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%%%
%%% ON THE PURITY OF THE LIMITING GIBBS STATE FOR THE ISING
%%% MODEL ON THE BETHE LATTICE.
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%%% Authors: P.M. BLEHER, J. RUIZ and V.A. ZAGREBNOV.
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\baselineskip=14pt
\parskip=5pt plus 1pt minus 1pt
\null
\def \b{\beta}
\def \sg{\sigma}
\def \ep{\varepsilon}
\def\t{\theta}
\def \sk{\sum_{j=1}^k}
\def \pr{\prod_{j=1}^k a(h_j)\nu_n(h_j)} \def \prv{\prod_{v\in S(x)}
a(h_v)\,\nu_v(h_v)} \def \ff{f(h_1)+\dots+f(h_k)}
\def \di{\displaystyle}
\def \artanh{\,{\rm artanh}\,}
\def \vi{\vskip 2mm\noindent}
\font\eightrm=cmr8
\font\ninerm=cmr9
\font\eightbf=cmbx8
\font\ninebf=cmbx9
\font\eighti=cmti8
%\rightline{\it Submitted to Journal of Statistical Physics}
\vskip 5mm
\centerline{\bf ON THE PURITY OF THE LIMITING GIBBS STATE FOR THE ISING }
\vskip .2 truecm
\centerline{\bf MODEL ON THE BETHE LATTICE}
\vskip 1 truecm
\centerline{ P.M. BLEHER\footnote{$^1$}
{\eightrm Department of Mathematical Sciences, Indiana University -- Purdue
University at Indianapolis,
402 N. Blackford Street, Indianapolis, IN 46202-3216 USA
\hfill\break
E-mail address:{\eighti bleher@math.iupui.edu} },
J. RUIZ\footnote{$^2$}
{{\eightrm E-mail address:}{\eighti ruiz@cpt.univ-mrs.fr} },
V.A. ZAGREBNOV\footnote{$^3$}
{\eightrm and D\'epartement de Physique, Universit\'e d'Aix-Marseille II
\hfill\break
E-mail address:{\eighti zagrebnov@cpt.univ-mrs.fr} } }
\vskip 1truecm
\centerline {Centre de Physique Th\'eorique}
\vskip .1 true cm
\centerline {CNRS-Luminy Case 907}
\vskip .1 true cm
\centerline {F-13288 Marseille Cedex 9 (France)}
\vskip 1.5 true cm
\hangindent= 1.5 true cm
\hangindent= - 1.5 true cm
\hangafter= - 5
\noindent
{\eightbf Abstract}
{\eightrm : We give a proof that for the Ising model on the Bethe lattice,
the limiting Gibbs state
with zero effective field (disordered state)
persists to be pure for temperature below ferromagnetic critical
temperature
$T_c^F$ till the critical temperature $T_c^{SG}$
of the corresponding spin-glass model.
This new proof revises the one proposed in [1].}
\hangindent= 1.5 true cm
\hangindent= - 1.5 true cm
\hangafter= - 2
\noindent
{\eightbf Key words}{\eightrm: Pure Gibbs state, Ising-Bethe model,
spin-glass.}
\vskip 2 truecm
\noindent
{\bf 1.}
The question about the possibility of the disordered limiting Gibbs state
(i.e., with zero effective field) of the ferromagnetic Ising model
on the Bethe lattice, to be pure below the
ferromagnetic critical temperature
$T_c$ has
been formulated
as an open problem in [2,3] (some non--homogeneous pure Gibbs states
were constructed in [4,5]). In [1] it was proved that the disordered
phase is pure up to the spin--glass critical temperature $T_c^{SG}$,
but as was found out later, in the correspondence of the author
with Hans-Otto Georgii, in the recursive formula used in [1] for the
probability measures $\nu_x$ [see the formula (3.7) below], the
multiplier $a(h_v)$ was missed both in numerator and denominator.
A corrected version of the proof [6] was based on an
inequality valid
for the Bethe lattice $\tau^k$ of degree $k \leq 6$.
In the present note we give a new proof valid for all $k$.
\vskip 2mm
\noindent
{\bf 2.}
Let $\tau^k$ be the Bethe lattice of degree $k \geq 2$ such that exactly
$(k+1)$ edges
come out of any of it vertex.
Then the Ising model is defined by the Hamiltonian
$$
H(\sg)
=
- \sum_{ x,y }
J_{xy} \; \sg (x) \sg (y)
\eqno(2.1)
$$
where the sum is over pairs of nearest neighbors $x,y$ and $\sg (x) = \pm
1$
The ferromagnetic model corresponds to $J_{xy} = J > 0$, while the
spin-glass model
corresponds to $J_{xy} = \pm J$ $(J > 0)$, where $\{ J_{xy} \}$ are
independent random variables with $\Pr\{J_{xy} = +J\} =\Pr\{J_{xy} = -J\}
=1/2$ for any pair $xy$.
Denote
$$
\theta = \tanh \b,\qquad \b=J / k_B T,
\eqno(2.2)
$$
where $T$ is the temperature.
Then for the ferromagnetic model the critical value $\theta^F_c = 1/k$
(see e.g. [3]), while
for the spin-glass model $\theta^{SG}_c = 1/\sqrt{k}$, see e.g. [7].
The main result of this note is the following theorem
\vskip 2mm
\noindent
{\bf Theorem.}
\it
The limiting Gibbs state with zero
effective field
(disordered phase) is pure for $0 < \theta \leq \theta ^{SG}_c$.
\rm
\vskip 2mm
\noindent
{\bf Remark.}
For
$0 < \theta \leq \theta ^{F}_c$ the limiting Gibbs state is unique, and so
trivially pure.
On the other hand, for $\theta > 1/\sqrt{k}$ the disordered phase is not
pure [2,4]. Thus
our main result concerns the interval $[1/k, 1/\sqrt{k}]$.
\vskip 2mm
\noindent
{\bf 3.}
Let $\tau^k=(V,L,i)$ be the Cayley tree of order $k$ with a root vertex
$x_*\in V$. Here $V$ is
the set of vertices, $L$ is the set of edges and $i$ is the incidence
function which
corresponds to each edge $l\in L$ its end-points $x_1,x_2\in V$. There is a
distance $d(x,y)$
on $V$ which is the length of the minimal path from $x$ to $y$, assuming
that the length of
any edge is 1. Denote by
$$
W_n=\{x\in V: \; d(x_*,x)=n\}
$$
the sphere of radius $n$ on $V$, and by
$$
V_n=\{x\in V: \; d(x_*,x)\le n\}
$$
the ball of radius $n$, so that
$$
V_n=\cup_{m=0}^n W_m.
$$
For any $x\in W_n,\; n=0,1,2,\dots,$ denote by
$$
S(x)=\{y\in W_{n+1}:\; d(x,y)=1\}.
$$
Let $\{h(x),\; x\in V\}$ be a set of real numbers satisfying for each $x\in
V$ the recursive
relation
$$
h(x)=\sum_{y \in S(x)} f_{\t}(h(y)),
\quad f_\t(h)={\rm artanh}\,(\theta\tanh(h)).
\eqno (3.1)
$$
Define the Gibbs probability distribution on the configurational space
$$
\Sigma(V_n)=\{\sigma_n=\{\sigma(x)=\pm 1,\; x\in V_n\}\}
$$
by the formula
$$
\mu_n(\sg_n)=Z_n^{-1}
\exp\left(\b\sum_{x,y\in V_n:\, d(x,y)=1}\sg(x)\sg(y) +\sum_{x\in W_n}
h(x)\sg(x)\right).
\eqno (3.2)
$$
Then (3.1) implies the consistency of $\mu_n$ for different $n$ so that
there exists a Gibbs measure $\mu$ on the infinite configurational space
$$
\Sigma(V)=\{\sigma=\{\sigma(x)=\pm 1,\; x\in V\}\}
$$
whose finite--dimensional distributions are $\mu_n$.
The function $h(x),\; x\in V,$ which satisfies (3.1), is called an
effective field.
Remark, that $h(x)\equiv 0$ satisfies (3.1), so that
the finite--dimensional distributions
$$
\mu_n^\#(\sg_n)=Z_n^{-1}
\exp\left(\b\sum_{x,y\in V_n:\, d(x,y)=1}\sg(x)\sg(y)
\right)
\eqno (3.3)
$$
are consistent and generate a Gibbs measure $\mu^\#$, the disordered phase.
To prove that $\mu^\#$ is pure we will prove that the spins
$\{\sg(x),\; x\in V_n\}$ and $\{\sg(x),\; x\in W_N\}$ are asymptotically
independent with respect to $\mu^\#$, when $N\to\infty$ and $n$ is fixed.
To that end we first fix $N>0$ and
define recursively for every $x\in V_{N-1}$ a random variable $h_x$ such
that
$$
h_x=\sum_{y\in S(x)} f_\t(h_y),\qquad \forall\, x\in V_{N-2},
\eqno (3.4)
$$
with the initial data
$$
h_x=\b\sum_{y\in S(x)} \sg(y),\qquad \forall\, x\in W_{N-1}.
\eqno (3.5)
$$
Eqs. (3.4), (3.5) define $h_x$ as a function of $\sg(y)$,
$y\in W_N\cap V_x$, where $V_x$ is the subtree growing from $x$.
It is to be noted that in general the random variables
$\{h_x,\; x\in V_{N-1}\}$ depend on $N$.
\vskip 3mm
\noindent
{\bf Lemma 3.1.}
{\it For every $n\le N-1$, the joint distribution of $\sg_n=\{\sg(x),\;x\in
V_n\}$ and
$h^{(n)}=\{h_x,\; x\in W_n\}$ with respect to $\mu^\#$ is given by
$$
\mu^\#(\sg_n,h^{(n)})=Z^{-1}\exp\left(-H_n(\sg_n) +
\sum_{x\in W_n}h_x\sg(x)\right)
\prod_{x\in W_n}\nu_x(h_x),
\eqno (3.6)
$$
where the probability distributions $\nu_x(h)$ are defined by the recursive
equations
$$
\nu_{x}(h)={\di\sum_{h_y,\dots,h_z:\,
f(h_y)+\dots+f(h_z)=h}
\left(\prv\right)\over
\di\sum_{h_y,\dots,h_z} \left(\prv\right)}\,, \qquad \forall\,x\in V_{N-2},
\eqno (3.7)
$$
where
$$
S(x)=(y,\dots,z),\qquad f(h)=f_\t(h),
\qquad a(h)=[1+(1-\t^2)\sinh^2h]^{1/2},
\eqno (3.8)
$$
and by the initial condition
$$
\nu_{x}(h)=Z^{-1}\sum_{\sg(y),\dots,\sg(z):\,
\b\,[\sg(y)+\dots+\sg(z)]=h} 1\,,
\qquad \forall\,x\in W_{N-1}.
\eqno (3.9)
$$}
\noindent
{\bf Remark.} Lemma 3.1 allows the following extension.
Let $\mu$ be a Gibbs measure generated,
according to (3.2),
by some set of real numbers $\{ h(x),\; x\in V\}$ which
satisfy the consistency equations (3.1), and let $\{ h_x,\;
x\in W_{N-1}\}$ be random variables defined through (3.4) and
(3.5).
Then the joint distribution $\mu (\sg_n,h^{(n)})$ has the
form (3.6) with probability distributions $\nu_x(h)$ defined by
the recursive equations (3.6) and by the initial condition
$$
\nu_{x}(h)=Z^{-1}\sum_{\b\,[\sg(y)+\dots+\sg(z)]=h}
\exp [h(y)\sg(y)+\dots+h(z)\sg(z)]\,,
\qquad \forall\,x\in W_{N-1}.
\eqno (3.10)
$$
When $h(x)\equiv 0$, (3.10) reduces obviously to (3.9).
If, in addition, the coupling constant $J_{xy}$ depend on
$xy$ then again the formulas (3.2)--(3.8), (3.10) are valid with
$$
\b=J_{xy}/k_BT,\qquad
\t=\tanh(J_{xy}/k_BT),
$$
and some natural modifications in these formulas.
The proof of this extension of
Lemma 3.1 is similar to the proof of Lemma 3.1.
\vskip 2mm
\noindent
{\it Proof of Lemma 3.1.} The proof of (3.7) is based on the
following identity:
$$
\sum_{\sg(y)=\pm 1}\exp [\b\sg(x)\sg(y)+h_y\sg(y)]
=Z\,a(h_y)\exp\left[f_\t(h_y)\,\sg(x)\right],\quad {\rm if}\quad\sg(x)=\pm
1,
\eqno (3.11)
$$
where $Z=\cosh\b$.
To prove this indentity notice that according to $\sg(x)=\pm 1$, it reduces
to
two identities:
$$\eqalign{
&\exp(\b+h_y)+\exp(-\b-h_y)=Z\,a(h_y)\exp f_\t(h_y),\cr
&\exp(-\b+h_y)+\exp(\b-h_y)=Z\,a(h_y)\exp [-f_\t(h_y)].\cr}
$$
If we divide the first identity by the second one then we obtain
$$
{\exp(\b+h_y)+\exp(-\b-h_y)\over
\exp(-\b+h_y)+\exp(\b-h_y)}=\exp [2f_\t(h_y)],
$$
which follows easily from the formula $\exp[2\artanh z]=(1+z)/(1-z)$.
If we multiply the first identity by the second one then we obtain
$$
\cosh^2\b+\sinh^2 h_y=Z^2a^2(h_y),
$$
which follows easily from the formula $a^2(h)=1+\sinh^2h/\cosh^2\b$.
Thus (3.11) is proved. Multiplying (3.11) over $y\in S(x)$ we obtain (3.7)
by induction.
(3.9) follows directly from (3.3). Lemma 3.1 is proved.
$\diamondsuit$
\vskip 3mm
\noindent
{\bf 4.}
By (3.9) and (3.7) the measures $\nu_x(h)$ coincide for all
$x\in W_n$ so we will denote them by $\nu_n(h)$. By (3.9) and (3.7)
$\nu_n(h)$ is symmetric, hence
$$
\sum_h\sinh h\,\nu_n(h)=0.
$$
Let
$$
D_n=\sum_h\sinh^2 h\,\nu_n(h).
$$
\vskip 2mm
\noindent
{\bf Lemma 4.1.} {\it If $k\t^2\le 1$ then
$$
D_{n-1}\le k\t^2D_n.
\eqno (4.1)
$$}
\vskip 2mm
\noindent
{\it Proof.} It is convenient to rewrite (3.4) and (3.7) in the
form
$$
h=\sk f(h_j),\qquad f(h)=f_\t(h),
$$
and
$$
\nu_{n-1}(h)=
{\di\sum_{h_1,\dots,h_k:
f(h_1)+\dots+f(h_k)=h} \left(\pr\right)\over \di\sum_{h_1,\dots,h_k}
\left(\pr\right)}. \eqno (4.2)
$$
Denote
$$
s=\sinh h,\qquad s_j=\sinh h_j,\qquad t_j=\tanh h_j.
$$
We have
$$
s^2=\sinh^2h=\sinh^2(\ff)={\tanh^2(\ff)\over 1-\tanh^2(\ff)},
$$
and
$$
\tanh(x_1+\dots+x_k)=
{\sum_{{\rm odd}\; p}
\sum_{j_10,\qquad \sum_{x} c(x)d(x)>0.}
$$
Then
$$
{\sum_{x} b(x)c(x)d(x)\over
\sum_{x} c(x)d(x)}
\ge
{\sum_{x} b(x)d(x)\over
\sum_{x} d(x)}.\eqno (4.4)
$$}
\vi
{\it Proof.} We have:
$$\eqalign{
0&\le \sum_{x_1,x_2}[b(x_1)-b(x_2)]\,[c(x_1)-c(x_2)]d(x_1)
d(x_2)\cr
&=2\sum_{x_1}b(x_1)c(x_1)d(x_1)\sum_{x_2}d(x_2)
-2\sum_{x_1} b(x_1)d(x_1)\sum_{x_2} c(x_2)d(x_2),\cr
}$$
which implies (4.4). $\diamondsuit$
\vskip 2mm
Applying FKG inequality to the RHS of (4.3), we obtain
$$
D_{n-1}\le
{\di\sum_{h_1,\dots,h_k}
\left(
\sum_{{\rm odd}\; p}\t^{2p}
\sum_{j_10$,
$$
A(\t)=
{\di\sum_{{\rm odd}\; p}C_k^p\t^{2p}D_n^p(1+D_n)^{k-p} \over
k\t^2D_n[1+(1-\t^2)D_n]^k}
={\di\sum_{{\rm odd}\; p}C_k^p\t^{2(p-1)}D_n^{p-1} (1+D_n)^{k-p}
\over
k[1+(1-\t^2)D_n]^k}\,
$$
Then (4.6) is equivalent to
$$
A(\t)\le 1.
$$
Observe that $A(\t)$ is an increasing function of $\t$, so it is sufficient
to prove that
$$
A(k^{-1/2})\le 1.
\eqno (4.7)
$$
By the Newton binomial formula,
$$
\eqalign{
A(k^{-1/2})&=
{\di\sum_{{\rm odd}\; p}C_k^p k^{-p}D_n^p(1+D_n)^{k-p} \over
D_n[1+(1-k^{-1})D_n]^k}\cr
&={(1+D_n+k^{-1}D_n)^k
-(1+D_n-k^{-1}D_n)^k
\over
2D_n[1+(1-k^{-1})D_n]^k}\cr
&={\di\left(
{1+(1+k^{-1})D_n
\over
1+(1-k^{-1})D_n}\right)^k-1
\over
2D_n}\,.
\cr}
\eqno (4.8)
$$
Put
$$
z=
{1+(1+k^{-1})D_n
\over
1+(1-k^{-1})D_n}-1
={2D_n\over k+(k-1)D_n}\,.
$$
Then
$$
0From (4.5) and Lemma 4.3 we obtain (4.1). Lemma 4.1 is proved.
$\diamondsuit$
Now we observe that (4.9) can be strengthened to
$$
kz+{k(k-1)\over 2}z^2+{k(k-1)(k-2)\over 6}z^3+\dots
\le { kz+\di{k(k-1)\over 2}z^2+{k(k-1)^2\over 4}z^3+\dots \over
1+\di{(k-1)^2\over 12}\,z^2}\,,
$$
which leads to
$$
(1+z)^k-1\le{2D_n\over
\di 1+{(k-1)^2\over 12}\,\left( {2D_n\over k+(k-1)D_n}\right)^2}
$$
and then we get the following
\vskip 3mm
\noindent
{\bf Lemma 4.4.}
$$
D_{n-1}\le {k\t^2 D_n
\over
\di 1+{(k-1)^2\over 3}\,\left({D_n\over k+(k-1)D_n} \right)^2}\,.
\eqno (4.10)
$$
This is useful when $k\t^2=1$.
\vskip 3mm
\noindent
{\bf 5.} {\it Proof of the Theorem.}
The inequality (4.10) says that
$$
\lim_{N-n \to \infty} D_n =0
\eqno (5.1)
$$
for $ 0< \theta \leq 1/ \sqrt{k}$.
Following the line of reasoning of Section 3 from [1], one can now check
that (5.1) implies the extremality of the disordered state $\mu^\#$.
This means that for any $\varepsilon >0$, $n>0$ and any configuration
$\sg_n \subset \Sigma(V_n)$,
there exist $N>n$ and a set $\Omega_N \subset \Sigma(W_N) $ such that
\vskip 2mm
\begingroup
\item{(i)}
$\mu^\#(\Omega_N) > 1- \varepsilon$
\item{(ii)}
$\vert \mu^\#(\sg_n \mid \sg^{(N)}) -\mu^\#(\sg_n) \vert < \varepsilon$,
$\forall \sg^{(N)} \in \Omega_N$
\endgroup
\vskip 2mm
\noindent
This gives our main result. $\diamondsuit$
One of the open problems remains about the purity of the limiting Gibbs
states in the case of nonzero external field.
Another open problem is the characterization of the limiting states for
the random field model.
The ground states for this model for binary distribution were examined in
[8].
\vskip 3mm
\noindent
\noindent {\bf Acknowledgements.}
One of us (P.M.B.) would like to express his sincere gratitude
to Hans-Otto Georgii for his help in finding out that in [1]
the multipliers $a(h_v)$ were missed
in the recursive formula for the probability measures $\nu_x$
(cf. the formula (3.7) above).
A corrected version of the proof for $k\le 6$ was given then
in the letter of P.M.B. to H.-O. Georgii [6] and the general case
is considered here. P.M.B. thanks
the Centre de Physique
Th\'eorique--Marseille for
hospitality and the Universit\'e de Provence for financial support
during his visit to CPT--Marseille where this work was done.
\vskip 3mm
\noindent
\noindent {\bf References.}
\medskip
\begingroup
\item{1.} P.M. Bleher,
{\it Extremity of the disordered phase in the Ising model on the Bethe
lattice},
Commun. Math. Phys. {\bf 128}, 411-419 (1990).
\smallskip
\item{2.} T. Moore, J.L. Snell,
{\it A branching process showing a phase transition},
J. Appl. Prob. {\bf 16}, 252-260 (1979).
\smallskip
\item{3.} H.-O. Georgii,
{\it Gibbs Measures and Phase Transitions}
(De Gruyter Studies in Math., vol. 9)
Berlin, New York: De Gruyter 1988.
\smallskip
\item{4.}
Y. Higuchi,
{\it Remarks on the limiting Gibbs states on a ($d+1$)-tree},
Publ. RIMS Kyoto University
{\bf 13}, 335-348 (1977).
\smallskip
\item{5.} P.M. Bleher and N.N. Ganihodgaev,
{\it On pure phases of the Ising model on the Bethe lattice},
Theory Prob. Appl. {\bf 35}, 216-227 (1990).
\smallskip
\item{6.} P.M. Bleher,
{\it Letter to H.-O. Georgii from January 21, 1992}.
\smallskip
\item{7.} J.M. Carlson, J.T. Chayes, L. Chayes, J.P. Sethna, D.J. Thouless,
{\it Critical behavior of the Bethe lattice spin-glass},
Europhys. Lett. {\bf 5}, 355-360 (1989).
\smallskip
\item{8.} R. Bruinsma,
{\it Random field Ising model on a Bethe lattice},
Phys. Rev. B {\bf 30}, 289-299 (1984).
\endgroup
\end