>1$. There exists a number $a_0=a_0(\lambda)>0$ such that this sequence satisfies Condition 1 for all $00$ such that this sequence satisfies Condition 4 for all $00$ such that this sequence satisfies Condition 5 for all $01$ there exists a number $$ a_3=a_3(\lambda,\k,\bar\eta,L)= \min\{a_0(\la),a_1(\k,\la),a_2(\bar\eta,L)\}>0 $$ such that for all $00$ is a norming factor, and $$ \| \e(t)\|_{C^4(\R^1)}<0.01. \tag1.18 $$ Now we formulate our main results. We denote by $p_n(x,T)$ the distribution of the average spin $2^{-n}[\sigma(1)+\dots+\sigma(2^n)]$ with respect to the Gibbs measure $\mu_n(dx;T)$ and put $$ \bar M_n(T)=\(\int_{\R^r}x^2p_n(x,T)\,dx\)^{1/2} \tag1.19 $$ By $\bar p_n(x,T)$ we denote the rescaled density function $$ \bar p_n(x,T)=\bar M_n^r(T)p_n(\bar M_n(T)x,T) \tag1.20 $$ and by $\bar\nu_{n,T}(dx)$ the corresponding probability distribution $$ \bar\nu_{n,T}(dx)=\bar p_n(x,T)\,dx. \tag1.21 $$ \medskip\noindent {\it Formulation of the main results.}\/ We fix a sufficiently small positive number $\eta$ which will be the same through the whole paper. For instance, $\eta=10^{-100}$ is a good choice. Define the following number $N=N(\eta)$: $$ N=\min\{ n\: l_n>\eta^{-1}\}.\tag1.22 $$ Assume that an arbitrary number $\bar\eta$ in the interval $0<\bar\eta\le\eta$ is fixed (it is used in Condition 5). \proclaim {Theorem 1.3} {\rm (Necessity of Dyson's condition).} {\it Assume that $$ \sum_{n=1}^\infty l_n^{-1}=\infty. \tag1.23 $$ Then there exists a number $\k_0=\k_0(N)$ such that for all $0<\k<\k_0$ the following statements hold. Assume that the density $p(x)=\dfrac{\nu(dx)}{dx}$ belongs to the class $\Cal P_\k$ and the sequence $\{l_n,\;n\ge 0\}$ satisfies Conditions 1 --- 3. Then there exists a constant $L=L(\bar\eta,\k)$ such that if the sequence $\{l_n,\;n\ge 0\}$ satisfies Condition 5, then for all $T>0$, there exists the limit, $$ \lim_{n\to\infty} 2^n \bar M_n^2(T)=\chi(T)>0. \tag1.24 $$ In particular, the spontaneous magnetization satisfies the relation $$ M(T)=\lim_{n\to\infty}\bar M_n(T)=0. \tag1.25 $$ In addition, the distribution $\bar\nu_{n,T}(dx)$ tends weakly to the standard normal distribution as $n\to\infty$.} \endproclaim To formulate our results for the case when the Dyson condition (1.3) holds, we define a function $\hat p_n(t,T)$ such that $$ p_n(x,T)=\hat p_n(|x|,T), \tag1.26 $$ and introduce the notations $$ V_n(T)=\(\int_0^\infty (t-\bar M_n(T))^2\hat p_n(t,T)\,dt\)^{1/2} \tag1.27 $$ and $$ \pi_n(t,T)=L_n^{-1}(T)\hat p_n\(\bar M_n(T)+V_n(T)\,t,T\), \qquad t\ge -\frac{\bar M_n(T)}{V_n(T)}, \tag1.28 $$ where $$ L_n(T)=\int_{-\bar M_n(T)/V_n(T)}^\infty \hat p_n\(\bar M_n(T)+ V_n(T)\,t,T\)\,dt. $$ Thus, by (1.26) and (1.28) $$ p_n(x,T)=L_n(T)\,\pi_n\({|x|-\bar M_n(T)\over V_n(T)}\,,T\)\,. \tag1.29 $$ Our aim is to prove that in the case when the Dyson condition (1.3) holds, there exists a critical temperature $T_c$ such that the spontaneous magnetization $M(T)=\lim\limits_{n\to\infty}\bar M_n(T)$ is positive for $T0. \tag1.31 $$ \endproclaim For a proof of Proposition 1.4 see the proof of Lemmas~12 and 13 in [BM3]. \proclaim {Theorem 1.5} {\it Assume that $$ \sum_{n=1}^\infty l_n^{-1}<\infty. $$ Then there exists a number $\k_0=\k_0(N)$ such that for all $0<\k<\k_0$ the following statements hold. Assume that the density $p(x)=\dfrac{\nu(dx)}{dx}$ belongs to the class $\Cal P_\k$ and the sequence $\{l_n,\;n\ge 0\}$ satisfies Conditions 1 --- 4. Then there exists a constant $L=L(\bar\eta,\k)$ such that if the sequence $\{l_n,\;n\ge 0\}$ satisfies Condition 5, then there exists a critical temperature $T_c>0$ with the following properties. \medskip \item {1)} If $T>T_c$ then $$ \lim_{n\to\infty} 2^n \bar M_n^2(T)=\chi(T)>0, \tag1.32 $$ and the distribution $\bar\nu_{n,T}(dx)$ approaches weakly as $n\to\infty$ a standard normal distribution. The function $\chi(T)$ in (1.32) satisfies the following estimates near the critical point. There exists a temperature $T_0>T_c$ and numbers $C_2>C_1>0$ such that for all $T_0>T>T_c$ there exists a number $\bar n(T)$ such that $$\aligned &C_1\sum_{k=\bar n(T)}^\infty l_k^{-1}< T-T_c\le C_2 \sum_{k=\bar n(T)}^\infty l_k^{-1},\\ &C_1{2^{\bar n(T)}\over l_{\bar n(T)}}<\chi(T)< C_2{2^{\bar n(T)}\over l_{\bar n(T)}},\quad \text{for all }T_c0, \tag1.36 $$ and $$ C_1|T-T_c|^{1/2}0, \tag1.38 $$ and $$ \lim_{n\to\infty} \pi_n(t,T)=\pi(t)\equiv Ce^{-2t/3}g\(t-a\), \tag1.39 $$ where $g(t)$ is a probability density which satisfies the fixed point equation (1.30), and the quantities $C$ and $a$ are determined from the equations $$ \int_{\R^1}\pi(t)\,dt=1,\qquad \int_{\R^1}t\,\pi(t)\,dt=0. \tag1.40 $$} \endproclaim Let us make some remarks about Theorem 1.5. Relations (1.32) and (1.34) imply that $$ M(T)=\lim_{n\to\infty} \bar M_n(T)=0,\quad \text {for all } T\ge T_c, \tag1.41 $$ i.e., the spontaneous magnetization $M(T)$ vanishes at $T\ge T_c$. By (1.37), $$ \lim_{T\to T_c^-}M(T)=0, $$ hence there is no Thouless effect (by $\lim_{t\to a^{\pm}}f(t)$ we denote, as usually, limits of $f(t)$ as $t\to a$ from the right and from the left, respectively). The number $\bar n(T)$ in (1.33) is very %@ important for our investigation in the subsequent sections. It shows how many iterations of the recursive equation (renormalization group transformation) is needed to reach the ``high temperature region'' (see Section 3 below for precise definitions). The quantity $\xi(T)=2^{\bar n(T)}$ is the {\it correlation length.} Usually the correlation length has a power-like asymptotics $\xi(T)\asymp |T-T_c|^{-\nu}$ as $T\to T_c$ where $\nu$ is the critical exponent of the correlation length (see, e.g., [Fi] or [WK]). It follows from (1.33) that in the case under consideration, $\xi(T)$ grows super-polynomially as $T\to T_c^+$. For instance, if $l_n$ is a sequence determined by equation (1.16) then $\xi(T)$ grows like $\exp \[C_0(T-T_c)^{1/(\la-1)}\]$. Similarly, (1.33) implies that the magnetic susceptibility $\chi(T)$ diverges super-polynomially as $T\to T_c^+$. The estimates (1.37) correspond to the value of the critical exponent of spontaneous magnetization $\beta =1/2$. Relation (1.38) shows that the mean square deviation of the average spin along the radius behaves, when $n\to\infty$, as $$ V_n(T)\sim {T\over 3M(T)l_n}, \qquad T0$ is increasing. Moreover, after finitely many steps $k$ the high temperature approximation is already applicable, and the number of steps we need to get into this situation can be bounded by a constant independent of the parameter~$T$. The proof given in Section~5 contains arguments essentially different from the rest of the paper. Here we heavily exploit that the numbers $c_n=\di\frac{l_n}{l_{n-1}}$ are very close to one. Informally speaking, the sequence of numbers $c_n-1$ behaves like a small parameter, and this ``small parameter" enables us to handle our model near the critical temperature. The setup of the rest of the paper is the following. In Section 2 we give an analytic reformulation of the problem and connect Dyson's condition (1.3) with an approximate recursive formula for some quantities $M_n(T)$ related to the spontaneous magnetization (see (2.28) below). In Section 3 we introduce a notion of low and high temperature regions together with an intermediate region. Then we formulate the basic auxiliary theorems about the characterization of these regions. In Sections~4,~5, and~6 we prove the main estimates concerning the low temperature region, the intermediate region, and the high temperature region, respectively. In Section~7 we prove the convergence of the recursive iterations to the fixed point for all $T0$ is very large. In this case the low temperature expansion can be applied. As we shall see later, the approximation of $\bar {\bold Q}_{n,M_n(T)}^{\bar c}$ by $\bar {\bold T}_{n,M_n(T)}^{\bar c}$ yields that $$ M_{n+1}(T)\sim M_n(T)-\frac{r-1}{4c^{(n)}M_n(T)}\, , \tag2.28 $$ which, in turn, implies that $$ M_{n+1}^2(T)\sim M_n^2(T)-{r-1\over 2c^{(n)}} \tag2.29 $$ It follows from (2.4) and (1.10) that $$ 2\le A_n\le 2.03,\qquad \lim_{n\to\infty}A_n=1, \tag2.30 $$ hence by (2.11), $$ 3\le{c^{(n)}\over l_n}\le 3.03,\qquad \lim_{n\to\infty} {c^{(n)}\over l_n}=3. \tag2.31 $$ This allows us to rewrite (2.29) as $$ M_{n+1}^2(T)\sim M_n^2(T)-{r-1\over 6l_n} \tag2.32 $$ This formula underlines the importance of the Dyson condition (1.3). Namely, if the series $$ B=\sum_{n=1}^\infty l_n^{-1} \tag2.33 $$ converges then $M_n(T)$ remains large for all $n$ if $T>0$ is small. Indeed, assume that $TC(\k T^2)^{-1}$, hence by (2.32), $$ M_n^2(T)\ge M^2_0(T)-{r-1\over 6}\sum_{n=0}^\infty l_n^{-1}\ge C(\k T^2)^{-1}-C_1\gg 1 $$ for all $n$ if $T>0$ is small, which was stated. On the other hand, if the series (2.33) diverges, then for some $n$, $M_n(T)$ becomes small, and the approximation (2.28) becomes inapplicable. The low temperature approximation can be applied when $M_n(T)$ is large. When $M_n(T)$ is small a different approximation is natural. If the function $q_n(x,T)$ is essentially concentrated in a ball whose radius is much less than $\(c^{(n)}\)^{-1/2}$, then a small error is committed if the kernel function $e^{-c^{(n)}u^2}$ in formula (2.12) is omitted. This means that the formula expressing $q_{n+1}(x)$ by $q_n(x)$ can be well approximated through the convolution $q_{n+1}(2x)=q_n*q_n(2x)$. This approximation will be called the {\it high temperature approximation}. If the high temperature approximation can be applied for $q_n(x,T)$, then the function $q_{n+1}(x,T)$ is even more strongly concentrated around zero. Hence, as a detailed analysis will show, if at a temperature $T$ it can be applied for a certain $n_0$, then it can be applied for all $n\ge n_0$. Finally, there are such pairs $(n,T)$ for which the function $f_n(x,T)$ can be studied neither by the low nor by the high temperature approximation. We call the set of such pairs an {\it intermediate region.} We shall prove that if the sequence $c^{(n)}$ sufficiently slowly tends to infinity and the function $f_n(x,T)$ is out of the region where the low temperature approximation is applicable, then the density function $f_{n+1}(x,T)$ will be more strongly concentrated around zero than the function $f_n(x,T)$. Moreover, in {\it finitely many}\/ steps the function $f_{n+k}(x,T)$ will be so strongly concentrated around zero that after this step the high temperature approximation is applicable. It is important that the number of steps $k$ needed to get into the high temperature region can be bounded independently of the parameter $T$. The main part of the paper consists of an elaboration of the above heuristic argument. \beginsection 3. Formulation of Auxiliary Theorems To describe the region where the low temperature approximation will be applied we define some sequences $\beta_n(T)$ which depend on the temperature~$T$. Define recursively, $$ \aligned \beta_N(T)&=\frac{\( c^{(N)}\)^2}{2^N}\,,\cr \beta_{n+1}(T)&=\(\frac{c_{n+1}^2}2+\sqrt{\frac{\beta_n(T)}{c^{(n)}}}\) \beta_n(T) +\frac{10}{M_n^2(T)}\quad \text{\rm for }n> N, \endaligned \tag3.1 $$ where the number $N$ is defined in (1.22) and $M_n(T)$ in (2.15). As it will be seen later, these numbers measure how strongly the functions $f_n(x,T)$ are concentrated around zero. We define the low temperature region, where low temperature approximation will be applied. \proclaim {Definition of the low temperature region} {\it A pair $(n,T)$ is in the low temperature region if $0N$ and $\dfrac{\beta_{n-1}(T)}{c^{(n-1)}}\le\eta$. The temperature $T$ is in the low temperature region if the pair $(n,T)$ is in the low temperature region for all numbers~$n$.} \endproclaim Let us remark that by (2.6) and (1.10) $$ 1\le l_n=\prod_{j=1}^n c_j\le 1.01^n, \tag3.2 $$ hence by (2.31), $$ 3\le c^{(n)}\le 3.03\cdot 1.01^n. \tag3.3 $$ Therefore, by (3.1), $$ \dfrac{\beta_N(T)}{c^{(N)}}={c^{(N)}\over 2^N}\le {1\over c^{(N)}}\le \eta \tag3.4 $$ hence the pair $(N+1,T)$ is in the low temperature region if $T\le c_0 A_0/2$. Since $\beta_{n+1}(T)\ge\dfrac{10}{M_n^2(T)}$ the pair $(n,T)$ can get out of the low temperature region only if $M_n(T)$ becomes very small. To define the high temperature region introduce the notations $$ \aligned h_n(x,T)&=\(c^{(n)}\)^{-r/2}q_n\(\frac x{\sqrt{c^{(n)}}},T\),\\ D_n^2(T)&=\int_{\R^r}x^2h_n(x,T)\,dx. \endaligned\tag3.5 $$ where the function $q_n(x,T)$ is defined in (2.10). Let us also introduce the probability measure $H_{n,T}$, $$ H_{n,T}(\bold A)=\int_{\bold A} h_n(x,T)\,dx,\quad \bold A\subset \R^r \tag 3.6 $$ on $\R^r$. \proclaim {Definition of the high temperature region} {\it A pair $(n,T)$ is in the high temperature region if $D^2_n(T)0$, there exist pairs $(n,T)$ which do not belong to the low temperature region. Let $\bar n(T)\ge 0$ be the smallest such number. Assume that the pair $(\bar n(T),T)$ does not belong to the high temperature region. (In this case $(\bar n(T),T)$ is in the intermediate region.) Then there exist numbers $K=K(\bar\eta,t)>0$, $\tilde\eta=\tilde\eta(\bar\eta,t)$, and $k=k(\bar\eta,t)\in\N$ such that $$ D^2_{\bar n(T)}(T)0$ belong to either the low or the high temperature region. If the Dyson condition (1.3) holds, then all sufficiently low temperatures belong to the low and all sufficiently high temperatures to the high temperature region. If the Dyson condition (1.3) is violated, then all temperatures~$T>0$ belong to the high temperature region.} \endproclaim Next theorem concerns the {\it low temperature region.} \proclaim{Theorem 3.2} {\it There exists a number $\k_0=\k_0(N)$ such that for all $0<\k<\k_0$ the following is true. Assume that the Dyson condition (1.3) and Conditions 1 and 2 hold. Assume that the temperature $T$ is in the low temperature region. Then the limit $$ \lim_{n\to\infty} M_n(T)=M_\infty(T) \tag3.8 $$ exists, and $$ \lim_{n\to\infty}\frac{M^2_n(T)-M^2_\infty(T)}{\dfrac{r-1}2 \sum\limits_{k=n}^\infty \dfrac1{c^{(k)}}}=1. \tag 3.9 $$ In addition, $$ \lim_{n\to\infty} \left\|\frac1{M_n(T)} f_n\(\frac t{M_n(T)},T\)- g\(t-\frac{r-1}4\)\right\|=0\,, \tag3.10 $$ where $$ \| f(t) \|=\sum_{j=0}^2\,\sup_{t\ge -c^{(n)}M^2_n(T)}e^{|t|} \left| {d^jf(t)\over d\,t^j} \right| \tag3.11 $$ and the probability density $g(t)$ is defined as a solution of the fixed point equation (1.30).} \endproclaim \demo {\it Remark} Observe that the value ${r-1\over 4}$ of the shift of the function $g(t)$ in (3.10) fits the equation $$ \int_{\R^1}t\, g\(t-\frac{r-1}4\)dt=0. $$ \enddemo {}From this theorem, the Part 3) of Theorem 1.5 follows, with %@ the exception of estimate (1.37). Indeed, we can express the function $p_n(x,T)$ in terms of $f_n(t,T)$. Namely, by (2.10), (2.14), and (2.17), $$ p_n(x,T)=L_n^{-1}(T)\, \exp\(- {A_nl_n|x|^2\over 2T}\) f_n\({c^{(n)}\over \sqrt T}\(|x|- \sqrt T\,M_n(T)\),T\) \tag3.12 $$ Let us write that $|x|^2=\(\sqrt T\,M_n(T)+|x|-\sqrt T\,M_n(T)\)^2$, hence $$\aligned \exp\(- {A_nl_n|x|^2\over 2T}\)&= \exp\left\{- {A_nl_n\over 2T}\big[ TM_n^2(T)+2\sqrt T\,M_n(T)(|x|-\sqrt T\,M_n(T))\right.\\ &\qquad +(|x|-\sqrt T\,M_n(T))^2\big]\bigg\}, \endaligned\tag3.13 $$ and substitute it into (3.12). This leads to the equation $$ p_n(x,T)=\tilde L_n^{-1}(T)\tilde f_n\({|x|-\tilde M_n(T)\over \tilde V_n(T)}\,,T\), \tag3.14 $$ where $$\aligned &\tilde M_n(T)=\sqrt T\, M_n(T),\qquad\tilde V_n(T)={\sqrt T\over c^{(n)}M_n(T)}\\ &\tilde f_n(t,T)=f_n\( {t\over M_n(T)},T\)\exp\(-{A_nl_nt\over c^{(n)}}-\e_n(t,T)\)\\ &\e(t,T)={A_nl_nt^2\over 2(c^{(n)})^2M^2_n(T)}. \endaligned\tag3.15 $$ Observe that by (2.30) and (2.31) $$ \lim_{n\to\infty} {A_nl_n\over c^{(n)}}={2\over 3}\,, \qquad \lim_{n\to\infty}{A_nl_n\over 2(c^{(n)})^2M^2_n(T)}=0, \tag3.16 $$ hence (3.10) implies that there is some $C_0>0$ such that $$ \lim_{n\to\infty}\left\|\tilde f_n(t,T))-C_0 g\(t-{r-1\over 4}\)e^{-2t/3}\right\|'=0, \tag3.17 $$ where $$ \| f(t) \|'=\sum_{j=0}^2\,\sup_{t\ge -c^{(n)}M^2_n(T)}e^{|t|/3} \left| {d^jf(t)\over d\,t^j} \right|. \tag3.18 $$ It remains to shift $\tilde f_n(t)$ to secure the mean value to be zero. Consider $\pi(t)$ defined in (1.31). Put $$ \tilde \pi_n(t,T)=C_n(T) \tilde f_n(t-a_0,T),\qquad a_0=a-{r-1\over 4}, \tag3.19 $$ where $C_n(T)$ is a norming factor such that $$ \int_{\R^1}\tilde \pi_n(t)\,dt=1. \tag3.20 $$ Then $$ p_n(x,T)=L_n^{-1}(T)\tilde \pi_n\({|x|-M_{n0}(T)\over \tilde V_n(T)}\,,T\)\quad\text{with } M_{n0}(T)=\tilde M_n+a_0\tilde V_n(T) \tag3.21 $$ and $$ \lim_{n\to\infty}\int_{\R^1}t\pi_n(t)\,dt=0. \tag3.22 $$ Comparing this formula with (1.29) we obtain (1.36), (1.38), and (1.39) with $$ M(T)=\sqrt T\, M_\infty (T),\qquad \g(T)={\sqrt T\over 3M_\infty (T)}={T\over 3M(T)}, \tag3.23 $$ where $M_\infty (T)$ is the limit (3.4). Now we formulate a theorem about the {\it high temperature region}. Put $$ \tilde h_n(x,T)=2^{-rn/2}q_n\(2^{-n/2}x,T\)=\(\frac{c^{(n)}}{2^n}\)^{r/2} h_n\(\sqrt{\frac{c^{(n)}}{2^n}}x,T\), \tag3.24 $$ and define the probability measures $$ \tilde H_{n,T}(\bold A)=\int_{\bold A} \tilde h_n(x,T)\,dx,\quad \bold A\subset\R^r \tag 3.25 $$ on $\R^r$. \proclaim{Theorem 3.3} {\it There exists a number $\k_0=\k_0(N)$ such that for all $0<\k<\k_0$ there exists a number $L=L(\bar\eta,\k)$ such that the following is true. Assume that Conditions 1 and 5 hold and that $T$ is in the high temperature region. Then the measures $\tilde H_{n,T}$ defined in (3.25) converge weakly to a normal distribution on $\R^r$ with expectation zero and variance $\sigma^2(T)\bold I$ with some $\sigma^2(T)>0$, where $\bold I$ denotes the identity matrix. If $T$ belongs to the high temperature region, but the pair $n=(0,T)$ does not belong to it, (i.e.\ the temperature $T$ is not too high) then the inequality $$ C_1\frac{2^{\bar n(T)}}{c^{(\bar n(T))}}\le \sigma^2(T)\le C_2\frac{2^{\bar n(T)}}{c^{(\bar n(T))}} \tag3.26 $$ also holds with some $C_2>C_1>0$, where $\bar n(T)$ is defined in Theorem~3.1.} \endproclaim \demo {Remark} Not only the convergence of the measures $\tilde H_{n,T}$ but also the convergence of their density functions $\tilde h_n(x,T)$ could be proved. But the proof of the convergence of the distribution is simpler, and it is also sufficient for our purposes. \enddemo \proclaim {Corollary} {\it Let $\bar H_{n,T}$ denote the probability measure on $\R^r$ with the density function $2^{-rn/2}T^rp_n(2^{-n/2}\sqrt{T}x,T)$. Under the conditions of Theorem 3.3 the measures $\bar H_{n,T}$ have the same Gaussian limit as the measures $\tilde H_{n,T}$ defined in Theorem~3.3 as $n\to\infty$.} \endproclaim Our last theorem concerns the {\it critical point}. We want to show that there is a critical temperature $T_{\text {cr.}}$ above which all temperatures belong to the high and below which all temperatures belong to the low temperature region. We also want to describe the situation in the neighbourhood of the critical temperature in more detail. In Theorem 3.4 we prove such a result. \proclaim {Theorem 3.4} {\it There exists a number $\k_0=\k_0(N)$ such that for all $0<\k<\k_0$ there exists a number $L=L(\bar\eta,\k)$ such that the following is true. Assume that Conditions~1 --- 4 are satisfied. Then for a fixed $n$ the set of temperatures $T$ for which $(n,T)$ belongs to the low temperature region forms an interval $(0,T_n]$, and the sequence $T_n$, $n=1,2,\dots$, is monotone decreasing in $n$. Define the critical temperature $T_{\text{cr.}}$ as the limit $T_{\text{cr.}}=\lim\limits_{n\to\infty}T_n$. Then $c_0A_0/4> T_{\text{cr.}}>0$. The function $M_\infty(T)=\lim\limits_{n\to\infty}M_n(T)$ exists in the interval $(0,T_{\text{cr.}}]$, and for fixed $n$ the function $M_n(\cdot)$ is strictly decreasing in the interval $(0,T_n]$. The relation $M_\infty (T_{\text{cr.}})=0$ holds. If $T_{\text{cr.}}+\e>T>T_{\text{cr.}}$ with some $\e>0$, then the inequality $$ C_1\sum_{k=\bar n(T)}^\infty \frac1{c^{(k)}}C_1>0$, where $\bar n(T)$ is defined in Theorem 3.1. If $T_{\text{cr.}}-\e0$, then $$ C_1(T_{\text{cr.}}-T)^{1/2}< M_\infty(T)0$ is small (exact conditions on the smallness of $\k$ will be given later). In this case the function $q_0(x,T)$ has its maximum in the points $\pm \hat M_0(T)$ (see (2.13)) where $$ \hat M_0(T)=\(\frac{A_0 c_0-T}{\k T^2}\). \tag4.1 $$ is a large number. From (2.13) we obtain that $$ \frac1{c^{(0)}} q_0\(\hat M_0(T)+\frac x{c^{(0)}},T\)=\frac1{Z_0(T)} \exp\left\{-\(A_0 c_0-T\)\(\frac{x}{c^{(0)}}\)^2\(1+\frac x{2c^{(0)}\hat M_0(T)}\)^2 \right\} \tag4.2 $$ where $$ Z_0(T)=\int_{-\hat M_0(T)}^\infty \exp\left\{-(A_0 c_0-T)\(\frac x{c^{(0)}}\)^2\(1+\frac x{2c^{(0)}\hat M_0(T)}\)^2 \right\}\,dx . \tag$4.2'$ $$ It can be proved by means of the identity $$ M_0(T)-\hat M_0(T)=\frac{\dsize \int_{-\hat M_0(T)}^\infty x\exp\left\{-(c_0A_0-T)\(\frac x{c^{(0)}}\)^2\(1+\frac x{2c^{(0)}\hat M_0(T)}\)^2 \right\}\,dx} {\dsize\int_{-\hat M_0(T)}^\infty \exp\left\{-(A_0 c_0-T)\(\frac x{c^{(0)}}\)^2\(1+\frac x{2c^{(0)}\hat M_0(T)}\)^2 \right\}\,dx} \tag 4.3 $$ that $$ \left|M_0(T)-\hat M_0(T)\right|\le\frac{\const}{M_0(T)}\le \const \sqrt \k\,T \tag4.4 $$ where $M_0(T)$ is defined (2.15). This shows that $\hat M_0(T)$ is a very good approximation to $M_0(T)$. Some calculation yields, with the help of formulas (4.1) and (4.3), that $$ \left|Z_0(T)- \frac{c^{(0)}\sqrt \pi}{\sqrt{(A_0 c_0-T)}}\right|\le\const \sqrt \k\,T \tag4.5 $$ and from (4.1)--(4.5) we obtain that $$ \aligned &\left|\frac{\partial^j}{\partial x^j}\(f_0(x,T)-\frac{\sqrt{A_0 c_0-T}}{c^{(0)}\sqrt\pi}\exp\left\{-(A_0 c_0-T)\(\frac x{c^0}\)^2\right\}\)\right| \le\const \k^{1/4}e^{-2|x|/c^{(0)}}\\ &\qquad\qquad\qquad\qquad \text{if }|x|<\log \k^{-1},\quad j=0,1,2, \endaligned \tag4.6 $$ and $$ \aligned &\left|\frac{\partial^j}{\partial x^j} f_0(x,T)\right|\le C\exp\left\{ -\frac{(A_0 c_0-T)}{4c^{(0)}}\left|2x +\frac{x^2}{c^{(0)}M_0^2(T)}\right|\right\}\\ &\qquad\qquad\qquad\qquad \text{for }x\ge -c^{(0)} M_0(T),\quad j=0,1,2. \endaligned \tag4.7 $$ A relatively small error is committed if $M_n$ is very large and the arguments $\ell_{n,M_n}^\pm(x,u,v)$ (defined in formula (2.9)) of the function $f_n$ in the operator $\bar{\bold Q}_{n,M}^{\bold c}f_n$ are replaced by $x\pm u$. Exploiting this fact one can prove, using a natural adaptation of the proof of Proposition~1 of paper~[BM3], the following \proclaim {Proposition 4.1} {\it There exists a number $\k_0=\k_0(N)$ such that if %\hfill\break (i) $0<\k<\k_0$, (ii) $0 N \endaligned \tag4.11 $$ To formulate the inductive hypotheses we also introduce a regularization of the functions $f_n(x,T)$. \proclaim {Definition of the regularization of the functions $f_n(x,T)$} {\it Let us fix a $C^\infty$-function $\varphi(x)$, $-\infty0$. One could choose e.g.\ $C=10^{20}$.} \endproclaim \proclaim {Hypothesis $J(n)$} {\it $$ |\tilde \varphi_nf_n(t+is,T)|\le \frac{e^{\beta_n(T)s^2}}{1+\alpha_n(T)t^2} \quad\text{\rm if } |s|\le\frac2{\sqrt{\beta_{n+1}(T)}} $$ with $\tilde \varphi_nf_n(t+is,T)=\int e^{(it-s)x} f_n(x,T)\,dx$, i.e.\ it is the Fourier transform (with a different norming constant) of the function $f_n(x,T)$ together with its analytic continuation.} \endproclaim We need %@ Proposition~4.1 because of its consequence formulated below. Its proof can also be found in~[BM3]. \proclaim{Corollary of Proposition 4.1} {\it Under the conditions of %@ Proposition~4.1 the inductive hypotheses $I(n)$ and $J(n)$ hold for $n=N$ with a universal constant $C>0$ in hypothesis $I(n)$. (For instance one can choose $C=10^5$.)} \endproclaim Before formulating the main result of this Section we introduce the operators $\bold T_n$. They are appropriate scaling of the operators $\bar{\bold T}_{n,M_n(T)}^{\bold c}$ defined in formula~(3.11), but these operators will be applied only for the regularization of the functions $f_n(x,T)$ and not for the functions $f_n(x,T)$ themselves. Put $$ \aligned {\bold T}_{n}\varphi_n(f_n(x,T))=\frac2{c_{n+1}\sqrt \pi}\int &e^{-v^2} \varphi_n\(f_n\(\frac{x}{c_{n+1}}-\frac1{4M_n(T)} +u+\frac{v^2}{2M_n(T)},T\)\) \\ &\varphi_n\(f_n\(\frac x{c_{n+1}}-\frac1{4M_n(T)}- u+\frac{v^2}{2M_n(T)},T\)\) \,du\,dv. \endaligned\tag4.12 $$ The main result of this section is the following \proclaim {Proposition 4.2} {\it There exists $\k_0=\k_0(N)>0$ such that if (i) $0<\k<\k_0$, %@ (in formula (1.4)) (ii) Condition~1 holds, and (iii) the pairs $(m,T)$ belong to the low domain region for all $0\le m\le n$, then the function $f_{n+1}(x,T)$ satisfies the inductive hypotheses $I(n+1)$ and~$J(n+1)$ with the same universal constant $C>0$ (independent of $\k$,~$n$,~$\eta$ and~$T$). Also the relation $$ M_{n+1}(T)=M_n(T)-\frac1{4c^{(n)}M_n(T)}+\frac{\gamma_n(T)}{c^{(n)}} \quad \text{with } |\gamma_n(T)|\le C_1\bbeta{n+1} \sqrt{\beta_{n+1}(T)} \tag4.13 $$ holds with a universal constant $C_1>0$ together with the inequalities $$ 1\le\frac {\beta_{n+1}(T)}{\alpha_{n+1}(T)}\le K_1, \tag4.14 $$ $$ \aligned &\left|\frac{\partial^j}{\partial x^j}\(f_{n+1}(x,T)-\bold T_n\varphi_n(f_n(x,T))\)\right|\le \frac{K_2C^4}{\beta_{n+1}^{(j+1)/2}(T)} \bbeta{n} \\ &\qquad \left[\exp\left\{-\frac1{\sqrt{\beta_{n+1}(T)}} \left|2x+\frac{x^2} {c^{(n+1)}M_{n+1}(T)}\right| \right\}+\exp \left\{-\frac{2|x|}{\sqrt{\beta_{n+1}(T)}}\right\}\right] \\ &\qquad\qquad\qquad\qquad\qquad\qquad x>-c^{(n+1)}M_{n+1}(T),\quad j=0,1,2 \endaligned \tag4.15 $$ and $$ \aligned \left|\frac{\partial^j}{\partial x^j}\bold T_n\varphi_n(f_n(x,T))\right|&\le \frac{K_3C^2}{\beta_{n+1}^{(j+1)/2}(T)} \exp\left\{-\frac{2|x|}{\sqrt{\beta_{n+1}(T)}}\right\},\\ &\qquad x\in \R^1,\quad j=0,1,2,3,4 \endaligned \tag4.16 $$ with some universal constants $K_1$,~$K_2$ and~$K_3$. }\endproclaim The proof of Proposition~4.2 is based on the observation that the operator $\bold T_n$ approximates the operator $\bold Q_{n,M_n(T)}^{\bold c}$ very well, it has a relatively simple structure, it is actually a convolution. More explicitly, it can be written in the form $$ \align \bold T_n\varphi_{n}(f_n(x,T))&=\frac4{c_{n+1}\sqrt \pi}\int_0^\infty e^{-v^2} \varphi_{n}(f_n)*\varphi_{n}(f_n)\(\frac{2x}{c_{n+1}} +\frac{v_2}{M_n(T)}-\frac1{2M_n(T)},T\)\\ &=\frac2{c_{n+1}}\varphi_n(f_n)*\varphi_n(f_n)*k_{M_n(T)} \(\frac{2x}{c_{n+1}}\), \endalign $$ where $*$ denotes convolution, and $k_{M_n(T)}(x)=M_n(T)k\(M_n(T)x\)$ with $k(x)=\dfrac1{\sqrt{\pi x}}e^{-x}$ for $x>0$ and $k(x)=0$ for $x\le0$. The operator $\bold T_n$ has a certain contraction property which can be expressed in the Fourier space. The Fourier transform of $\tilde{\bold T}_{n} \tilde\varphi_n( f_n(\xi,T))$ can be expressed as $$ \tilde{\bold T}_{n} \tilde\varphi_n( f_n(\xi,T))= \frac{\exp\left\{i\dfrac {c_{n+1}}{4M_n(T)}\xi \right\}}{\sqrt{1+i\dfrac {c_{n+1}}{2M_n(T)}\xi}} \tilde\varphi_n\(f_n^2\(\frac {c_{n+1}}{2}\xi,T\)\), \tag4.17 $$ where $\tilde f(\xi)=\int e^{i\xi x}f(x)\,dx$. These facts are explained in paper~[BM3]. Also the proof of Proposition~4.2 is a natural adaptation of the proof of the corresponding result (of Proposition~3) in paper~[BM3]. Hence we only explain the main points and the necessary modifications. First we remark that in the regularization of the functions $f_n(x,T)$ the same normalization could be applied as in paper~[BM3]. Because of the inductive property $I(n)$ $f_n(x,T)$ is essentially concentrated in a neighbourhood of the origin of size $\sqrt{\beta_n(T)}$, and if $(n,T)$ is in the low temperature domain and $\eta>0$ is chosen sufficiently small, then $\dfrac{|x|}{100\sqrt{c^{(n)}}}\le\dfrac{\eta}{10}$ for $|x|\le \sqrt{\beta_n(T)}$, and the function $f_n(x,T)$ (disregarding the scaling with the numbers $A_n$ and $B_n$) is not changing in the typical region by the regularization of the function $f_n(x,T)$. This is the reason why such a regularization works well. The proof of Proposition~4.2 contains several estimates. First we list those results whose proof apply the bound on $f_n(x,T)$ formulated in the Inductive hypothesis~$I(n)$. One can bound the differences $$ \dfrac{\partial^j}{\partial x^j} (\bar{\bold Q}_{n,M_n(T)}^{\bold c}f_n(x,T)- \bar{\bold Q}_{n,M_n(T)}^{\bold c}\varphi_n(f_n(x,T))) \quad \text{(Lemma~4 in [BM3]),} $$ $$ \dfrac{\partial^j}{\partial x^j} (\bar{\bold Q}_{n,M_n(T)}^{\bold c}\varphi_n(f_n(x,T))- \bar{\bold T}_{n,M_n(T)}^{\bold c}\varphi_n(f_n(x,T))) \quad \text{(Lemma~5 in [BM3]),} $$ with the help of Property~$I(n)$ similarly to paper~[BM3]. The absolute value of these expressions can be bounded for all $\e>0$ by $$ \bbeta{n} \frac{C_1(\e)C^2}{\beta_n^{(j+1)/2}(T)} \exp\left\{-\frac{2(1-\e)}{c_{n+1}\sqrt{\beta_{n}(T)}} \left|2x+\frac{x^2} {c^{(n+1)}M_{n}(T)}\right|\right\} $$ with some appropriate constant $C_1(\e)>0$ if $f_n(x,T)$ satisfies Condition $I(n)$. The main difference between these estimates and the analogous results in paper~[BM3] is that the upper bounds given for the above expressions contain a small multiplying factor $\dfrac{\beta_n(T)}{c^{(n)}}$. In paper~[BM3] the multiplying factors $2^{-n}$ and $1/c^{(n)}$ appear instead of this term. In the proof of this paper we had to make some modifications, because while in paper~[BM3] only very low temperatures were considered when $M_n(T)$ is strongly separated from zero, now we want to give an upper bound under the weaker condition formulated in the definition of the low temperature region. The proofs are very similar. The only essential difference is that in the present case the typical region, where a good asymptotic approximation must be given is chosen as the interval $|x|<10\sqrt{c^{(n)}}$, i.e.\ it does not depend on the value of $M_n(T)$. Also the expression $\bar{\bold Q}_{n,M_n(T)}^{\bold c}f_n(x,T)$ can be bounded together with their first two derivatives with the help of Property~$I(n)$ in the same way as in Lemma~3 of paper~[BM3]. But this estimate is useful only for large $x$. It can be proved, similarly to the proof of the corresponding result in paper~[BM3] (lemma~7) that the scaling constants which appear in the formulas expressing $\bold Q_{n,M_n(T)}^{\bold c}$ through $\bar{\bold Q}_{n,M_n(T)}^{\bold c}$ and $\bold T_n$ through $\bar{\bold T}_{n,M_n(T)}^{\bold c}$ are very close to each other. Here again the multiplying factor $\dfrac{\beta_n(T)}{c^{(n)}}$ appears in the error term instead of the multiplying factor $1/c^{(n)}$ in paper~[BM3]. This Lemma~7 in~[BM3] is a technical result which expresses the difference of the functions $\bar{\bold T}_{n,M_n(T)}^{\bold c}F_1(x)$ and $\bar{\bold T}_{n,M_n(T)}^{\bold c}F_2(x)$ together with its derivatives if we have a control on the difference of the original functions $F_1(x)$ and $F_2(x)$. %@ We gain such kind of information from the inductive hypothesis~$I(n)$. They give a good control on the difference $f_{n+1}(x,T)-\bold T_n\varphi_n(f_n(x,t))$. The consequences of these results are formulated in Proposition~2 in paper~[BM3]. These results also imply an estimate on the Fourier transforms $\tilde\varphi_{n+1}(f_{n+1}(\xi,T))-\tilde{\bold T}_n\tilde\varphi_n(f_n(\xi,T)))$ and $\tilde{\bold T}_n\tilde\varphi_n(f_n(\xi,T))$ and also on their analytic continuation. This is done in lemma~8 in paper~[BM3]. Now again the analogous result holds under the conditions of the present paper with the difference that the term $c^{-n}$ must be replaced $\dfrac{\beta_n(T)}{c^{(n)}}$. The estimate obtained for $\tilde{\bold T}_n\tilde\varphi_n(f_n(\xi,T))$ in such a way is %@ relatively weak, it is useful only for large $\xi$. The above results are not sufficient to prove Proposition~4.2. In particular, they do not explain why the right scaling was chosen in the definition of the function $f_n(x,T)$. Their role is to bound the error which is committed when $\bold Q_{n,M_n(T)}^{\bold c}f_n(x,T)$ is replaced by $\bold T_n\varphi(f_n(x,T))$. The function $\bold T_n\varphi_n(f_n(x,T))$ together with its derivatives and Fourier transform can be well bounded by means of formula (4.17) and the inverse Fourier transform. In the estimations leading to such bounds the inductive hypothesis $J(n)$ plays a crucial role. The proof of Lemma~9 in paper~[BM3] can be adapted to the present case without any essential difficulty. But, the parameters $\alpha_n$, $\beta_n$ and~$c$ must be replaced by $\alpha_n(T)$, $\beta_n(T)$ and $c_{n+1}$ in the present case. Proposition 4.2 can be proved similarly to its analog, Proposition~3 in paper~[BM3]. The notation must be adapted to the notation of the present paper. Beside this, the small coefficient $c^{-n/2}$ appearing in the proof of Proposition~3 in~[BM3] must be replaced by $\sqrt{\dfrac{\beta_n(T)}{c^{(n)}}}$. There is one point where a really new argument is needed in the proof. This argument requires a more detailed discussion. It is the proof of relation (4.14), i.e.\ of the fact that $\alpha_n(T)$ and $\beta_n(T)$ have the same order of magnitude. Their ratio must be bounded by a number independent of $\eta$. The proof of the analogous result in paper~[BM3] exploited the fact that in the model of that paper the sequence~$c^{(n)}$ tended to infinity exponentially fast. In the present case this property does not hold any longer, hence a different argument is needed. The validity of relation (4.14) has a different cause for relatively small and large indices~$n$. For large $n$ it can be shown that both $\beta_n(T)$ and $\alpha_n(T)$ have the same order of magnitude as $M_n^{-2}(T)$, and for large $n$ these relations imply~(4.14). If $n$ is relatively small and $M_0(T)$ is large, then $M_n^{-2}(T)$ is much less than $\alpha_n(T)$ and $\beta_n(T)$. In this case the above indicated argument does not work, but it can be proved that for such indices~$n$ the numbers $\beta_n(T)$ are decreasing exponentially fast, and the proof of relation~(4.14) for such~$n$ is based on this fact. To distinguish between small and large indices $n$ define the number $$ \aligned N_1(T)&=\left\{\min n\: n\ge N,\text{ and } \beta_{n+1}(T)\le \frac{100}{M_n^2(T)}\right\}, \\ &\qquad(N_1(T)=\infty \text{ if there is no such }n). \endaligned \tag4.18 $$ where the number $N$ was defined in formula~(1.22). We shall later see that $N_1(T)<\infty$ for all $00$ in~(1.4) is sufficiently small and $T\le c_0A_0/2$ imply that $NN_1(T)$ we can prove by induction with respect to $n$ together with the inductive proof of Proposition~4.2 that $$ \aligned \beta_{n+1}(T)\le\frac{100}{M_n^2(T)}\quad \text{if }n\ge N_1(T) \text{ and $(n,T)$ is in the low temperature region.} \endaligned \tag4.20 $$ By applying formula (4.20) for $n-1$ and the fact that $(n,T)$ is in the low temperature region we get that the term $\gamma_{n-1}(T)$ in formula (4.13) can be bounded as $$ |\gamma_{n-1}(T)|\le\frac{\beta_{n}(T)}{c^{(n)}}\sqrt{\beta_{n}(T)}\le\eta \frac{10}{M_{n-1}(T)}\le\frac1{8C_1M_{n-1}(T)} \tag4.21 $$ with the same number $C_1$ which appears in (4.13) if the number $\eta>0$ was chosen sufficiently small. Then formula (4.13) implies that $M_n(T)\le M_{n+1}(T)$. Hence we get by applying again formula~(4.20) with $n-1$ that $M_n(T)N_1(T)$ if $(n,T)$ is in the low temperature domain. With the help of this argument Proposition~4.2 can be proved by an adaptation of the proof of the corresponding result in~[BM3]. We formulate and prove a lemma which describes some properties of the numbers $\beta_n(T)$ in the cases when $n\le N_1(T)$ or $n\ge N_1(T)$. Several parts of it were already proved in the previous arguments. \proclaim{Lemma 4.3} {\it Let $00$ in formula~(1.4) is sufficiently small, then the number $N_1(T)$ defined in (4.18) is finite, and $N_1(T)>N$. The pair $(N_1(T),T)$ is in the low temperature domain. The relations (4.19), (4.20), $$ \aligned &M_{n}(T)-\frac3{8c^{(n)}M_{n}(T)}\le M_{n+1}(T)\le M_{n}(T)-\frac1{8c^{(n)}M_{n}(T)}\\ &\qquad \text{if }n\ge N_1(T) \text{ and $(n,T)$ is in the low temperature region,} \endaligned \tag4.22 $$ %@ Also the relations $$ \aligned M_{n}(T)&-\frac1{4c^{(n)}M_{n}(T)}-\eta\(\frac23\)^{(n-N)/2} \le M_{n+1}(T)\\ &\qquad \le M_{n}(T)-\frac1{4c^{(n)}M_{n}(T)}+\eta\(\frac23\)^{(n-N)/2} \quad \text{if } N\le n\le N_1(T), \endaligned \tag4.23 $$ and $$ N_1(T)-N\le 10 \log (1/\k T^2)\quad\text{with the parameter %@ $\k$ appearing in (1.17).} \tag4.24 $$ hold. If $M_n(T)<10$ then $n\ge N_1(T)$.} \endproclaim \demo{Proof} Formulas (4.19) and (4.20) were already proved in the previous argument, and since $(N,T)$ is in the low temperature region, i.e. $\beta_N(T)\ge\eta c^N$, relation (4.19) implies that $(n,T)$ is in the low temperature region for all $N\le n\le N_1(T)$. Formula~(4.22) follows from formula~(4.21) with the replacement of $n-1$ by $n$ and formula~(4.13). By relation (4.19) $\beta_n(T)\le\(\dfrac23\)^{n-N}$ if $N\le n\le N_1(T)$. Hence it follows from~(4.13) that $$ M_{n+1}(T)\le M_n(T)+\dfrac{\beta_{n+1}(T)} {c^{(n)}}\frac{\sqrt{\beta_{n+1}(T)}}{c^{(n)}} \le M_n(T)+\eta\(\frac23\)^{(n-N)/2}, \tag4.25 $$ and even relation (4.23) holds in this case. Relation~(4.25) and the estimate obtained for $\beta_n(T)$ imply that $M_{n}^2(T)\le (M_N(T)+1)^2 \le 2M_N^2(T)$ and $\beta_{n+1}(T)M^2_n(T)\le 2M_N^2(T)\(\dfrac{2}{3}\)^{n-N}$ if $n\le N_1(T)$. This relation together with the definition of the index $N_1(T)$ defined in (4.18) imply that $2M_N^2(T)\(\dfrac{2}{3}\)^{n-N}\ge 100$ if $n0$ which will be useful in the subsequent part of the paper. \proclaim{Lemma 4.4} {\it Let $00$ in formula~(1.17) be sufficiently small. If the Dyson condition (1.3) is not satisfied, then for all $T>0$ there is some $n=n(T)$ for which $(n,T)$ does not belong to the low temperature region. If, on the other hand, condition (1.3) is satisfied, then $T$ belongs to the low temperature region for sufficiently small~$T>0$. In this case relation (3.8) and under the additional Condition~2 also relation (3.9) (with $r=2$) hold.} \endproclaim \demo {Proof} It follows from formulas (4.22) and (4.23) that $$ -\frac1{c^{(n)}}\le M_{n+1}^2(T)-M_n^2(T)\le -\frac1{8c^{(n)}} \tag4.26 $$ if $n\ge N_1(T)$ and the pair $(n,T)$ is in the low temperature region, and $$ \aligned -\frac1{2 c^{(n)}}-10\(\frac23\)^{n-N}(M_N(T)+1)&\le M_{n+1}^2(T)-M_n^2(T)\\ &\le -\frac1{2 c^{(n)}} +10\(\frac23\)^{n-N}(M_N(T)+1) \endaligned\tag$4.26'$ $$ if $N\le n\le N_1(T)$. Formula (4.26) can be obtained by taking square in formula (4.22) and observing that $c^{(n)}M_n(T)^2>10\eta^{-1}$. Formula $(4.26')$ can be deduced similarly from (4.23) by observing first that the right-hand side of (4.23) implies that $M_n(T)\le M_N(T)+1$ for $N\le n\le N_1(T)$. Formulas (4.26) and $(4.26')$ imply that if a temperature $T>0$ is in the low temperature region, then $$ \sum_{k=N}^n\frac 1{c^{(n)}}\le 8(M_N^2(T)-M_n^2(T))+30(M_N(T)+1)\le 8M_N^2(T)+30(M_N(T)+1) $$ for all~$n\ge N$, where the number $N$ is defined in (1.22). Since the right-hand side of the last formula does not depend on $n$, this implies that (1.3) holds. In the other direction, if (1.3) holds, then since by Proposition~4.1 $\lim\limits_{T\to\infty}M_0(T)=\lim\limits_{T\to\infty}M_N(T)=\infty$, there is some number $\bar T\le c_0A_0/2$ such that for all temperatures $08\sum\limits_{n=N}^\infty \dfrac1{c^{(n)}}+30M_n(T)+31$. If $T>0$ satisfies the above inequality, then the left-hand side of the inequalities (4.26) and $(4.26')$ imply that if the pair $(n,T)$ is in the low temperature domain and $n\ge N_1(T)$, then $$ M_n^2(T)>M_N^2(T)-8\sum\limits_{n=N}^n\frac1{c^{(n)}}30(M_n(T)+1))\ge1. $$ Hence $M_n^2(T)>1$ for all $n$, and $T$ is in the low temperature region. Let $T>0$ be in the low temperature region. If $n>m>N_1(T)$, then by (4.26) $$ \left| M_n^2(T)-M_m^2(T)\right|\le \sum_{k=m}^n\frac1{c^{(k)}}. $$ Since in this case Condition~1 holds, the last relation implies that $M_n^2(T)$, $n=1,2,\dots$, is a Cauchy sequence, and relation (3.8) holds. We claim that if Condition~2 also holds, then for any $\e>0$ $$ -\frac{1+\e}{2 c^{(n)}}\le M_{n+1}^2(T)-M_n^2(T)\le -\frac{1-\e}{2c^{(n)}} \tag4.27 $$ if $n\ge n(\e)$. Relation $(3.9)$ is a consequence of (4.27). Relation (4.27) can be deduced from (4.13) and (4.20) if we show that for any temperature $T>0$ in the low temperature region $$ \lim\limits_{n\to\infty}\dfrac{\beta_n(T)}{c^{(n)}}=0. \tag4.28 $$ Relation (4.28) holds under Condition~2, since by (4.26) in this case for all $n>N_1(T)$ $$ M_n^2(T)\ge\lim_{k\to\infty}\(M_n^2(T)-M_k^2(T)\) \ge\frac18\sum_{k=n}^\infty\frac1{c^{(k)}}, $$ and $\dfrac{\beta_n(T)}{c^{(n)}}\le\dfrac{100}{M_{n-1}^2(T)c^{(n)}}\le800 \(c^{(n)}\sum\limits_{k=n-1}^\infty\dfrac1{c^{(k)}}\)^{-1}$. Under Condition~2 the last expression tends to zero as $n\to\infty$. This implies formula~(4.27). Lemma~4.4 is proved. \beginsection 5. Estimates in the Intermediate Region. Proof of Theorem 3.1 In this section we give some estimates on $q_n(x,T)$ when the pair $(n,T)$ belongs neither to the low nor to the high temperature region and prove Theorem~3.1 with their help. Let us consider the number $\bar n=\bar n(T)$ introduced in the formulation of Theorem~3.1. We shall prove some estimates about a scaled version of the function $q_{\bar n(T)}(x,T)$ in Lemmas~5.1 and~5.2. In Lemma~5.1 the case $T\le c_0A_0$, in Lemma~5.2 the case $T\ge c_0A_0$ will be considered. Lemmas 5.1 and 5.2 yield some estimates on the tail-behaviour of a scaled version of the function $q_{\bar n(T)}(\cdot,T)$. This will be needed to start an inductive %@ procedure for all $n\ge \bar n(T)$ which state that the functions $q_n(x,T)$ become more and more strongly concentrated around zero as the index~$n$ is increasing. This procedure is based on Lemmas~5.3 and~5.4. The role of Lemma~5.3 is to give an appropriate lower bound for the norming constant $Z_n(T)$ in the definition of the function $q_n(x,T)$. Then in Lemma~5.4 we prove some contraction property of the operator which maps an appropriate scaled version of the distribution function with density function $\const q_{n-1}(|x|,T)$ to an appropriate scaled version of the distribution function with density $\const q_n(|x|,T)$, $x\in\R^2$. The proof of Lemma~5.4 will exploit the rotation symmetry of the model. Theorem~3.1 will be proved by means of these lemmas. To formulate these results we introduce some notations. Let us introduce the functions $$ \hat h_n(x,T)= \dfrac1{\sqrt{c^{(\bar n(T))}}}q_n\(\dfrac x{\sqrt{c^{\bar n(T)}}},T\), \quad x\in \R^2 \tag5.1 $$ and measures $$ \hat H_{n,T}(\bold A)=\int_{\bold A}\hat h_n(x,T)\,dx,\quad \bold A\subset\R^2 \tag5.2 $$ in the space $\R^2$. Define also the function $$ \hat H_{n,T}(R)=\hat H_{n,T}(\{x\: |x|\ge R\}) \quad\text{for }R\ge0. \tag5.3 $$ The functions $\hat h_{n,T}$ and measures $\hat H_{n,T}$ are similar to the functions $h_{n,T}$ and measures $H_{n,T}$ defined in (3.5) and $(3.6)$. The only difference is that the scaling of $q_n(x,T)$ in (5.2) and (5.3) is made by means of $c^{(\bar n(T))}$ instead of $c^{(n)}$. If Condition~5 is satisfied with a sufficiently small $\bar\eta$ and sufficiently large $L(\bar\eta,t)$, and $n-\bar n(T)$ is not too large, then %@ the approximation of $c^{(n)}$ by $c^{(\bar n(T))}$ is sufficiently good for our purposes. Hence it will be enough to have a good control on the measure $\hat H_{n,T}$. In Lemma~5.3 we give a bound on it for large $|x|$ and in Lemma~5.4 we prove an estimate which enables to bound $\hat H_{n,T}(x)$ for small $x$ too. With the help of these results we can prove that starting from $\bar n=\bar n(T)$ after finitely many steps $k$ the pair $(\bar n+k,T)$ is in the high temperature region. Moreover, this number $k$ can be bounded from above independently of the temperature~$T$. First we formulate Lemma~5.1. \proclaim{Lemma 5.1} {\it Under the conditions of Proposition~4.2 the function $h_{\bar n(T)}(x,T)$ defined in~(3.5)satisfies the inequality $$ h_{\bar n(T)}(x,T)\le \exp\left\{\frac K{\eta}-\frac{|x|^2}{10}\right\}\quad \text{if } T\le c_0A_0/2 \tag5.4 $$ with an appropriate $K>0$. For $ T\le c_0A_0/2$ the pair $(\bar n(T),T)$ does not belong to the high temperature region, and there exists some $\tilde \eta=\tilde \eta(\eta)$ such that the function $\hat H_{n,T}(\cdot)$ defined in (5.3) satisfies the inequality $$ \hat H_{\bar n(T),T}\(\tilde\eta^{-1}\)\le 1/2\quad\text{if } T\le c_0A_0/2, \tag5.5 $$ i.e. for $T\le c_0A_0/2$ there is a circle with its center in the origin whose radius depends only on $\eta$, and whose $\hat H_{\bar n(T),T}$ measure is greater than $1/2$.} \endproclaim \demo{Proof} Let us introduce the function $$ \bar h_n(x,T)=\dfrac1{\sqrt{c^{(n)}}}\bar q_n\(\dfrac x{\sqrt{c^{(n)}}},T\),\quad x\ge0 %@. $$ %@ with the function $\bar q_n$ introduced in (2.17). This function is very similar to the intersection of the function $h_n(x,T)$ with the coordinate axis $y=0$. Only the norming of the two functions is different, since $\int_{\R^2} h_n(x,T)\,dx=1$, and $\int_0^\infty\bar h_n(x,T)\,dx=1$. We can apply %@ Proposition~4.2 with the choice $n=\bar n(T)-1$. Since hypothesis $I(n)$ holds for $n=\bar n$ $$ \align f_{\bar n(T)}(x,T)\le \frac K{\beta_{\bar n(T)-1}^{1/2}(T)}\exp&\left\{-\frac1{\sqrt{\beta_{\bar n(T)}(T)}} \left|2x+\frac{x^2} {c^{(\bar n(T))}M_{\bar n(T)}(T)}\right|\right\} \\ &\qquad\qquad \text{if } x>-c^{(\bar n(T))}M_{\bar n(T)}(T) \endalign $$ with some universal constant $K>0$. It follows from this relation that the function $\bar h_{\bar n(T)}(x,T)=\sqrt{c^{(\bar n(T))}}f_{\bar n(T)}\(\sqrt{c^{(\bar n(T))}}x- c^{(\bar n(T))}M_n(T)),T\)$ satisfies the inequality $$ \bar h_{\bar n(T)}(x,T)\le K\( \frac {c^{(\bar n(T))}}{\beta_{\bar n(T)-1}(T)}\)^{1/2} \!\! \exp\left\{\frac1{\sqrt{\beta_{\bar n(T)}(T)}} \(c^{(\bar n(T))}M_{\bar n(T)}(T)-\frac{x^2} {M_{\bar n(T)}(T)}\)\right\} $$ The inequalities $\beta_{\bar n(T)}(T)>\eta c^{(\bar n(T))}$ and $\beta_{\bar n(T)-1}(T)\le\eta c^{(\bar n(T)-1)}$ hold. %@ Lemma~4.3 implies that the fractions $\dfrac{\beta_{\bar n(T)}(T)}{\beta_{\bar n(T)-1}(T)}$, \ $\dfrac{M_{\bar n(T)}(T)}{M_{\bar n(T)-1}(T)}$ and %@ $\beta_{\bar n(T)}(T)M_{\bar n(T)}(T)^2$ are separated both from zero and infinity. Hence $\dfrac {c^{(\bar n(T))}}{\beta_{\bar n(T)-1}(T)}\le \dfrac{\const}\eta$, \ $\dfrac{c^{(\bar n(T))}M_{\bar n(T)}(T)} {\sqrt{\beta_{\bar n(T)}(T)}}\le \dfrac{\const}\eta$ and $\dfrac1 {M_{\bar n(T)}(T) \sqrt{\beta_{\bar n(T)}(T)}}\ge \dfrac1{20}$. These inequalities together with the last relation imply that $$ \bar h_{\bar n(T)}(x,T)\le e^{\bar K/\eta}e^{-x^2/20} \tag5.6 $$ with an appropriate $\bar K>0$. Since the relation $$ h_{\bar n(T)}(x,T)= C(T)\bar h_{\bar n(T)}(x,T) \tag5.7 $$ holds between the functions $h_{\bar n(T)}$ and $\bar h_{\bar n(T)}$ with an appropriate number $C(T)$, formula (5.4) can be deduced from (5.6) if we give a good upper bound for the constant $C(T)$ in~(5.7). Observe that $$ \int_{\R^2} \bar h_{\bar n(T)}(|x|,T)\,dx=2\pi \int_0^\infty x \bar h_{\bar n(T)}(x,T)\,dx\ge 2\pi R\(1-\int_0^R \bar h_{\bar n(T)}(x,T)\,dx\) $$ for any $R>0$, and by formula (5.6) $$ \int_0^R \bar h_{\bar n(T)}(x,T)\,dx\le \frac12 $$ if $00$ such that if $0<\k<\k_0$ and $T\ge c_0A_0/2$, then %@ $\bar n(T)=0$, and $$ \align h_{\bar n(T)}(x,T)&\le10 T \exp\left\{-10Tx^2+\frac{100}\k\right\} \quad \text{if } T\ge c_0A_0/2 \tag5.9 \\ h_{\bar n(T)}(x,T)&\le 10 T \quad \text{if } T\ge c_0A_0/2 \tag$5.9'$ \\ h_{\bar n(T)}(x,T)&\le100 e^{-Tx^2/4}\quad \text{if } T\ge 10A_0 \text{ and }|x|\ge T^{-1/3}. \tag$5.9''$ \endalign $$ The pair $(\bar n(T),T)$ belongs to the high temperature region if $T$ is very large, e.g.\ if $T\ge e^{-1/\eta^9}$, and it does not belong to it if $T>0$ is relatively small, e.g.\ if $T\le \eta^{-100}$. If $(\bar n(T),T)$ does not belong to the high temperature region, then the function $h_{\bar n(T)}(x,T)$ defined in formula~(3.5) satisfies the inequality $$ h_{\bar n(T)}(x,T)\le \exp\{K(\eta,\k)-\alpha|x|^2\} \tag5.10 $$ with a %@ constant $\alpha=\alpha(\eta)>0$ and an appropriate number $K(\eta,\k)$ depending only on $\k$ and $\eta$. In this case there is a constant $B=B(\eta,\k)>0$ in such a way that the quantity $\hat H_{\bar n(T),T}(\cdot)$ defined in (5.3) satisfies the inequality $$ 1-\hat H_{\bar n(T),T}(B)\le \frac12. \tag5.11 $$ This means that if the pair $(\bar n(T),T)$ is not in the high temperature region (and $T\ge c_0A_0/2$), then there is a radius $B=B(\eta,\k)$ such that the $\hat H_{\bar n(T),T}$ measure of the circle $\{x\: |x|\le B(\eta,\k)\}$ which is bigger than half. If $(\bar n(T),T)=(0,T)$ is in the high temperature region, then $$ \hat H_{\bar n(T),T}(x)\le K_1e^{-K_2\eta^2 x^2}\quad\text{for all } x>0 \tag5.12 $$ with some universal constants $K_1>0$ and $K_2>0$.} \endproclaim \demo{Proof} First we estimate the norming factor $Z_0(T)$ from below. Let us observe that $\(A_0-\dfrac T{ c_0}\)\dfrac{x^2}2 -\k T^2\dfrac{x^4}{4 c_0^2}\ge -10Tx^2$ if $\k Tx^2\le1/100$ and $ c_0A_0/2\le T$. Hence $$ Z_0(T)\ge 2\pi\int_0^{1/10\sqrt{\k T}} xe^{-10 Tx^2}\,dx =2\pi\int_0^{1/10\sqrt{\k}} \frac{xe^{-10x^2}}{ T}\,dx \ge\dfrac1{10 T}. \tag5.13 $$ If $T\ge c_0A_0/2$, then $\(A_0-\dfrac T{\bar c_0}\)\dfrac {x^2}2-\k T^2\dfrac{x^4}{4 c_0^2}\le \dfrac{100}\k-10Tx^2$. This relation together with (5.13) imply formula (5.9), and formula $(5.9')$ follows from (5.13). If $T\ge 10A_0$ then $\(A_0-\dfrac T{ c_0}\)\dfrac {x^2}2-\k T^2\dfrac{x^4}{4 c_0^2}\le-\dfrac T2x^2\le-\dfrac T4x^2-\dfrac{T^{1/3}}4$ for $|x|\ge T^{-1/3}$. This relation together with (5.13) imply relation~$(5.9'')$. Formula $(5.9'')$ implies that if $T>e^{-1/\eta^9}$, then the pair $(0,T)$ belongs to the high temperature region. To see that for $T<\eta^{-100}$ the pair $(0,T)$ does not belong to the high temperature domain it is enough to observe that in this case by formula $(5.9')$ the $H_{0,T}$ measure of the circle $\{x\:|x|\le \eta^{100}\}$ is less than $10\pi T\eta^{200}\le 1/2$. Hence in this case the variance $D_0^2(T)$ is larger than in the high temperature region. Inequality (5.9) together with the fact that if the pair $(0,T)$ does not belong to the high temperature region then $T\le e^{-1/\eta^{9}}$ imply relations (5.10) and (5.11). Since $T>\eta^{-100}$ if the pair $(0,T)$ is in the high temperature region, relation $(5.9'')$ implies relation (5.12). Lemma~(5.2) is proved. \enddemo To prove Lemmas 5.3 and 5.4 we rewrite formula (2.12) for the functions $\hat h_n(x,T)$ defined in (5.1). It has the form $$ \hat h_{n+1}(x,T)=\frac2{Z_n(T)}\int_{\R^2} \exp\left\{-\frac{c^{(n)}}{c^{(\bar n(T))}} u^2\right\} \hat h_n(x-u,T)\hat h_n(x+u,T)\,du \tag5.14 $$ with $$ {Z_n(T)}=2\int_{\R^2\times\R^2} \exp\left\{-\frac{c^{(n)}}{c^{(\bar n(T))}} u^2\right\} \hat h_n(x-u,T)\hat h_n(x+u,T)\,du\,dx \tag$5.14'$ $$ for all $n\ge \bar n(T)$. Let us also introduce the moment generating function of the measures $\hat H_{n,T}$ defined in (5.2) $$ \varphi_{n,T}(u)=\int_{\R^2} e^{ux}\hat h_{n,T}(x)\,dx, \quad u=(u_1,u_2)\in \R^2, $$ where $ux$ denotes scalar product. By studying the properties of the moment generating function $\varphi_{n,T}(u)$ in Lemma~5.3 we give an upper bound for the function $\hat H_{n,T}(R)$ for large values $R$. \proclaim{Lemma 5.3} {\it There exists $\k_0=\k_0(N)$ %@ with the number $N$ defined in (1.22) such that for all $0<\k<\k_0$ a number $L=L(\k,\bar\eta)$ can be chosen in such a way that if Conditions 1 and 5 are satisfied, then the following relations hold. For all temperatures $T>0$ for which the number $\bar n(T)$ exists, and the pair $(\bar n(T),T)$ does not belong to the high temperature region, the inequality $$ \hat H_{\bar n(T)+l,T}(x)\le e^{-2^l\alpha |x|^2/5} \quad\text{if }|x|\ge D\text{ and }0\le l\le L \tag5.15 $$ holds with appropriate constants $\alpha>0$ and $D>0$. Also the norming factor $Z_n(T)$ in $(5.14')$ can be estimated as $$ Z_{\bar n(T)+l}(T)\ge 2D_1\quad \text{for } 0\le l\le L \tag5.16 $$ with some constant $D_1>0$. Here %@ $\alpha=\alpha(\eta)$, and the numbers $D>0$ and $D_1>0$ do not depend on the temperature $T$.} \endproclaim \demo{Proof} It follows from formulas (5.4) and (5.10) that $$ \varphi_{\bar n(T),T}(u)\le \exp\left\{ K_0+\frac {u^2}\alpha\right\}\quad\text{for all }u\in\R^2 $$ with some $K_0=K_0(\eta,\k)>100$ and %@ $\alpha=\alpha(\eta)>0$. It can be seen by induction with respect to $l$ that $$ \varphi_{\bar n(T)+l,T}(u)\le \exp\left\{2^l K_l+\frac {u^2}{2^l\alpha}\right\}\quad\text{for all } 0\le l \le L \text{ and } u\in\R^2 \tag5.17 $$ with $$ K_l=K_{l-1}-\frac{\log \frac{Z_{\bar n(T)+l-1,T}}2}{2^l}\,. \tag$5.17'$ $$ %@ Indeed, the function $\hat h_{\bar n(T)+l+1,T}(x)$ is increased if the kernel term $\exp\left\{-\dfrac{c^{(n)}}{c^{(\bar n(T))}} u^2\right\}$ is omitted from the integral in (5.14), and the integral turns into the convolution $2\hat h_{\bar n(T)+l,T}*\hat h_{\bar n(T)+l,T}(2x)$ after this change. %@ By computing this convolution with the help of the inductive hypothesis and dividing it by $\dfrac{Z_{\bar n(T)+l+1}}2$ we get an upper bound for $\varphi_{\bar n(T)+l+1,T}(u)$. Formulas (5.17) and $(5.17')$ follow from these calculations. We shall prove formulas (5.15) and (5.16) from these relations by induction for $l$ together with the inductive hypothesis that $$ K_l\le B\quad\text{for all }0\le l\le L \tag5.18 $$ with some constants $B>10$ depending only on $\k$ and $\bar \eta$. By applying a standard technique for the estimation of probabilities by means of moment generating functions we get with the help of formula (5.17) that the function $\hat H_{\bar n(T)+l,T}(R)$ defined in formulas %@ (5.2) and (5.3) satisfies the inequality $$ \align \hat H_{\bar n(T)+l,T}(R)&\le 4 \hat H_{\bar n(T)+l,T} %@ \(\{x=(x_1,x_2)\in \R^2,\; x_1>\dfrac R{\sqrt2}\)\\ &\le4\exp\left\{-\frac{uR}{\sqrt 2}+2^lK_l+\frac{u^2}{2^l\alpha}\right\} \endalign $$ for all real numbers~$u$. In particular, %@ $$ \hat H_{\bar n(T)+l,T}(R) \le4\exp\left\{2^l\(K_l-\frac {R^2\alpha}{8}\)\right\} \tag5.19 $$ with the choice %@ $u=2^{l-3/2}R\alpha$. Hence %@ $$ \hat H_{\bar n(T)+l,T}\(4\sqrt{\frac{B}\alpha}\) \le4e ^{-2^l B} \le\frac12 \tag5.20 $$ with the number $B>0$ appearing in (5.18). Formula (5.20) implies that %@ $$ \hat H_{\bar n(T)+l,T}\(\left\{x\: x\in\R^2,\,|x|\le 4\sqrt{\frac{B}\alpha} \right\}\)\ge \frac 12. \tag$5.20'$ $$ For $z\in \R^2$ and $u>0$ let $K(z,u)=\{x\: x\in \R^2, |x-z|\le u\}$ denote the circle with center $z$ and radius~$u$. Since the circle %@ $\left\{x\: x\in\R^2,\,|x|\le 4\sqrt{\frac{B}\alpha} \right\}$ can be covered by $64B(\alpha \bar\eta)^{-1}$ circle of radius $\sqrt{\bar\eta}$ there is a circle $K\(z,\sqrt{\bar\eta}\)$ of radius $\sqrt{\bar\eta}$ whose $\hat H_{n,T}$ measure (this measure was defined in (5.2)) is greater than $\dfrac{\alpha\bar\eta}{128 B}$. Hence $$ \hat H_{\bar n(T)+l,T}\times\hat H_{\bar n(T)+l,T} \(K(z,\sqrt{\bar\eta})\times K(z,\sqrt{\bar\eta})\)\ge\frac{\alpha^2\bar\eta^2}{4096 B^2}, $$ and because of %@ Condition 5 the expression $2Z_n(T)$ defined in $(5.14)$ can be bounded %@ by means of the estimation %@ $$ \aligned \frac{Z_{\bar n(T)+l}(T)}2&\ge \int_{x+u\in K\(2z,2\sqrt{\bar\eta}\),\, u\in K\(z,\sqrt{\bar\eta}\)} \exp\left\{-\frac {c^{(\bar n(T)+l)}}{ c^{\bar n(T)}}\frac{(x-u)^2}4\right\} \\ &\qquad\qquad\qquad\qquad \hat h_{\bar n(T)+l,T}(x) \hat h_{\bar n(T)+l,T}(u)\,dx\,du \\ &\ge e^{-5\bar\eta}\hat H_{\bar n(T)+l,T}\times\hat H_{\bar n(T)+l,T} \(K\(z,\sqrt{\bar\eta}\)\times K\(z,\sqrt{\bar\eta}\)\)\ge e^{-1}\frac{\alpha^2\bar\eta^2}{4096 B^2} \\ &\qquad\ge\frac{\alpha^2\bar\eta^2}{15000 B^2}. \endaligned $$ The last relation implies (5.16) with $D_1=\dfrac{\alpha^2\bar\eta^2}{15000 B^2}$. We get from $(5.17')$ and the inductive hypothesis %@ (5.18) that $$ K_l\le (1-2^{-l}) B, $$ if the number $B$ is chosen as $B=\max(K_0, K^*)$, where $K^*$ is the larger solution of the equation $x=\log \dfrac{15000x^2} {\alpha^2\bar\eta^2}$. This implies validity of the inductive hypothesis (5.18) for $l$. Finally, relation (5.15) follows from (5.18) and (5.19). Lemma~5.3 is proved. \enddemo Formulas (5.14) and $(5.14')$ can be rewritten for the function $\hat H_{n,T}(R)$ defined in (5.3) as $$ \aligned \hat H_{n+1,T}(R)&=\frac2{Z_n(T)}\int_{|x|\ge R} \int_{u\in\R^2} \exp\left\{-\frac {c^{(n)}}{c^{(\bar n(T))}}u^2 \right\} \hat h_n(x-u,T)\hat h_n(x+u,T)\,du\,dx\\ &=\frac1{Z_n(T)} \int_{\left|\frac{x+u}2\right|\ge R}\int_{u\in\bold R^2} \exp\left\{-\frac { c^{(n)}}{c^{(\bar n(T))}} \frac{(x-u)^2}4\right\} \hat H_{n,T}(\,dx) \hat H_{n,T}(\,du) \endaligned \tag5.21 $$ with $$ Z_{n}(T)=\int_{x\in\R^2}\int_{u\in\R^2} \exp\left\{-\frac {c^{(n)}}{ c^{(\bar n(T))}}\frac{(x-u)^2}4\right\} \hat H_{n,T}(\,dx) \hat H_{n,T}(\,du). \tag$5.21'$ $$ for all $R\ge0$. We apply these formulas in the proof of the following Lemma~5.4. The proof of Lemma~5.4 also exploits the rotational invariance of the measure $\hat H_{n,T}$. \proclaim{Lemma 5.4} {\it Let the conditions of Lemma~5.3 hold. Then there exist some numbers $\delta=\delta\(\hat\eta,D_1\)>0$ and $M=M\(\hat\eta,D_1\)>0$ depending only on the numbers $D_1$ in formula (5.16) and $\bar\eta$ in %@ Condition~5 in such a way that $$ \aligned \hat H_{\bar n(T)+l+1,T}((1-\delta) R)&\le\frac12\hat H_{\bar n(T)+l+1,T}((1-\delta)R)+M\hat H_{\bar n(T)+l+1,T}(R)) \\ &\qquad\qquad \text{for all } R>0 \text { and } 0\le l\le L. \endaligned\tag5.22 $$ } \endproclaim \demo {Proof} Observe that $$ \align &\left\{\left|\frac{x+u}2\right|\ge (1-\delta)R\right\} \subset\{|x|\ge R\} \cup\{|u|\ge R\} \\ &\qquad \cup\{|x|\ge (1-\delta)R,\;\text{arg} %@ (x,u)\le \alpha\} %@ \cup\{|u|\le (1-\delta)R,\;\text{arg} (x,u)\ge \alpha\} \endalign $$ for all $R>0$ and $0<\delta<1$ with $\alpha=2\arccos(1-\delta)$. Indeed, if $\left|\dfrac{x+u}2\right|\ge (1-\delta)R$, then either $|x|>R$ or $|u|>R$ or both $|x|$ and $|u|$ is less than $R$, but in this case either $|x|>(1-\delta)R$ or $|u|>(1-\delta)R$, and the angle between the vectors $x$ and $u$ must be small. On the other hand, because of the rotational invariance of the measure $\hat H_{n,T}$ $$ \align &\hat H_{\bar n(T)+l,T}\times \hat H_{\bar n(T)+l,T} \(\{(x,y)\:|x|\ge(1-\delta)R, %@ \text{ arg}(x,u)\le\alpha\}\)\\ &\qquad\le \frac\alpha{\pi} \hat H_{\bar n(T)+l,T} (\{x\:|x|\ge(1-\delta)R\}) %@ =\frac\alpha{\pi}\hat H_{\bar n(T)+l,T}((1-\delta)R). \endalign $$ The last two relations together with (5.21) and the inequality %@ $\frac\alpha\pi\le\sqrt \delta$ imply that $$ \hat H_{\bar n(T)+l+1,T}((1-\delta)R)\le \frac1{Z_n(T)}\(2\sqrt \delta \hat H_{\bar n(T)+l,T}((1-\delta)R)+2\hat H_{\bar n(T)+l,T}(R)\). \tag5.23 $$ Relation (5.22) follows from (5.23) and (5.16) if we choose $\delta>0$ so small that the inequality $\dfrac{2\sqrt \delta} {D_1}\le \dfrac12$ holds. Lemma~5.4 is proved. \enddemo Put $P(j,l)=P(j,l,T)=\hat H_{\bar n(T)+l}((1-\delta)^j D)$, $j=0,1,\dots$, $0\le l\le L$ with the number $D$ appearing in (5.15) and $\delta$ in Lemma 5.4. Clearly, $P(j,l)\le 1$ for all $j$ and $l$. By Lemma~5.4 $$ P(j,l+1)\le\frac12 P(j,l)+MP(j-1,l),\quad j\ge1, \tag5.24 $$ and %@ by relation (5.15) $P(0,l)\le e^{-\alpha 2^l D^2/5}$ if $l\le L$. Hence there is a constant $k_0>0$ in such a way that $P(0,k_0+l) \le \(\dfrac23\)^{l}$ if $k_0+l\le L$. %@ Because of this relation, the inequality $P(j,l)\le1$ and formula (5.24) there is a constant $k_1\ge k_0$ in such a way that $P(1,k_1+l) \le \dfrac1{3M}\(\dfrac23\)^{l}$ and $P(1, k_1+l)\le \(\dfrac23\)^{l}$ if $k_1+l \le L$. %@ Similarly, there is a constant $k_2$ such that $P(2,k_2+l) \le \dfrac1{3M}\(\dfrac23\)^{l}$, and $P(2, k_2+l)\le \(\dfrac23\)^{l}$ if $k_2+l \le L$. This procedure can be continued, and we get a sequence $k_0\le k_1\le k_2\le\cdots$ in such a way that the inequality $P(p,k_p+l)\le \(\dfrac23\)^{l}$ holds if $k_p+l \le L$. The numbers $k_p$ depend only on the parameter $\kappa$ in (1.4) and the number $\bar\eta$ in %@ Condition~5. The above procedure can be continued till $k_p\le L$. %@ In such a way we have proved that for all fixed $j\ge0$ $$ \hat H_{\bar n(T)+l}\((1-\delta)^pD\)\le C(l)\(\frac23\)^l, $$ if $0\le l \le L$. The above relation together with formula (5.15) imply that if %@ Condition~5 holds with a sufficiently large constant $L=L(\bar\eta,t)$, then an integer $k>0$ can be chosen independently of the parameter $T$ in such a way that %@ $$ \hat H_{\bar n(T)+l,T}(R)\le 2\exp\left\{-\frac{e^{1/\eta^3}R^2}{\bar\eta}\right\} \quad\text{for all }R>0\text{ and }k\le l\le L(\bar\eta,t). \tag5.25 $$ Since the measure $H_{n,T}$ defined in (3.6) satisfies the relation $$ H_{\bar n(T)+l,T}\{x\: |x|>R\}= \hat H_{\bar n(T)+l,T}\(\sqrt{\frac{c^{(\bar n(T))}} {c^{(\bar n(T)+l)}}}R\) \le\hat H_{\bar n(T)+l,T}\(\sqrt{\bar\eta} R\) $$ relation (5.25) implies that %@ $$ H_{\bar n(T)+l,T}(R)\le 2\exp\left\{-e^{1/\eta^3}R^2\right\}\quad\text{for all }R>0,\text{ and }l^*\le l \le L \tag5.26 $$ with some appropriate $l^*\ge0$. Relation (5.26) implies in particular that $D^2_{\bar n(T)+l}(T)R_1>0$ let $\bold K(R_1,R_2)=\{x\: x\in \R^2,\, R_1\le |x|\le R_2\}$ denote the annulus between the concentrical circles with center in the origin and radii $R_1$ and $R_2$. We claim that for any $0\le l\le L$ there exist some positive numbers $R_1(l)=R_1(l,\bar \eta,t)$, $R_2(l)=R_2(l,\bar \eta,t)$ and %@ $A(l)=A(l,\bar \eta,t)>0$ such that the measure of the annulus determined by these numbers satisfies the inequality $$ \hat H_{\bar n(T)+l,T}(\bold K(R_1(l),R_2(l))\ge A(l),\quad 0\le l\le L \tag5.27 $$ if the pair $(0,T)$ does not belong to the high temperature region. Relation (5.27) implies the required lower estimate for $D^2_{\bar n(T)+k}(T)$ needed in %@ Theorem~3.1 if $k=k(T)$ is chosen as the smallest index $l$ for which $D^2_{\bar n(T)+l}(T)0$ are multiplied with an appropriate number. This implies that the variance $D^2_{\bar n(T)+k,T}$ can be bounded from below by a positive number which depends only on $k$ and~$\bar\eta$. We shall prove a slightly stronger statement than relation (5.27) which will be useful in later applications. We shall prove that $$ \hat H_{\bar n(T)+l,T}\(\bold K\(\frac1{2^l}R_1,\(\frac{\sqrt 3}2\)^lR_2\)\)\ge A(l),\quad 0\le l\le L. \tag$5.27'$ $$ with some numbers $R_2>R_1>0$ and $A(l)>0$ if the pair $(0,T)$ does not belong to the high temperature region. The numbers $R_j$ can be chosen in such a way that $R_j=R_j(\eta,\k)$, $j=1,2$. Let us first observe that relation $(5.27')$ holds for $l=0$ if $\bar n(T)$ is not in the high temperature region. This follows from relations (5.4) and (5.5) in the case $T\le c_0 A_0/2$ and from $(5.9')$ and (5.11) if $T\ge c_0 A_0/2$, but $(\bar n(T),T)$ does not belong to the high temperature region. Indeed, formulas (5.5) and (5.11) make possible to choose the number $R_2$ in such a way that the $H_{\bar n(T),T}$ measure of the circle with center in the origin and radius $R_2=R_2(\eta)$ is greater than $1/2$. By formulas (5.4) and $(5.9')$ we can choose the number $R_1=R_1(\eta)$ in such a way that by cutting out from this circle the circle with radius $R_2$ and center in the origin the remaining annulus $\bold K(R_1,R_2)$ has a measure greater than $1/4$. We claim that $$ \hat H_{\bar n(T)+l+1,T}\(\bold K\(\frac{\bar R_1}2,\frac{\sqrt 3}2 \bar R_2\)\)\ge B(\bar R_1,\bar R_2,\bar\eta)\hat H_{\bar n(T)+l,T}(\bold K(\bar R_1,\bar R_2))^2 \tag5.28 $$ for all $0\le l\le L$ and $\bar R_2>\bar R_1>0$ and an appropriate constant $B(\bar R_1,\bar R_2,\eta)>0$. Relation $(5.27')$ follows from (5.28) and the previous argument. In the proof of relation (5.28) we exploit the relation $$ \align &\left\{(u,x)\: u\in \R^2, x\in \R^2, \frac{\bar R_1}2\le \left|\frac{x+u}2\right|\le\frac{\sqrt 3}2\bar R_2,\; \frac \pi3\le \text{arg}\,(x,u)\le \frac \pi2\right\}\\ &\qquad\supset \left\{(u,x)\: u\in \R^2, x\in \R^2,\;\bar R_1\le |x|,|u|\le \bar R_2,\; \frac \pi3\le \text{arg}\,(x,u)\le \frac \pi2\right\}. \endalign $$ It follows from relation $(5.14')$ that $Z_{\bar n(T)+l+1}(T)\le1$, since we get an upper bound for it by omitting the kernel term $\exp\left\{-\dfrac{c^{(n)}}{c^{(\bar n(T))}} u^2\right\}$ from the integral in $(5.14')$. Hence the previous relation together with (5.21) and the rotational invariance of the measure $\hat H_{\bar n(T)+l,T}$ yield that $$ \allowdisplaybreaks \align &\hat H_{\bar n(T)+l+1,T}\(\bold K\(\frac{\bar R_1}2,\frac{\sqrt 3}2 \bar R_2\)\) =\frac1{Z_{\bar n(T)+l+1}(T)} \int\int _{\frac{\sqrt 3}2\bar R_2\ge\left|\frac{x+u}2\right|\ge \frac{\bar R_1}2,\;x,u\in\R^2} \\ &\qquad\qquad\qquad\qquad\exp\left\{-\frac {c^{(\bar n(T)+l)}}{\bar c^{(\bar n(T))}}\frac{(x-u)^2}4\right\} \hat H_{\bar n(T)+l,T}(\,dx)\hat H_{\bar n(T)+l,T}(\,du)\\ &\qquad\ge e^{-\bar R_2^2/\bar \eta} \int\int_{\frac{\sqrt 3}2\bar R_2\ge\left|\frac{x+u}2\right|\ge\frac{\bar R_1}2,\;x,u\in\bold R^2,\frac{\pi}3\le\arg(x,u)\le\frac \pi 2} \hat H_{\bar n(T)+l,T}(\,dx) \hat H_{\bar n(T)+l,T}(\,du)\\ &\qquad\ge e^{-\bar R_2^2/\bar \eta} \int\int_{\bar R_2\ge|x|,|u|\ge \bar R_1,\frac{\pi}3\le\arg(x,u)\le\frac\pi2} \hat H_{\bar n(T)+l,T}(\,dx) \hat H_{\bar n(T)+l,T}(\,du)\\ &\qquad=\frac1{12} e^{-\bar R_2^2/\bar \eta} \hat H_{\bar n(T)+l,T}(\bold K(\bar R_1,\bar R_2))^2. \endalign $$ The last estimate implies relation (5.28) with $B(\bar R_1,\bar R_2,\bar\eta)=\dfrac1{12} e^{-\bar R_2^2/\bar \eta}$. Theorem 3.1 is proved. \beginsection 6. Estimates in the High Temperature Region. Proof of Theorem 3.3 To study the behaviour of the function $f_n(x,T)$ in the high temperature region we need a starting index $n=\nn$ for which a good estimate is known about the tail behaviour of the measure $H_{\nn,T}$. We also need a lower bound for the variance $D_n^2(T)$ defined in (3.5) for $n\ge\nn$. This requirement will be also taken into consideration in the definition of $\nn$. Let us first define the number $$ l_0=l_0(T)=\min \left\{l\: \(\frac{\sqrt 3}2\)^l R_2\le \frac{\bar\eta}{10}\right\} \tag6.1 $$ if the pair $(0,T)$ is not in the high temperature region, where $\bar\eta$ appeared in condition~3, and the number $R_2$ was introduced in formula $(5.27')$. Now define $$ \nn=\left\{ \aligned & 0\quad\text{if $(0,T)$ is in the high temperature region}\\ & \bar n(T)+l \text{ with the smallest $l$ satisfying both (5.26) and the }\\ &\qquad\qquad\,\text{inequality $l\ge l_0$ with $l_0$ defined in (6.1)}\\ &\qquad\qquad\,\text{if $(0,T)$ is not in the high temperature region.} \endaligned \right. \tag6.2 $$ It follows from the results of the previous section that for a temperature~$T$ which is not in the low temperature region the inequality $0\le\nn-\bar n(T)\le L(\bar\eta,t)$ holds if the number $L$ in Condition~5 is chosen sufficiently large. The measure $H_{\nn,T}$ introduced in formula $(3.6)$ is strongly concentrated around the origin. Indeed, formulas $(5.12)$ and (5.26) give a good estimate for the $H_{\nn,T}$ measure of the sets $\{x\: |x|\le R\}$ for all $R\ge 0$. Let us introduce the moments of the functions $h_{\nn+l}(x,T)$ defined in (3.5). $$ M_k(l,T)=\int_{\R^2}|x|^k h_{\nn+l}(x,T)\,dx\quad l\ge0,\;k\ge1. $$ We shall estimate the moments $M_2(l,T)$ and $M_4(l,T)$. It follows from relations (5.12) and (5.26) that $$ M_2(0,T)\le\eta^*\quad\text{and}\quad M_4(0,T)\le\eta^*\quad\text{with } \eta^*=e^{-1/\eta^2} \tag6.3 $$ for all $T>0$ which is not in the low temperature region. To get lower bounds for the second moments $M_2(l,T)$ let us introduce the truncated second moments $$ M_{2,\text{tr.}}(l,T)=M_{2,\text{tr.}}\(\frac1{10},l,T\) =\int_{|x|\le \frac1{10}}|x|^2 h_{\nn+l}(x,T)\,dx.\tag6.4 $$ It follows from $(5.9'')$ if $(0,T)$ is in the high temperature region and from $(5.27')$ and the definition of $\nn$ if $(0,T)$ is not in the high temperature region that $$ \aligned &M_{2,\text{tr.}}(0,T)>0, \quad\text{for all }T\ge c_0A_0/2 \\ &M_{2,\text{tr.}}(0,T)>\tilde\eta, \quad \text{if $T\ge c_0A_0/2$ and $(0,T)$ is not in the high temperature region} \endaligned \tag6.5 $$ with some $\tilde\eta=\tilde\eta(\eta,\k)>0$. First we shall bound $M_2(l,T)$ and $M_4(l,T)$ from above in Lemma~(6.1) for all $l\ge0$. Then the second moment $M_2(l,T)$ will be bounded from below in Lemma~(6.2). These estimates enable us to prove the central limit theorem for $g_{\nn+l}(x,T)$ by means of the characteristic function technique. Simple calculation yields that $$ \aligned M_k(l+1,T)=\frac2{Z_l(T)}\int e^{-u^2} |x|^k &h_{\nn+l}\(\frac x{\sqrt{c_{\nn+l+1}}}-u,T\) \\ &\qquad h_{\nn+l}\(\frac x{\sqrt{c_{\nn+l+1}}}+u,T\)\,dx\,du \endaligned \tag6.6 $$ for all $l\ge0$ and $k\ge1$ with $$ Z_l(T)=2\int e^{-u^2} h_{\nn+l}\(\frac x{\sqrt{c_{\nn+l+1}}}-u,T\) h_{\nn+l}\(\frac x{\sqrt{c_{\nn+l+1}}}+u,T\)\,dx\,du. \tag$6.6'$ $$ These formulas will be used in the proof of the following \proclaim {Lemma 6.1} {\it Under the conditions of Theorem~3 the inequalities $$ \align M_2(l,T)&\le\eta^*\(\frac23\)^l, \tag6.7 \\ Z_l(T)&\ge c_{\nn+l}\(1-6\sqrt{\eta^*}\(\frac56\)^l\), \tag$6.7'$ \\ M_2(l+1,T)&\le \frac{c_{\nn+l+1}}2 \(1+10\sqrt{\eta^*}\(\frac5{6}\)^l\)M_2(l,T), \tag$6.7''$ \endalign $$ $$ M_2(l,T)\le 2\cdot 2^{-l}\frac{c^{(\nn+l)}}{c^{(\nn)}}\eta^* \quad \text{and}\quad M_4(l,T)\le 5\cdot 4^{-l}\(\frac{c^{(\nn+l)}}{c^{(\nn)}}\)^2\eta^* \tag6.8 $$ hold for all $l\ge0$ with the same number $\eta^*$ which appears in (6.3). }\endproclaim \demo{Proof} Relation (6.7) holds for $l=0$ by relation (6.3). We shall prove that if relation (6.7) holds for an integer $l$, then relations $(6.7')$ and $(6.7'')$ also hold for this~$l$. Then we prove that if relations (6.7) and $(6.7'')$ hold for some $l$, then relation (6.7) holds also for $l+1$. These statements imply relations (6.7)~---~$(6.7'')$. We prove them with the help of the following calculations. It follows from formulas (6.6) and $(6.6')$ that $$ \aligned M_k(l+1,T)&=\frac{c_{\nn+l+1}}{Z_l(T)}\int \exp\left\{-\frac{(x-u)^2}4\right\} \left|\frac{x+u}2\right|^k c_{\nn+l+1}^{k/2} \\ &\qquad\qquad\qquad\qquad\qquad\qquad h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du \\ &\le \frac {c_{\nn+l+1}^{k/2+1}}{2^k Z_l(T)}\int |x+u|^k h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du \endaligned \tag6.9 $$ for all $l\ge0$ and $k\ge1$, and $$ \aligned Z_l(T)&=c_{\nn+l+1}\int \exp\left\{-\frac{(x-u)^2}2\right\} h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du\\ &\ge c_{\nn+l+1}e^{-4\sqrt{M_2(l,t)}} \int_{|x|\le M_2(l,t)^{1/4},\,|u|\le M_2(l,t)^{1/4}}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad h_{\nn+l}(x,T) h_{\nn+l}\(u,T\)\,dx\,du \\ &\ge c_{\nn+l+1}e^{-4\sqrt{M_2(l,t)}} \biggl(1-\frac1{\sqrt{M_2(l,t)}}\int_{|x|\le M_2(l,t)^{1/4},\,|u|\le M_2(l,t)^{1/4}} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad x^2 h_{\nn+l}(x,T) h_{\nn+l}\(u,T\)\,dx\,du \biggl) \\ &\ge c_{\nn+l+1}e^{-4\sqrt{M_2(l,t)}}\(1-\sqrt{M_2(l,t)}\) \endaligned $$ The last relation and formula (6.7) for $l$ together imply that $$ \align Z_l(T)&\ge c_{\nn+l+1}\(1-5\sqrt{M_2(l,t)}\) \(1-\sqrt{M_2(l,t)}\)\\ &\ge c_{\nn+l+1}\(1-6\sqrt{M_2(l,t)}\) \ge c_{\nn+l}\(1-6\sqrt{\eta^*}\(\frac56\)^l\), \endalign $$ and this is relation $(6.7')$ for the number $l$. Relation (6.9) for $k=2$ and formula $(6.7')$ for $l$ together yield that $$ M_2(l+1,T)\le\frac{c_{\nn+l+1}^2}{2Z_l(T)}M_2(l,T)\le \frac{c_{\nn+l+1}}2\(1+10\sqrt{\eta^*}\(\frac5{6}\)^l\) M_2(l,T), $$ and this is formula $(6.7'')$ for $l$. Finally, if $\eta$ is chosen sufficiently small, then formulas (6.7) and $(6.7'')$ for $l$ imply (6.7) for $l+1$. Thus formulas (6.7)~---~$(6.7'')$ are proved. The first relation in $(6.8)$ follows from the first relation in (6.3) and $(6.7'')$. Formula (6.9) with the choice $k=4$, $(6.6')$ and the first formula in (6.8) imply that $$ \align M_4(l+1,T)&\le \frac {c_{\nn+l+1}^{3}}{8Z_l(T)} \(3M_2(l,T)^2+M_4(l,T)\)\\ &\le\frac18 c_{\nn+l+1}^2\(1+10\sqrt{\eta^*}\(\frac5{6}\)^l\) \(3M_2(l,T)^2+M_4(l,T)\)\\ &\le 4^{-l}\(\frac{c^{(\nn+l)}}{c^{(\nn)}}\)^2{\eta^*}^2+\frac{M_4(l,T)}8. \endalign $$ The second relation in (6.8) follows by induction from the last inequality and the second inequality in (6.3). Lemma~6.1 is proved. \enddemo \demo{Remark} The Corollary formulated after Theorem~3.1 follows from Theorem~3.1, formula (6.8) and Lemma 4.4. Indeed, if $T$ is not in the low temperature, then by Theorem~3.1 the pair $(\nn,T)$ with the definition of $\nn$ given in (6.1) is in the high temperature domain. By formula (6.8) all pairs $(n,T)$, $n\ge\nn$, are in the high temperature region, i.e.\ if $T>0$ is not in the low temperature region, then it is in the high temperature region. The remaining statements of the Corollary are contained in Lemma~4.4. \enddemo In the next lemma we prove an estimate from below for $M_2(l,T)$. \proclaim{Lemma 6.2} {\it Put $$ \sigma^2(l,T)=2^l\frac{c^{(\nn)}}{c^{(\nn+l)}} M_2(l,T), \quad l\ge0. \tag6.10 $$ Under the conditions of Theorem~3.3 the limit $$ \bar\sigma^2(T)=\lim_{l\to\infty} \sigma^2(l,T)>0 \tag6.11 $$ exists, and it is positive for all $T>0$. If $\nn\neq 0$, i.e.\ if $(0,T)$ is not in the high temperature region, then there exist two constants $C_2>C_1>0$ depending only on the parameter $\tilde\eta$ in formula $(6.3')$ in such a way that the inequalities $$ C_1\le \sigma^2(T)\le C_2 \tag$6.11'$ $$ hold. The upper bound in $(6.11')$ holds for all $T>0$ which is not in the low temperature region.} \endproclaim \demo{Proof} The hard part of the proof is to show that $\sigma^2(l,T)$ has a non-negative $\liminf$. It follows simply from formula $(6.6')$ that $Z_l(T)\le c_{\nn+l+1}$. A natural lower bound for $M_2(l,T)$ can be obtained in the following way. By formula (6.6) and the upper bound for $Z_l(T)$ $$ \allowdisplaybreaks \align M_2(l+1,T)&\ge c_{\nn+l+1}\int e^{-{(x-u)^2}/4} \left|\frac{x+u}2\right|^2 h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du\\ &\ge \frac {c_{\nn+l+1}}4\biggl(2M_2(l,T)-\int |x+u|^2\(1-e^{-(x+y)^2/4}\)\\ &\qquad\qquad\qquad\qquad\qquad h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du\biggr) \\ &\ge \frac {c_{\nn+l+1}}2\biggl(M_2(l,T)-\int\frac18 |x+u|^2|x-u|^2\\ &\qquad\qquad\qquad\qquad\qquad h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du\biggr)\\ &\ge \frac {c_{\nn+l+1}}2\biggl(M_2(l,T)-\int\frac12(|x|^4+|u|^4)\\ &\qquad\qquad\qquad\qquad\qquad h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du\biggr)\\ &=\frac {c_{\nn+l+1}}2\(M_2(l,T)-{M_4(l,T)}\). \tag6.12 \endalign $$ However, this estimate is useful only if we know that the right-hand side in it is non-negative. We do not know such an estimate for small $l$, hence in this case we apply a different argument. Clearly $$ M_2(l,T)\ge M_{2,\text{tr.}}(l,T), $$ where $M_{2,\text{tr.}}(l,T)$ is the truncated moment. On the other hand, we get by using an argument similar to the previous calculation and making the observation $$ \align &\left\{(x,u)\: x\in \R^2, u\in \R^2, c_{\nn+l+1} \left| \frac{x+u}2\right|\le \frac1{10} \right\}\\ &\qquad \supset\left\{(x,u)\: x\in \R^2, u\in \bold R^2, |x|\le\frac1{10}, |u|\le\frac1{10},\text{arg}\,(x,u)\subset I \right\} \endalign $$ with $I=\(\dfrac\pi{50},\dfrac{49\pi}{50}\)\cup \(\dfrac{51\pi}{50},\dfrac{99\pi}{50}\)$ that $$ \allowdisplaybreaks \align M_{2,\text{tr.}}(l+1,T) &\ge c_{\nn+l+1} \int_{c_{\nn+l+1}\left|\frac{x+u}2\right|\le \frac{1}{10}} e^{-{(x-u)^2}/4} \left|\frac{x+u}2\right|^2\\ &\qquad\qquad\qquad h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du\\ &\ge c_{\nn+l+1}e^{-1/100} \int_{ |x|\le\frac1{10}, |u|\le\frac1{10},\text{arg}\,(x,u)\subset I} \left|\frac{x+u}2\right|^2\\ &\qquad\qquad\qquad h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du\\ &=\frac{c_{\nn+l+1}}4e^{-1/100}\int_{ |x|\le\frac1{10}, |u|\le\frac1{10},\text{arg}\,(x,u)\subset I} (x^2+u^2)\\ &\qquad\qquad\qquad h_{\nn+l}(x,T)h_{\nn+l}\(u,T\)\,dx\,du\\ &=c_{\nn+l+1}e^{-1/100}\frac{12}{25}M_{2,\text{tr.}}(l,T)\ge\frac13 c_{\nn+l+1} M_{2,\text{tr.}}(l,T). \endalign $$ The last estimate implies that $$ \sigma^2(l,T)=2^l\frac {c^{(\nn)}}{c^{(\nn+l)}}M_2(l,T)\ge 2^l\frac {c^{(\nn)}}{c^{(\nn+l)}} M_{2,\text{tr.}}(l,T)\ge\( \frac23\)^l M_{2,\text{tr.}}(0,T). \tag6.13 $$ On the other hand, it follows from (6.12) and the second inequality in (6.8) that $$ \aligned \sigma^2(l+1,T)&\ge\sigma_2(l,T)-2^{l} \frac {c^{(\nn)}}{c^{\nn+l+1}}M_4(l,T) \\ &\ge\sigma^2(l,T)-\frac{5\eta^*}{2^lc_{\nn+l+1}} \frac{c^{(\nn+l)}} {c^{(\nn)}}\ge\sigma^2(l,T)-5\eta^*\(\frac34\)^l. \endaligned \tag6.14 $$ Because of (6.13) and (6.5) an index $\bar l\ge0$ can be chosen in such a way that $$ \sigma^2(\bar l,T)\ge 100\eta^*\(\frac34\)^{\bar l}. $$ Moreover, $\bar l\le K(\bar \eta,\k)$ with some appropriate $K(\bar \eta,\k)$, if the pair $(n=0,T)$ is not in the high temperature region. Hence, relation (6.14) implies that $$ \frac{\sigma^2(\bar l+l+1,T)}{\sigma^2(\bar l,T)} \ge \frac{\sigma^2(\bar l+l,T)}{\sigma^2(\bar l,T)} -\frac 1{20}\(\frac34\)^l . $$ This relation together with the bound on $\sigma^2(\bar l,T)$ imply that $\liminf\limits_{l\to\infty}\sigma^2(l,T)>0$, and this $\liminf$ can be bounded by a positive number depending only on $\bar\eta$ and $\k$ if $(0,T)$ is not in the high temperature region. The analogous result for $\limsup$ follows from $(6.7'')$. To complete the proof it is enough to show that the $\liminf$ is actually $\lim$. To prove this let us observe that for any $\e>0$ and $N>0$ there is some $m>N$ such that $\sigma^2(m,T)<\liminf\limits_{n\to\infty} \sigma^2(n,T)+\e$. Then by formula $(6.7'')$ $$ \sigma^2(n,T)\le\sigma^2(m,T)\prod_{l=m}^n \(1+10\sqrt{\eta^*}\(\frac5{6}\)^l\)\le\liminf_{n\to\infty} \sigma^2(n,T)+2\e,\quad n>m $$ for any $\e>0$ if $N=N(\e)$ is chosen sufficiently large. Lemma 6.2 is proven. \enddemo To prove Theorem 3.3 let us introduce the characteristic functions $$ \varphi_n(s,T)=\int_{\R^2} e^{i sx}\tilde h_n(x,T)\,dx,\quad s\in \R^2 \tag6.15 $$ and moments $$ \tilde M_k(n,T)=\int_{\R^2} |x|^k\tilde h_n(x,T)\,dx, $$ where the function $\tilde h(x,T)$ was defined in (3.24). Clearly, $$ \tilde M_k(n,T)=\(\frac{2^n}{c^{(n)}}\)^{k/2}M_k(n-\nn,T)\quad \text{if }n\ge \nn. $$ In particular, $\tilde M_2(n,T)=\dfrac {2^{\nn}}{c^{(\nn)}}\sigma^2(n-\nn,T)$. We shall prove Theorem~3.3 by means of the usual characteristic function technique. The following lemma plays a crucial role in the proof. \proclaim{Lemma 6.3} {\it Under the conditions of Theorem 3.3 the relation $$ \lim_{n\to\infty}\frac{c^{(\nn)}}{2^{\nn}}\tilde M_2(n,T)=\bar \sigma^2(T) \tag6.16 $$ holds with the constant $\bar\sigma^2(T)$ appearing in Lemma 6.2, and $$ \lim_{n\to\infty}\sup_{|s|\le A}\left|\log\varphi_n\(s,T\)+ \frac{2^{\nn}}{c^{(\nn)}}\bar\sigma^2(T) \frac{s^2}2\right|\to0 \tag6.17 $$ for all $A>0$.} \endproclaim \demo{Proof} Relation (6.16) follows from Lemma~6.2, and it follows from the second relation in (6.8) that $\tilde M_4(n,T)\le \(\dfrac{2^{\nn}}{c^{(\nn)}}\)^2\eta^*$. Hence the characteristic function $\varphi$ can be estimated as $$ \left|\varphi_n(s,T)-\(1- \tilde M_2(n,T)\frac{s^2}2\)\right|\le \(\frac{2^{\nn}}{c^{(\nn)}}\)^2 \eta^*|s|^4 \quad\text{for }n\ge\nn\text{ and }s\in \R^2. \tag6.18 $$ The coefficient of $|s|^4$ is bounded by a constant (depending on $T$), and the coefficient at $|s|^2$ converges to the positive constant $\dfrac{2^{\nn}}{c^{(\nn)}}\bar\sigma^2(T)$. Hence formula (6.18) implies that for any $\e>0$, $$ \left|\log\varphi_n\(s,T\)+\frac{2^{\nn}}{c^{(\nn)}}\bar\sigma^2(T) \frac{s^2}2\right|\le\e\quad\text{if } n>n_1 \text{ and }|s|\le\delta \tag6.19 $$ with some $n_1=n_1(\e,T)$ and $\delta=\delta(\e,T)$. By a rescaled version of the recursive formula (2.12) we can write $$ \align \varphi_{n+1}(\sqrt 2s,T)&=\frac1{Z_n(T)}\int \exp\left\{is(x+u) -\frac{c^{(n)}(x-u)^2}{4\cdot2^n}\right\}\tilde h_n(x,T)\tilde h_n(u,T)\,dx\,du\\ &=\frac1{Z_n(T)}\biggl[\varphi_{n}\(s,T\)^2-\int e^{is(x+u)}\(1-\exp\left\{ -\frac{c^{(n)}(x-u)^2}{4\cdot2^n}\right\}\)\\ &\qquad\qquad\qquad\qquad\qquad\tilde h_n(x,T)\tilde h_n(u,T)\,dx\,du\biggr] \endalign $$ with $$ Z_n(T)=\int \exp\left\{-\frac{c^{(n)}(x-u)^2}{4\cdot2^n}\right\} \tilde h_n(x,T)\tilde h_n(u,T)\,dx\,du. $$ The estimates $$ \align &\left|\int e^{is(x+u)}\(1-\exp\left\{ -\frac{c^{(n)}(x-u)^2}{4\cdot2^n}\right\}\) \tilde h_n(x,T)\tilde h_n(u,T)\,dx\,du\right|\\ &\qquad\le \int\frac{c^{(n)}(x-u)^2}{4\cdot2^n} \tilde h_n(x,T)\tilde h_n(u,T)\,dx\,du= \frac{c^{(n)}}{2^n}\tilde M_2(n,T) \endalign $$ and similarly $$ 1\ge Z_n(T)\ge1-\frac{c^{(n)}}{2^{n}}\tilde M_2(n,T) $$ hold. Hence $$ \varphi_{n}^2\(s,T\)-\frac{c^{(n)}}{2^n}\tilde M_2(n,T) \le\varphi_{n+1}(\sqrt2s,T)\le\frac{\varphi_{n}^2\(s,T\) +\frac{c^{(n)}}{2^n}\tilde M_2(n,T)}{1-\frac{c^{(n)}}{2^n}\tilde M_2(n,T)}. $$ The term $\dfrac{c^{(n)}}{2^n}\tilde M_2(n,T)$ is much less than $\(\dfrac23\)^n$ for large $n$. If we have a positive lower bound on $\varphi_n(s)$ then we get, by taking logarithm in the last relation, that $$ \left|\log\varphi_{n+1}\(\sqrt2s,T\)-2\log\varphi_n(s,T)\right| \le\(\frac23\)^n\quad\text{if }n>n_2\text{ and } \varphi_n(s,T)\ge\frac1K \tag6.20 $$ with some $n_2=n_2(K,T)$. Formula (6.17) can be deduced from (6.19) and (6.20). Indeed, define an index $k$ by the relation $A\le\delta2^{k/2}<\sqrt2A$ with the numbers $A$ and $\delta$ in (6.17) and (6.18). Put $K=2e^{-2^{\nn}\bar\sigma^2(T)A^2/c^{(\nn)}}$ and let $\e\le\dfrac1{8K}$. Choose a number $n_3$ such that $\(\dfrac23\)^{n_3}\le\e$, and let us consider such indices $n$ for which $n\ge \max(n_1(\e,T),n_2(K,T), n_3)$. Then simple induction yields that for $j\le k$ $$ \left|\varphi_{n+j}(s,T)+ \frac{2^{\nn}}{c^{(\nn)}}\bar\sigma^2(T) \frac{s^2}2\right|\le \e+3\(1-\(\frac23\)^{j+1}\)\le 4\e \quad\text {and }\left|\varphi_{n+j}(s,T)\right|\ge\frac1K $$ for $j\le k$. Since $\e$ can be chosen arbitrary small in the last relation, it implies relation (6.17). Lemma 6.3 is proved. \enddemo Theorem 3.3 simply follows from Lemmas 6.2 and 6.3. Indeed, Lemma~6.3 implies that the measures $\tilde H_{n,T}$ converge in distribution to the normal low with expectation zero and covariance $\dfrac{2^{\nn}}{c^{(\nn)}}\bar\sigma^2(T)\bold I$. The bounds obtained for the variance follow from Lemma 6.2 and the observation that the difference $\nn-\bar n(T)$ can be bounded by a number depending only on $\bar\eta$ and $\k$. Let us finally show that Corollary to Theorem~3.3 simply follows from Theorem ~3.3. By formulas (2.5), (2.10), and (3.8) we can write $$ 2^{-n}p_n(2^{-n/2}\sqrt Tx,T)=C(n)\exp\left\{-\frac{\bar c^{(n)}A_n}{2^{n+1}}x^2\right\}\tilde h_n(x,T) \tag6.21 $$ with an appropriate norming constant $C(n)$. Observe that the expressions at both sides of this identity are density functions, the measures with density function $\tilde h_n(x,T)$ have a limit as $n\to\infty$, the term $\left\{-\frac{\bar c^{(n)}A_n}{2^{n+1}}x^2\right\}$ is bounded, and it tends to 1 uniformly in any compact set as $n\to\infty$. These facts imply that $C(n)\to1$ in (6.21), and the measures with density functions $2^{-n}p_n(2^{-n/2}\sqrt Tx,T)$ have the same limit as the measures with density functions $\tilde h_n(x,T)$. Hence the Corollary of Theorem~3.3 holds. \beginsection 7. Estimates in the Low Temperature Region. Proof of Theorem 3.2 The proof of Theorem~3.2 heavily exploits the results of Section~4. These results show that by replacing the operator $\bold Q_n$, whose application makes possible to compute the function $f_{n+1}(x,T)$, with its linearization $\bold T_n$ only a negligible error is committed. Formula (4.17) enables one to investigate the operator $\bold T_n$ in the Fourier space. In such a way good estimates can be obtained for the Fourier transform of a regularized version of the function $f_{n+1}(x,T)$. The results of Theorem~3.2 can be proved by means of these estimates with the help of inverse Fourier transformation. Formulas (3.8) and (3.9) were already proved in Lemma~4.4. The proof of the statement that the fix point equation (1.30) has a unique non-zero solution which is a density function, is a simple adaptation of Lemma~12 in~[BM3]. It is enough to observe that in that proof the parameter $c$ which was taken there $10.\;\;j=0,1,2,\dots, \endaligned\right. \tag7.1 $$ and $$ |\tilde g(t+is)|\le \frac{C_j(\e)}{1+|t|^j} \text{ for }|s|\le2(1-\e) \text{ and arbitrary }t,\; j=1,2,\dots \tag7.2 $$ for all $\e>0$ with appropriate constants $A>0$, $C_j(\e)>0$ and $C_j(\e,\alpha)>0$. It is simpler to work with an appropriately scaled version of the functions $f_n(x,T)$. Put $$ \bar f_n(x,T)=\frac1{M_n(T)}f_n\(\frac x{M_n(T)},T\) $$ and $$ \bar\varphi_n(f_n(x,T))=\frac1{M_n(T)}\varphi_n\(f_n\(\frac x{M_n(T)},T\)\). $$ Let us also introduce the functions $$ \psi_{n+1}(f_n(x,T))=\frac1{M_n(T)}\bold T_n \varphi_n\(f_n\(\frac x{M_n(T)},T\)\). $$ The estimates of Proposition 4.2 and relation (4.17) can be rewritten for these new functions. We shall rewrite formulas (4.15) and (4.16) only in the case when $n>N_1(T)$ with the number $N_1(T)$ defined in formula (4.18), i.e.\ in the case when $\beta_n(T)$ and $M_n^{-1}(T)$ have the same order of magnitude. In this case $M_n(T)\sqrt{\beta_{n+1}(T)}\le10$, $$ \aligned &\left|\frac{\partial^j}{\partial x^j}\(\bar f_{n+1}(x,T)-\psi_{n+1} (f_n(x,T))\)\right|\\ &\qquad\le K_1 \bbeta{n} \left[\exp\left\{-\frac1{10} \left|2x+\frac{x^2} {c^{(n+1)}}\right| \right\}+\exp \left\{-\frac{|x|}{5}\right\}\right]\le K_2 \bbeta{n} e^{-|x|/10}\\ &\qquad\qquad\qquad\qquad\qquad\qquad x>-c^{(n+1)}M_{n+1}^2(T),\quad j=0,1,2 \endaligned \tag7.3 $$ and $$ \left|\frac{\partial^j}{\partial x^j}\psi_{n+1} (f_n(x,T))\right|\le K_3e^{-|x|/5}, \quad x\in \R^1,\quad j=0,1,2,3,4 \tag7.4 $$ with some universal constants $K_1$,~$K_2$ and~$K_3$. Formula (4.17) can be rewritten as $$ \tilde\psi_{n+1}(f_n(\xi,T))=\tilde{\bold T}_{n} \tilde\varphi_n( f_n(M_n(T)\xi,T))= \frac{\exp\left\{i\dfrac {c_{n+1}}{4}\xi \right\}}{\sqrt{1+i\dfrac {c_{n+1}}{2}\xi}} \tilde{\bar\varphi}_n^2\(f_n\(\frac {c_{n+1}}{2}\xi,T\)\). \tag7.5 $$ We claim that under the conditions Theorem~3.2, $$ \lim_{n\to\infty}\sup_{x\ge-c^{(n)}M_n^2(T)}\left|\frac{\partial^j}{\partial x^j}\(\bar f_n(x,T)-\bar\varphi(f_n(x,T))\)\right|e^{|x|/20} =0,\quad j=0,1,2. \tag7.6 $$ Indeed, by relations (7.3) and (7.4) $$ \left|\frac{\partial^j}{\partial x^j}\bar f_n(x,T)\right|\le e^{-|x|/10},\quad j=0,1,2,\quad\text{if }x\ge -c^{(n)}M_n^2(T) \tag7.7 $$ and $\bar\varphi_n(f_n(x,T))$ is the appropriate scaling of the function $$ \varphi\(\dfrac x{\sqrt{c^{(n)}}M_n(T)}\)f_n\(\dfrac x{\sqrt{c^{(n)}}M_n(T)}\). $$ Under the conditions of Theorem~3.2, formula (4.28) holds, which implies that $$ \lim_{n\to\infty}\sqrt{c^{(n)}}M_n(T)=\infty $$ This fact together with (7.7) allow us to give a good bound on the difference between the functions $\bar\varphi_n(f_n(x,T))$ and $\varphi\(\dfrac x{\sqrt{c^{(n)}}M_n(T)}\)f_n\(\dfrac x{\sqrt{c^{(n)}}M_n(T)}\)$. Relation (7.6) can be deduced from this bound and formula~(7.7). It follows from Lemma~4.4 that $\lim\limits_{n\to\infty}\dfrac{M_{n+1}(T)}{M_n(T)}=1$. Relations (7.3) and (7.6) together with this fact imply that $$ \lim_{n\to\infty}\sup_{|x|<\infty}\left|\frac{\partial^j}{\partial x^j}(\psi_{n}(f_{n-1}(x,T))-\bar\varphi_n(f_n(x,T))) \right|e^{|x|/20} =0, \quad j=0,1,2. \tag7.8 $$ The fix point equation (3.5) can be rewritten for the Fourier transform of the function $g_1(x)=g\(x-\frac14\)$ as $$ \tilde g_1(\xi)=\frac{\exp\left\{\frac{i}4\xi\right\}} {\sqrt{1+\frac{i}2\xi}} \tilde g_1^2\(\frac\xi2\). \tag7.9 $$ (We work with the function $g_1(x)$ instead of $g(x)$ because $\int g_1(x)\,dx=0$.) The right-hand side of formulas (7.5) and (7.9) are very similar. Let us recall that by Condition~1 $c_n\to1$ as $n\to\infty$. Now we prove, using an adaptation of the proof of Lemmas 14 and~15 in [BM3], that the Fourier transforms of the functions $\psi_{n+1}(f_n(x,T))$ converge to the Fourier transform of the function $g_1(x)$, and this convergence is uniform in all compact domains. First we prove a modified version of this statement, where $\psi_n$ is replaced with $\bar\varphi_n$ in a small neighbourhood of the origin. We want to work with the functions $\log\tilde {\bar\varphi}_{n}(f_n(\xi,T)))$. To do this, observe first that for $n>N_1(T)$ there is some constant $A>0$ such that all functions $\tilde{\bar\varphi}_{n+1}(f_n(\xi,T)))$ are separated from zero in the interval $|\xi|\le A$. Indeed, $$ \align |1-\tilde{\bar \varphi}_{n}(f_n(\xi,T)))|&\le \int |e^{ix\xi}-1|\bar\varphi_{n}(f_n(x,T)))\,dx\\ &\le \int|\xi| |x|\bar\varphi_{n}(f_n(x,T)))\,dx\le \const |\xi|. \endalign $$ Similarly, $$ \left|\frac{\partial^j}{\partial \xi^j} \tilde{\bar \varphi}_{n}(f_n(\xi,T))\right|\le C(j) \quad\text{for all }j\ge0 \text{ and }n\ge N_1(T). $$ Hence a constant $A>0$ can be chosen in such a way that $$ \sup_{|\xi|\le 2A}\max\( |1-\tilde g_1(\xi)|, \sup_{n\ge N_1(T)}|1-\tilde{\bar\varphi}_{n}(f_n(\xi,T))| \)\le \frac12. $$ These estimates imply that $$ \sup \sup_{|\xi|\le A}\left|\frac{\partial^2}{\partial \xi^2} \log\tilde{\bar \varphi}_{n}(f_n(\xi,T)))\right|\le C(T) \tag7.10 $$ with a constant $C(T)<\infty$ independent of $n$. We claim that $$ \sup_{|\xi|\le A}\left|\frac{\partial^2}{\partial \xi^2} \log\tilde{\bar \varphi}_{n}(f_n(\xi,T)))-\frac{d^2}{d^2\xi}\log \tilde g_1(\xi)\right|\to0 \quad \text{as }n\to\infty. \tag7.11 $$ To prove (7.11) let us first observe that $\lim\limits_{n\to \infty}c_n=1$ by Condition~1. By (7.8), $$ \lim_{n\to\infty}\sup_{|\xi|\le A}\left|\frac{\partial^2}{\partial \xi^2}\(\log\tilde \psi_{n+1}(f_{n}(\xi,T))-\log\tilde{\bar\varphi}_{n+1} (f_{n+1}(\xi,T))\)\right|=0, $$ and because of the estimates obtained for the derivatives of $\tilde{\bar\varphi}_n(\xi,T)$ $$ \align \left|\frac{\partial^2}{\partial \xi^2}\log\tilde{\bar \varphi}_{n}(f_n(\xi_1,T)))-\frac{\partial^2}{\partial \xi^2} \log\tilde{\bar \varphi}_{n}(f_n(\xi_2,T)))\right|&\le \const|\xi_1-\xi_2|\\ &\quad\text{if } |\xi_1|\le A\text{ and }|\xi_2|\le A \endalign $$ for all large indices $n$ with a constant independent of $n$. Taking logarithm and then differentiating twice in formulas (7.5) and (7.9) we get with the help of the above observations that $$ \align &\sup_{|\xi|\le A}\left|\frac{\partial^2}{\partial \xi^2} \log\tilde{\bar \varphi}_{n+1}(f_{n+1}(\xi,T)))-\frac{d^2}{d^2\xi}\log \tilde g_1(\xi)\right|\\ &\qquad\le\frac12 \sup_{|\xi|\le A}\left|\frac{\partial^2}{\partial \xi^2} \log\tilde{\bar \varphi}_{n}(f_n(\xi,T)))-\frac{d^2}{d^2\xi}\log \tilde g_1(\xi)\right|+\delta_n(T) \endalign $$ with some sequence $\lim\limits_{n\to\infty}\delta_n(T)=0$. This relation together with (7.10) imply (7.11). Since $$ \left.\frac{\partial}{\partial \xi} \log\tilde{\bar \varphi}_{n}(f_{n}(\xi,T)))\right|_{\xi=0}=\left. \frac{d}{d\xi}\log \tilde g_1(\xi)\right|_{\xi=0}=0 \text{ and } \log\tilde{\bar \varphi}_{n}(f_{n}(0,T)))=\log\tilde g_1(0)=0, $$ relation (7.11) also implies that $$ \lim_{n\to\infty}\sup_{|\xi|\le A}\left|\tilde{\bar \varphi}_{n}(f_{n}(\xi,T)))-\tilde g_1(\xi)\right|=0. \tag7.12 $$ Moreover, relation (7.12) holds for all $A>0$. This can be proved similarly to the argument of Lemma~15 in~[BM3]. One has to observe that because of the structure of formulas (7.5) and (7.9), the relation $c_n\to1$ as $n\to\infty$, the continuity of the function $\tilde g_1(\xi)$ and the relation $$ \lim_{n\to\infty}\sup_{|\xi|<\infty}\left|\tilde \psi_{n+1}(f_{n}(\xi,T))-\tilde{\bar\varphi}_{n+1} (f_{n+1}(\xi,T)))\right|=0, $$ which also follows from (7.9), the validity of relation (7.12) in an interval $|\xi|\le A$ also implies its validity in the interval $|\xi|\le(2-\e)A$ for any $\e>0$. In relation (7.12) the function $\tilde{\bar\varphi}_{n}(f_{n}(\xi,T)))$ can be replaced by $\tilde \psi_{n+1}(f_{n}(\xi,T))$, i.e.\ the relation $$ \lim_{n\to\infty}\sup_{|\xi|\le A}\left|\tilde\psi_{n+1} (f_{n}(\xi,T))) -\tilde g_1(\xi)\right|=0 \tag$7.12'$ $$ holds for all $A>0$. It can be proved from $(7.12')$ by means of inverse Fourier transformation that $$ \lim_{n\to\infty}\sup_{|x|<\infty}\left|\frac{\partial^j}{\partial x^j} \psi_{n+1} (f_{n}(x,T))) -\frac{d^j}{dx^j} g_1(x)\right|=0\quad j=0,1,2. \tag7.13 $$ To prove 7.13 we need, beside the estimate $(7.12')$, some bound about the decrease of the functions $\tilde g_1(\xi)$ and $\tilde\psi_{n+1} (f_{n}(\xi,T)))$ as $\xi\to\pm\infty$. The estimate (7.2) gives a good bound for the Fourier transform of the function $g_1(x)$. We can get a good estimate for the Fourier transform of $\psi_{n+1}(f_n(x,T))$ with the help of the inductive hypothesis $J(n)$ in Section~4 and relation (7.5). Rewriting the inductive hypothesis $J(n)$ for the function $\bar \varphi_n(f_n(x,T))$ we get with the help of some standard calculation that the Fourier transform $\tilde\psi_{n+1}(f_n(\xi,T))$ decreases at infinity faster than $|\xi|^{-4}$. These estimates are sufficient for the proof of (7.13). Relation (7.13) and (7.3) together give an estimate on the function $\bar f_n(x,T)$ and its derivatives which is equivalent to (3.10). Theorem~3.2 is proved. \beginsection 8. Estimates Near the Critical Point. Proof of Theorems 3.4, 1.3, and 1.5 Our previous results suggest that $M_{n+1}^2(T)\sim M_n^2(T)-\dfrac1{2c^{(n)}}$, hence the derivative $\dfrac {dM_n^2(T)}{dT}$, as a function of $n$, changes very little if the pair $(n,T)$ is in the low domain region (observe that $c^{(n)}$ does not depend on $T$). Therefore, it is natural to expect that $\dfrac{M^2_\infty(T)}{dT}$ is of constant order below the critical value $T_{\text cr.}$, and $M_\infty^2(T)-M_\infty^2(T_{\text{cr.}})\sim \const(T_{\text{cr.}}-T)$ for $TC_2>C_1>0. $$ \item{b.)} If $n\ge N$, then $$ \align \frac{dM_{n+1}(T)}{dT}&= \frac{dM_n(T)}{dT}\(1+\frac1{4c^{(n)}M_n^2(T)} +\frac{\delta_n(T)}{c^{(n)}}\),\\ &\qquad \text{where }|\delta_n(T)|\le C\frac{\beta_{n+1}(T)}{c^{(n+1)}}\beta_n(T) \text{ with some appropriate } C>0. \endalign $$ }\endproclaim \parindent=13pt We shall prove Theorem A in Appendix A below. This result can be interpreted in an informal way as the ``differentiation" of the asymptotic identity~(4.13). In this formal differentiation we bound the absolute value of $\dfrac{d\sqrt{\beta_n(T)}}{dT}$ by $\const \left| \dfrac{M_n(T)}{dT}\right|\beta_n(T)$. The main difficulty in the proof of Theorem~A is to bound the error caused by the linear approximation of the operator $\bold Q_n$ by $\bold T_n$ when differentiating with respect to $T$. To overcome this difficulty we need a good control not only on the functions $f_n(x,T)$ but also on their derivatives $\dfrac{\partial}{\partial T}f_n(x,T)$. Hence we have to work out the estimation of these derivatives. In particular, we have to find the inductive hypotheses describing their behaviour. These are the analogs of the inductive hypotheses $I(n)$ and $J(n)$ formulated in Section~4. It demands fairly much work to work out the details, but after the formulation and proof of these inductive hypotheses the proof of Theorem~A is simple. {\it Proof of Theorem 3.4.} We prove with the help of Theorem~A that if the conditions of Theorem~3.4 hold, $0<\bar TN$ first we show that $$ \aligned -\dfrac{dM_n^2(T)}{dT} \exp\left\{-K\(\frac{\beta_{n+1}(T)}{c^{(n)}}\)^2\right\} &\le-\dfrac{dM_{n+1}^2(T)}{dT}\\ &\le -\dfrac{dM_n^2(T)}{dT}\exp \left\{K\(\frac{\beta_{n+1}(T)}{c^{(n)}}\)^2\right\} \endaligned \tag8.2 $$ for all $T<\bar T$ and $n\ge N$ with an appropriate $K>0$. Relation (8.2) is a consequence of Part b.) of Theorem~A, formula (4.13), the inequality $\beta_{n+1}M_n^2(T)\ge10$ and the relation $\dfrac{\beta_{n+1}(T)}{c^{(n)}}\le \eta$ if $(n,T)$ is in the low temperature domain. Indeed, these relations imply that $$ \align &-\(1-\frac{M_n(T)}{4c^{(n)}}-C_1\frac{\beta_{n+1}^{3/2}(T)} {{c^{(n)}}^2M_n(T)}\)\(1+\frac1{4c^{(n)}M_n^2(T)} -C\frac{\beta^2_{n+1}(T)}{(c^{(n)})^2}\)\frac{dM^2_n(T)}{dT}\\ &\qquad\le-\frac{dM^2_{n+1}(T)}{dT}\\ &\qquad\le-\(1-\frac{M_n(T)}{4c^{(n)}}+C_1\frac{\beta_{n+1}^{3/2}(T)} {{c^{(n)}}^2M_n(T)}\)\(1+\frac1{4c^{(n)}M_n^2(T)} +C\frac{\beta^2_{n+1}(T)}{{c^{(n)}}^2}\)\frac{dM^2_n(T)}{dT}. \endalign $$ The left and right-hand side of this inequality can be bounded by $$ -\(1\pm K\dfrac{\beta_{n+1}^2(T)}{{c^{(n)}}^2}\)\dfrac{dM^2_n(T)}{dT}, $$ and formula (8.2) can be deduced from these relations. For $N\le n\le N_1(T)$ with the number $N_1(T)$ defined in relation (4.18) relation (8.1) follows from (8.2) and (4.19). Since by (4.20) $\beta_{n+1}M_n^2(T)\le100$ if $n\ge N_1(T)$ and the pair $(n,T)$ is in the low temperature domain, to prove formula (8.1) with the help of (8.2) for $n>N_1(T)$ it is enough to show that $$ \sum_{k=N_1(T)}^n\frac1{\(c^{(k)}M_k^2(T)\)^2}\le L\text{ if }n\ge N_1(T) \text{ and }(n,T) \text{ is in the low temperature domain} $$ with a constant $L>0$ independent of $T$ and $n$. Since $M_n^2(T)\ge\dfrac1{10\beta_{n+1}(T)}\ge\dfrac1{10\eta c^{(n)}}$ and $M_k^2(T)=M_n^2(T)+(M_k^2(T)-M_n^2(T))\ge\dfrac1{10\eta c^{(n)}} +\sum\limits_{j=k}^{n-1}\dfrac1{8c^{(j)}}$, $$ \sum_{k=N_1(T)}^n\frac1{\(c^{(k)}M_k^2(T)\)^2}\le \const \sum_{k=N_1(T)}^n\frac1{\(c^{(k)}\sum\limits_{j=k}^{n}\dfrac1{c^{(j)}}\)^2} \le L $$ because of Condition 3. Hence formula (8.1) holds. It follows from (1.3), Condition 4, and the results of Section~4 that all $T>c_0A_0/4$ belong to the high temperature region. Indeed, it follows from formulas (4.26), $(4.26')$, (4.1), (4.4) and (4.10), that if $T>0$ is in the low temperature region, then $$ 0\le M_n^2(T)\le M_N^2(T)-30(M_N(T)+1)-\sum_{n=1}^\infty\frac1{8c^{(n)}} \le \frac3{\k T^2}-\sum_{n=1}^\infty\frac1{8c^{(n)}} $$ for all $n\ge N$, and $T\le \(\sum\limits_{n=1}^\infty\dfrac \k {24c^{(n)}}\)^{-1/2}$. Hence Condition~4 implies that $T\le c_0A_0/4$. It follows from (8.2) that the for fixed $n$ the functions $M_n^2(T)$ is a strictly monotone decreasing, hence a simple induction with respect to $n$ yields that the function $\beta_n(T)$ is a monotone increasing, continuous function of~$T$. Put $$ T_n=\sup\{ T\: (T,n) \text{ is in the low temperature region}\}. $$ The sequence $T_n$ is monotone decreasing, hence the limit $T_{\text{cr.}}=\lim\limits_{n\to\infty}T_n$ exists, and by Lemma~4.2 $T_{\text{cr.}}>0$ under Dyson's condition (1.3). We want to show that $$ {C_1}\sum_{k=n}^\infty\frac1{c^{(k)}}\le T_n-T_{\text{cr.}}\le {C_2}\sum_{k=n}^\infty\frac1{c^{(k)}}. \tag8.3 $$ Since we can handle the sequence $M_n(T)$ better than the sequence $\beta_{n}(T)$ we also introduce the sequence $T(n)$ $$ T(n)=\sup \left\{ T\: M^2_n(T)\ge \frac{100}{c^{(n)}\eta}\;\right\}. $$ We will show that $$ T_{n+K}\le T(n)\le T_n \tag8.4 $$ for all sufficiently large $n$ with an appropriate $K>0$, and $$ \frac{C_1}{c^{(n)}}\le T(n)-T(n+1)\le \frac{C_2}{c^{(n)}} \tag8.5 $$ with some appropriate $C_2>C_1>0$ for all sufficiently large $n$. Because of Condition~5 relation (8.3) follows from (8.4) and (8.5) together with the relation $\lim\limits_{n\to\infty}T_n=T_{\text{cr.}}$. If $T\le T(n)$, and $m\le n$ then either $m\le N_1(T)$ with the number $N_1(T)$ defined in (4.19) or $\beta_{m+1}(T)\le \dfrac{100}{M_m^2(T)} \le \dfrac{100}{M_n^2(T)} \le c^{(n)}\eta$. This implies that for $T\le T(n)$ the pair $(m,T)$ is in the low temperature region for all $m\le n$, and $T(n)\le T_n$. This is the right-hand side of relation (8.4). To prove its left-hand side observe that because of Condition~5 there is some $K$ such that $$ \sum_{k=n}^{n+K-1}\frac1{8c^{(n)}}> \frac{100}{c^{(n)}\eta} $$ for all sufficiently large $n$ with appropriate $K>0$. We claim that $T\ge T(n)$ the pair $(n+K,T)$ is not in the low temperature region. This relation implies the left-hand side of (8.4). If $(n+K,T)$ were in the low temperature region, then we would get with the help of formula (4.26) that $$ M_{n+K}^2(T)\le M_n^2(T)-\sum_{k=n}^{n+K-1}\frac1{8c^{(n)}}< \frac{100}{c^{(n)}\eta}-\sum_{k=n}^{n+K}\frac1{8c^{(n)}}< \frac{100}{c^{(n)}\eta}-\frac{100}{c^{(n)}\eta}=0, $$ and this is a contradiction. To prove formula (8.5) let us first observe that because of the continuity and strict monotonicity of the function $M_n^2(T)$, \ $M_n^2(T(n))=\dfrac{100}{c^{(n)}\eta}$. It follows from the last statement of %@ Lemma~4.3 and formula (8.1) that $N_1(T)\le n$ for all $T(n)-\e0$. (The number $N_1(T)$ was defined in (4.18).). Hence we get with the help of formula (8.1) that for sufficiently large~$n$ and $T(n)-\e\bar C_1>0$. Hence the solution of the equation $M_{n+1}^2(T)=\dfrac{100}{c^{(n+1)}\eta}$ satisfies the inequality $K_1K_1>0$. Since the solution of this equation is $T(n+1)$, this fact implies relation~(8.5). It is not difficult to see that $T_{\text{cr.}}$ is in the low temperature region. Since the inequality $M_n^2(T_{\text{cr.}})\le \const (T(n)-T_{\text{cr.}})+\dfrac{100}{c^{(n)}\eta}$ holds for all large~ $n$, $\lim\limits_{n\to\infty}M_n(T_{\text{cr.}})=0$. Then relation (8.1) implies that $$ C_1\(T_{\text{cr.}}-T)\)\le M_n^2(T_{\text{cr.}})-M_n^2(T)\le C_2\(T_{\text{cr.}}-T\) $$ with some positive constants $C_2>C_1>0$ if $T_{\text{cr.}}\ge T\ge T_{\text{cr.}}-\e$. Letting $n$ tend to infinity in the last relation we get formula (3.28). Since formula (8.3) is equivalent to (3.27) Theorem~3.4 is proved. {\it Proof of Theorem 1.3.} By Corollary of Theorem 3.1, if the Dyson condition (1.3) is violated then all temperatures $T>0$ belong to the high temperature region. By Corollary of Theorem 3.3, (1.24) holds and the measures $\bar\nu_{n,T}(dx)$ approaches the standard normal distribution as $n\to\infty$. Theorem 1.3 is proved. {\it Proof of Theorem 1.5, Part 1).} The convergence of $\bar\nu_{n,T}(dx)$ to a standard Gaussian distribution and relation (1.32) follow from Corollary of Theorem 3.3. The asymptotics (1.33) follows from (3.26) and (3.27). {\it Part 2).} Formula (1.34) follows from (3.9), and the convergence of $\bar\nu_{n,T_c}(dx)$ to the uniform distribution on the sphere follows from Theorems 3.2 and 3.4. Namely, Theorem 3.4 tells us that the critical temperature $T_c$ belongs to the {\it low temperature region}. Then formula (3.10) proves that the probability distribution $\bar\nu_{n,T_c}(dx)$ converges to the uniform distribution on the sphere. As a matter of fact, (3.10) proves much more: it proves the %@ convergence at $T=T_c$ of the distribution of normalized fluctuations of the average spin along the radius to a limit. {\it Part 3).} By (2.10), (2.14), and (2.17), $$ p_n(x,T)=L_n^{-1}(T)\, \exp\(- {A_nl_n|x|^2\over 2T}\) f_n\({c^{(n)}\over \sqrt T}\(|x|- \sqrt T\,M_n(T)\),T\) \tag8.6 $$ Let us write that $|x|^2=\(\sqrt T\,M_n(T)+|x|-\sqrt T\,M_n(T)\)^2$, hence $$\aligned \exp\(- {A_nl_n|x|^2\over 2T}\)&= \exp\left\{- {A_nl_n\over 2T}\big[ TM_n^2(T)+2\sqrt T\,M_n(T)(|x|-\sqrt T\,M_n(T))\right.\\ &\qquad +(|x|-\sqrt T\,M_n(T))^2\big]\bigg\}, \endaligned\tag8.7 $$ and substitute it into (8.6). This leads to the equation $$ p_n(x,T)=\tilde L_n^{-1}(T)\tilde f_n\({|x|-\tilde M_n(T)\over \tilde V_n(T)}\,,T\), \tag8.8 $$ where $$\aligned &\tilde M_n(T)=\sqrt T\, M_n(T),\qquad\tilde V_n(T)={\sqrt T\over c^{(n)}M_n(T)}\\ &\tilde f_n(t,T)=f_n\( {t\over M_n(T)},T\)\exp\(-{A_nl_nt\over c^{(n)}}-\e_n(t,T)\)\\ &\e(t,T)={A_nl_nt^2\over 2(c^{(n)})^2M^2_n(T)}. \endaligned\tag8.9 $$ Observe that by (2.30) and (2.31), $$ \lim_{n\to\infty} {A_nl_n\over c^{(n)}}={2\over 3}\,, \qquad \lim_{n\to\infty}{A_nl_n\over 2(c^{(n)})^2M^2_n(T)}=0, \tag8.10 $$ hence (3.10) implies that there is some $C_0>0$ such that $$ \lim_{n\to\infty}\left\|\tilde f_n(t,T))-C_0 g\(t-{r-1\over 4}\)e^{-2t/3}\right\|'=0, \tag8.11 $$ where $$ \| f(t) \|'=\sum_{j=0}^2\,\sup_{t\ge -c^{(n)}M^2_n(T)}e^{|t|/3} \left| {d^jf(t)\over d\,t^j} \right|. \tag8.12 $$ It remains to shift $\tilde f_n(t)$ to secure the mean value to be zero. Define $$ \tilde \pi_n(t,T)=C_n(T) \tilde f_n(t-a_0,T),\qquad a_0=a-{r-1\over 4}, \tag8.13 $$ where $C_n(T)$ is a norming factor such that $$ \int_{\R^1}\tilde \pi_n(t)\,dt=1. \tag8.14 $$ Then $$ p_n(x,T)=L_n^{-1}(T)\tilde \pi_n\({|x|-M_{n0}(T)\over \tilde V_n(T)}\,,T\)\quad\text{with } M_{n0}(T)=\tilde M_n+a_0\tilde V_n(T) \tag8.15 $$ and $$ \lim_{n\to\infty}\int_{\R^1}t\tilde\pi_n(t)\,dt=0. \tag8.16 $$ Comparing these formulae with (1.29) we obtain (1.36), (1.38), and (1.39) with $$ M(T)=\sqrt T\, M_\infty (T),\qquad \g(T)={\sqrt T\over 3M_\infty (T)}={T\over 3M(T)}, \tag8.17 $$ where $M_\infty (T)$ is the limit (3.4). The estimates (8.17) follow from (3.28). Theorem 1.5 is proved. \beginsection Appendix A. Proof of Theorem A To prove Theorem A we need good estimates on the partial derivatives $g_n(x,T)=\dfrac{\partial}{\partial T}f_n(x,T)$, on the derivatives of a scaled version of the functions $q_n(x,T)$. This can be done similarly to the estimation of the functions $f_n(x,T)$, done in Section~4. First we give estimates for the starting function $g_0(x,T)$, then prove that similar estimates hold for small indices $n$, more explicitly for $n\le N$ with the index $N$ defined in (1.22). Then inductive hypotheses can be formulated and proved for the functions $g_n(x,T)$. In Section~4 we have introduced certain operators $\bar{\bold Q}_n$, their normalization $\bold Q_n$ and the linearization of these operators denoted by $\bar{\bold T}_n$ and $\bold T_n$. The inductive hypotheses formulated there were closely related to the properties of these operators. Now we want to work similarly. To do this we have to introduce some new operators. We introduce certain operators $\bar {\bold R}_n$ and $\bold R_n$ which are the derivatives of the operators $\bar{\bold Q}_n$ and $\bold Q_n$ with respect to the variable~$T$. We also need their linear approximation which we shall denote by $\bar{\bold U}_n$ and $\bold U_n$. We have to study the action of these operators on the functions $g_n(x,T)=\dfrac{\partial}{\partial T}f_n(x,T)$ and their Fourier transform. Although the proofs are not hard, it demands much time to work out the details even if only a brief explanation is given as in this Appendix. An appropriate description of the asymptotic behaviour of the starting functions $f_0(x,T)$ and numbers $M_0(T)$ were already given in formulas (4.2)~---~(4.7). Some more calculation yields, with the help of some formulas in Section~4, the following estimates for the derivatives of the magnetization $M_0(T)$ and the norming constant $Z_0(T)$ if $TC_2>C_1>0. $$ and $$ \left|\frac {dZ_0(T)}{dT}- \frac{\sqrt \pi}{2(A_0-T)^{3/2}}\right|\le\const \sqrt \k. $$ The derivatives of the functions $\bar q_0(x,T)$ and $f_0(x,T)$ satisfy the inequalities $$ \aligned \left|\frac{\partial q_0(x+M_0(T),T)}{\partial T}-\frac{\sqrt{A_0-T}}{\sqrt\pi}\(x^2-\frac1{2(A_0-T)}\)e^ {-(A_0-T)x^2}\right|&\le\const \k^{1/4}\\ &\quad \text{if }|x|<\log \k^{-1}, \endaligned \tag A1 $$ and $$ \left|\frac{\partial q_0(x+M_0(T),T)}{\partial T}\right|\le C\exp\left\{ -\frac{(A_0-T)}4\left|2x+\frac{x^2}{M_0^2(T)}\right|\right\} \quad \text{for }x\ge - M_0(T). \tag A2 $$ We shall apply the notation $$ g_n(x,T)=\dfrac {\partial f_n(x,T)}{\partial T},\quad n=0,1,\dots. \tag A3 $$ Since $f_0(x,T)=q_0(x+M_0(T),T)$ the previous estimates together with the results of Section~4 yield a sufficiently good control on $g_0(x,T)$. The functions $g_n(x,T)$, $n=1,2,\dots$, can be estimated inductively with respect to the parameter~$n$. Put $$ \bold {\bar R}_n f_n(x,T)=\dfrac{\partial}{\partial T}\bold {\bar Q}_{n,M_n(T)}^{\bold c} f_n(x,T) $$ and $$ \bold R_n f_n(x,T)=\dfrac{\partial}{\partial T}\bold Q_n f_n(x,T)=g_{n+1}(x,T). $$ Then $$ \bar {\bold R}_{n}f_n(x,T)=\bar {\bold R}_{n}^{(1)}f_n(x,T)+ \bar {\bold R}_{n}^{(2)}f_n(x,T) $$ with $$ \align \bar {\bold R}_{n}^{(1)}f_n(x,T)=2 \int\exp\left\{-\frac{u^2}{c^{(n)}}-v^2\right\} &f_n(\ell_{n,M_n(T)}^{+}(x,u,v),T)\\ &\qquad g_n(\ell_{n,M_n(T)}^{-}(x,u,v),T)\,du\,dv, \endalign $$ where the functions $g_n(x,T)$ and $\ell_{n,M_n(T)}^{\pm}(x,u,v),T)$ were defined in (A3) and (2.9), and $$ \align \bar {\bold R}_{n}^{(2)}f_n(x,T)=-2 \int\exp\left\{-\frac{u^2}{c^{(n)}}-v^2\right\}& f_n(\ell_{n,M_n(T)}^{+}(x,u,v),T)h_n(x,u,v,T)\\ &\qquad\frac\partial{\partial x}f_n(\ell_{n,M_n(T)}^-(x,u,v),T) \,du\,dv \endalign $$ with $$ \align h_n(x,u,v,T)&=-\frac{\partial \ell_{n,M_n(T)}^-(x,u,v)}{\partial T}\\ &=\frac{M_n'(T)v^2}{\sqrt{\(M_n(T)+\frac x{c^{(n+1)}} -\frac u{ c^{(n)}} \)^2+\frac{v^2}{c^{(n)}}}}\\ &\qquad\frac1{\sqrt{\(M_n(T)+\frac x{c^{(n+1)}} -\frac u{ c^{(n)}} \)^2+\frac{v^2}{c^{(n)}}}+ \(M_n(T)+\frac x{c^{(n+1)}}-\frac u{c^{(n)}}\)}. \endalign $$ The function $g_{n+1}(x,T)$ can be expressed as $$ \align g_{n+1}(x,T)=\bold R_n f_n(x,T)&=\frac{\bar{\bold R}_nf_n(x+m_n(T),T)}{Z_n(T)}+\frac{\dfrac{\partial}{\partial x} \bar{\bold Q}_n f_n(x+m_n(T),T)}{Z_n(T)}\frac{dm_n(T)}{dT}\\ &\qquad -\frac{\bar{\bold Q}_n f_n(x+m_n(T),T)}{Z_{n}^2(T)}\frac{dZ_n(T)}{dT} \endalign $$ with $$ Z_n(T)=\int_{-c^{(n)}M_n(T)}^\infty \bar{\bold Q}_n f_n(x,T)\,dx. $$ If the parameter $\k>0$ in formula (2.13) $q_0(x,T)$ is sufficiently small and $n\le N$, then the function $g_n(x,T)$ can be estimated similarly to the proof of Proposition~4.1 or Proposition~1 in [BM3]. Relation (A5) formulated below can be deduced from formula (A2) similarly to the proof of Lemma~1 of that paper. Then an argument similar to the proof of Lemma~2 in~[BM3] enables one to prove formula (A4) formulated below. In this argument one can observe that a negligible error is committed if in the integrals appearing in the definition of $\bar{\bold R}_nf_n(x,T)$ the arguments $\ell^\pm_{n,M_n(T)}(x,u,v)$ defined in formula (2.25) are replaced by $\dfrac x{c_{n+1}}\pm u$. Some calculation also shows that we commit a negligible error by replacing $\bold R_n f_n(x,T)$ with $\dfrac{\bar{\bold R}_n^{(1)}f_n(x,T)}{Z_n(T)}$. In such a way we get that $$ \aligned &\left|g_n(x,T)- \frac{\sqrt{A_0-T}}{\sqrt\pi}\frac{2^{n/2}}{c^{(n)}} \(x^2-\frac1{2(A_0-T)}\frac{{c^{(n)}}^2}{2^n}\)\exp\left\{-(A_0-T) \frac{2^nx^2}{{c^{(n)}}^2}\right\}\right|\\ &\qquad\qquad\quad \le C(n) \k^{1/4} \exp\left\{-\frac{(A_0-T)}4 \frac{2^n}{c^{(n)}} \left|2x+\frac{x^2}{M_n^2(T)}\right|\right\} \quad \text{if }|x|<2^{-n}\log \k^{-1}, \endaligned \tag A4 $$ $$ |g_n(x,T)|\le C(n) \exp\left\{-\frac{(A_0-T)}4 \frac{2^n}{c^{(n)}} \left|2x+\frac{x^2}{M_n^2(T)}\right|\right\} \quad \text{for }x\ge - M_n(T), \tag A5 $$ $$ |M_n(T)- M_0(T)|\le C(n)\k^{1/2},\qquad \left|Z_n(t)-\frac{\sqrt \pi}{\sqrt{A_0-t}}\right|\le C(n)\k^{1/2} $$ with some constant $C(n)$ which may depend on $n$ but not on the parameter $\k$ of the model. The previous results are sufficient to handle the functions $g_n(x,T)$ for small indices $n\le N$. To work with indices $n\ge N$ we have to introduce, similarly to the argument in Section~4, the regularization of the functions $g_n(x,T)$, the linearization $\bar{\bold U}_n$ and $\bold U_n$ of the operators $\bar{\bold R}_n$ and $\bold R_n$ and to describe their action in the Fourier space. Define the regularization of the function $g_n(x,T)$ as $\varphi_n(g_n(x,T))=\dfrac{\partial \varphi_n(f_n(x,T))}{\partial T}$. We want to approximate the operator $\bold R_n$ with a simpler operator $\bold U_n$ in analogy with the approximation of $\bold Q_n$ by $\bold T_n$. Then we formulate and prove some inductive hypothesis about the behaviour of the operators $\bold R_n$ and $\bold U_n$. A natural approximation of the operators $\bar{\bold R}_n$ and $\bold R_n$ by some operators $\bar{\bold U}_n$ and $\bold U_n$ can be obtained by differentiating $\bar{\bold T}_n\varphi(f_n(x,T))$ and ${\bold T}_n\varphi_n(f_n(x,T))$ with respect to the variable $T$. These considerations suggest the definition of the operators $$ \align \bar{\bold U}_n\varphi_n(f_n(x,T))&=2\int e^{-v^2} \varphi_n\(f_n\(\frac x{c_{n+1}}+u+\frac{v^2}{2M_n(T)},T\)\)\\ &\qquad\biggl\{\varphi_n\(g_n\(\frac x{c_{n+1}}-u+\frac{v^2}{2M_n(T)},T\)\)\\ &\qquad\quad-v^2\frac{M_n'(T)}{2M_n(T)^2} \frac\partial{\partial x}\varphi_n\(f_n\(\frac x{c_{n+1}}-u+\frac{v^2}{2M_n(T)},T\)\) \Biggr\} \,du\,dv \endalign $$ with the function $g_n(x,T)$ defined in (A3) and $$ \bold U_n\varphi_n(f_n(x,T))=\bold U_n^{(1)}\varphi_n(f_n(x,T))+ \bold U_n^{(2)}\varphi_n(f_n(x,T)) $$ with $$ \align \bold U_n^{(1)}\varphi_n(f_n(x,T))&= \frac4{c_{n+1}\sqrt\pi} \int e^{-v^2}\varphi_n\(f_n\(\frac x{c_{n+1}}+u-\frac1{4M_n(T)}+ \frac{v^2}{2M_n(T)},T\)\)\\ &\qquad\varphi_n\(g_n\(\frac x{c_{n+1}}-u-\frac1{4M_n(T)}-\frac{v^2}{2M_n(T)},T\)\)\,du\,dv, \endalign $$ and $$ \align \bold U_n^{(2)}\varphi_n(f_n(x,T))&=\frac4{c_{n+1}\sqrt\pi} \int e^{-v^2}\(\frac{M'_n(T)}{4M_n^2(T)}-v^2 \frac{M_n'(T)}{2M_n(T)^2}\)\\ &\qquad\qquad \varphi_n\(f_n\(\frac x{c_{n+1}}-u-\frac1{4M_n(T)}+\frac{v^2}{2M_n(T)},T\)\)\\ &\qquad\qquad\frac\partial{\partial x}\varphi_n\(f_n\(\frac x{c_{n+1}}-u-\frac1{4M_n(T)}+\frac{v^2}{2M_n(T)},T\)\)\,du\,dv. \endalign $$ The Fourier transform of $\bar{\bold U}_n\varphi_n(f_n(x,T))$, ${\bold U}_n^{(1)}\varphi_n(f_n(x,T))$ and ${\bold U}_n^{(2)}\varphi_n(f_n(x,T))$ can be expressed as $$ \align \tilde{\bar{\bold U}}_{n} \tilde\varphi_n (f_n(\xi,T))&= {c_{n+1}\sqrt\pi} \frac{\tilde\varphi_n\( f_n\(\dfrac {c_{n+1}}{2}\xi,T \)\)} {\sqrt{1+i\dfrac {c_{n+1}}{2M_n(T)}\xi}} \tilde\varphi_n \(g_n\(\dfrac {c_{n+1}}{2}\xi,T \)\)\\ &\qquad -\frac{c^2_{n+1}\sqrt\pi}{2}\frac{M_n'(T)}{M_n(T)^2} \xi \frac{\tilde\varphi_n^2\( f_n\(\dfrac {c_{n+1}}{2}\xi,T \)\)} {\(1+i\dfrac {c_{n+1}}{2M_n(T)}\xi\)^{3/2}}, \endalign $$ $$ \tilde{\bold U}_{n}^{(1)} \tilde\varphi_n (f_n(\xi,T))= 2\frac{\exp\left\{\dfrac{ic_{n+1}\xi}{4M_n(T)}\right\}} {\sqrt{1+i\dfrac {c_{n+1}}{2M_n(T)}\xi}} \tilde\varphi_n\( f_n\(\dfrac {c_{n+1}}{2}\xi,T \)\) \tilde\varphi_n \(g_n\(\dfrac {c_{n+1}}{2}\xi,T \)\) \tag A6 $$ and $$ \aligned \tilde{\bold U}_{n}^{(2)} \tilde\varphi_n (f_n(\xi,T))=& \frac{c_{n+1}M_n'(T)}{4M_n(T)^2} \frac{\exp\left\{\dfrac{ic_{n+1}\xi}{4M_n(T)}\right\}\xi} {\sqrt{1+i\dfrac {c_{n+1}}{2M_n(T)}\xi}}\\ &\qquad\tilde\varphi_n^2\( f_n\(\dfrac {c_{n+1}}{2}\xi,T \)\) \(1-\frac1{1+i\dfrac {c_{n+1}}{2M_n(T)}\xi}\). \endaligned \tag A7 $$ The above relation can also be extended to a larger set of the variables $\xi$ in the complex plane by means of analytic continuation. Now we formulate the inductive hypotheses we want to prove in the Appendix. \proclaim {Property $K_1(n)$} {\it \medskip $$ -\frac{dM_n(T)}{dT}>0. $$ }\endproclaim \proclaim {Property $K_2(n)$} {\it $$ \align |g_n(x,T)|=\left|\frac{\partial}{\partial T} f_n(x,T) \right| -c^{(n)}M_n(T) \endalign $$ with a universal constant $K$.} \endproclaim \proclaim{Property $K_3(n)$} {\it $$ \align &|g_n(x,T)-\bold U_{n-1}\varphi_{n-1}(f_{n-1}(x,T))|\\ &\qquad-c^{(n)}M_n(T) \endalign $$ with a universal constant $K$. The inequality remains valid if the function $g_n(x,T)$ is replaced by its regularization $\varphi_n( g_n(x,T))$.} \endproclaim The following property $K_4(n)$ which gives a bound on the Fourier transform of $\varphi_n(g_n(x,T))$ is an analog of Property~$J(n)$. \proclaim{Property $K_4(n)$} {\it $$ \align \left|\tilde\varphi_n(g_n(-is,T)\right|&=\left|\int e^{sx}\varphi_n(g_n(x,T)\,dx\right|\le \beta_n^{3/2}(T)s^2\left|\frac{dM_n(T)}{dT}\right| e^{\beta_n(T)s^2}\\ &\qquad\qquad \text{if } |s|< \frac{2}{\sqrt{\beta_{n+1}(T)}}. \endalign $$ }\endproclaim In Property $K_4(n)$ we formulated a weaker estimate than in $J(n)$. It is enough to have a good bound on the moment generating function, i.e.\ on the analytic continuation of the Fourier transform to the imaginary axis together with the trivial estimate $|\tilde\varphi_n (g_n(-is+t,T)|\le \tilde\varphi_n(g_n(-is,T)$ for all~$t$. The main result of the Appendix is the following Proposition~A. \proclaim{Proposition A} {\it Let the properties $K_1(m)$, $K_2(m)$, $K_3(m)$ and $K_4(m)$ hold in a neighbourhood of a parameter $T$ together with the property $\dfrac{\beta_m(T)}{c^m}\le \eta$ (with the same small number $\eta>0$ which appeared in the proof of %@ Propositions~4.1 and~4.2) for all $N\le m\le n$, and let also the inductive hypotheses $I(n)$ and $J(n)$ be also satisfied. Then the properties $K_{1}(n+1)$, $K_{2}(n+1)$, $K_{3}(n+1)$ and $K_{4}(n+1)$ also hold for this parameter $T$. The expression $$ \delta_n(T)=\frac{d}{dT}\(m_n(T)-\frac1{4M_n(T)}\) =\frac{dm_n(T)}{dT}+\frac1{4M^2_n(T)}\frac{dM_n(T)}{dT}, $$ satisfies the inequality $$ |\delta_n(T)|\le C\left|\frac{dM_n(T)}{dT}\right| \frac{\beta_{n+1}(T)}{c^{(n+1)}}\beta_{n+1}(T) \tag A8 $$ with an appropriate $C>0$, where $m_n(T)$ was defined in (2.22).} \endproclaim If we want to apply Proposition~A, then first we have to show that properties $K_1(n)$, $K_2(n)$, $K_3(n)$ and $K_4(n)$ hold for $n=N$ if $Tn/2$ and $n\ge j\ge k$ we estimate $$ l_kl_j^{-1}\ge C_0>0 $$ hence $$ l_k\sum_{j=k}^n l_j^{-1}\ge C_0(n-k+1) $$ Thus, $$ \sum_{k=1}^n\(l_k\sum_{j=k}^n l_j^{-1}\)^{-2}\le C^{-2}\sum_{k=1}^{n/2} (1+ak)^{-2} +C_0^{-2}\sum_{k=n/2}^n(n-k+1)^{-2}\le C_1 $$ Condition 3 is checked. \medskip\noindent Conditions 4 and 5 are obvious. \bigskip\noindent {\it Acknowledgements.} An essential part of this work was done at the Mathematisches Forschungsinstitut Oberwolfach, where the authors enjoyed their participation in the program ``Research in Pairs''. 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