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{\bf Weak versus strong uniqueness of Gibbs measures:}\\
{\bf a regular short-range example}\\
\mbox{}\\
\mbox{}\\
M. Campanino\\
{\em Dipartimento di Matematica}\\
{\em Universit\'a degli Studi di Bologna}\\
{\em piazza di Porta S. Donato 5}\\
{\em I-40127 Bologna}\\
{\em Italy}\\
A.C.D. van Enter\\
{\em Institute for Theoretical Physics}\\
{\em Nijenborgh 4}\\
{\em 9747 AG Groningen}\\
{\em The Netherlands}
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\vspace{3cm}
\noindent
{\bf Abstract:} We provide an example of a nearest neighbor random model on a
 regular lattice which has a unique (disordered) Gibbs state for every
 boundary condition, almost surely, although by choosing interaction-dependent
 boundary conditions one can obtain different Gibbs states.

\newpage

\noindent
The correct treatment and interpretation of boundary conditions for
 statistical mechanical systems is a subtle issue, especially for random
 models [COvE, vE1, vE2, vEF, vEG, FH1, FH2, FH3, GNS,  HF, N, NS, Oz, Z].
 For instance, the notion of a plurality of pure states with an ultrametric
 structure as in Parisi's proposal, as well as the mechanism by which they can
 be obtained, requires a careful treatment of boundary conditions. This issue
 is of great interest in spin-glass theory [MPV]. Another example is the
 distinction between ,,weak'' and ,,strong'' uniqueness of Gibbs measures for
 random spin systems. We say that strong uniqueness applies to a model if for
 almost all choices of the interaction only one Gibbs measure exists, while
 weak uniqueness holds if for any boundary condition chosen independently of
 the interaction, the thermodynamic limit of the Gibbs measure is the same for
 almost all interactions.  Weak uniqueness was first explicitly introduced and discussed
in [COvE] where it was proved to hold for one-dimensional Ising models with
random interactions decaying as $1/r^{\alpha}$ with $\alpha > 1$;
strong uniqueness is known to hold only for $\alpha > 3/2$ ([K]). The
 distinction between interaction-dependent versus interaction-independent
 boundary conditions was earlier discussed in [vEF, vEG, CCST], in a more
 implicit way.


Up till now there have been examples of systems which are weakly, but not
 strongly, unique for spin-glass models on the Bethe lattice ([CCST, BRZ]) in
 the temperature range between the ferromagnetic and the spin-glass transition
 temperatures and for extreme-long-range (square summable, but non-summable)
 spin-glass models ([GNS, FZ]) at high temperatures. The usual interpretation 
 of these results is that in these examples interaction-dependent boundary
 conditions should be dismissed as ,,unphysical''.

Here we present for the first time a nearest neighbor example on a regular
 lattice of this phenomenon. The interpretation of interaction-dependent
 boundary conditions as ,,unphysical'' seems somewhat tenuous in this case.

The model we consider is the $q$-state nearest-neighbor Potts model on ${\mathbb{Z}}^d$
 at $T_c$, for high $q$ (and $d \geq 2$) with a one-pattern Hopfield-type
site disorder
\[
H=- \sum_{\langle i,j \rangle} \delta (\xi_i \sigma_i, \xi_j \sigma_j)
\hspace{4cm} (1)
\]
where $\sigma_i \in \{ 1, ..., q\}$ and $\xi_i$ is a random permutation of the
 $q$ Potts states at site $i$ [Ka, Ga].

After a gauge transformation the model is equivalent to a Potts ferromagnet,
 for which it is well known [KS, BKL] that at $T_c$ there is a first-order
 phase transition in the temperature variable. At $T_c$ $q$ ordered states
 (Gibbs measures) coexist with a disordered one. Any fixed boundary condition,
 due to the random gauge transformation, is equivalent to taking a random
 choice for the boundary condition for the ferromagnet. Thus it is sufficient
 to argue that a sequence of finite volume Gibbs measures for the
 ferromagnetic model with randomly chosen boundary conditions converges to the
 disordered state. To do this we invoke the proof of the first-order
 transition due to Bricmont-Kuroda-Lebowitz [BKL]. This proof is based on
 Pirogov-Sinai theory, that is a sophisticated contour argument. The
 disordered state is a small perturbation of the ,,restricted ensemble'' of
 configurations in which all neighboring spins are different (small for $q$
 high enough). We observe that the density of pairs of neighboring spins on
 the boundary which are equal is close to  $\frac{1}{q}$ with large probability,
and they are ``sparse'', if the spins on the boundary
 are independently chosen on each boundary site, with probability
 $\frac{1}{q}$ for each of the $q$ Potts states. This means that the boundary
 condition is a small random perturbation of a ,,purely restricted ensemble''
 boundary condition. Therefore, a probabilistic contour argument, as in [CCF],
 will lead to the desired result, that almost surely with respect to the
 random boundary conditions the thermodynamic limit measure will be the
 disordered state. 

As we mentioned before, this implies the weak uniqueness of the
 gauge-transformed random Potts-Hopfield model.

Intuitively, it is of course plausible that uncorrelated random boundary
 conditions are the most ,,disordered'' ones. We note that for any extremal
 Gibbs measure it is true that almost all (with respect to this {\em same\/}
 Gibbs measure) boundary conditions recover the original measure [G]. What we
 have considered here, is almost all boundary conditions with respect to
 symmetric product measure which is the most random prescription in some sense
 (for example, this measure has the largest entropy).

\bigskip
\noindent
{\bf Acknowledgements:}\
This paper
was begun during a visit of M. C. to the university of Groningen. 
Part of the research of A. C. D. van Enter was made possible by a 
fellowship of the Royal Dutch Academy of Arts and Sciences (K.N.A.W.).
 This work was supported by EU-contract CHRX-CT93-0411.
\bigskip
\noindent
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