Gibbs measures of disordered lattice systems with unbounded spins

Abstract

The Gibbs measures of a spin system on $Z^d$ with unbounded pair interactions $J_{xy} \sigma (x) \sigma (y)$ are studied. Here $\langle x, y \rangle \in E $, i.e. $x$ and $y$ are neighbors in $Z^d$. The intensities $J_{xy}$ and the spins $\sigma (x) , \sigma (y)$ are arbitrary real. To control their growth we introduce appropriate sets $J_q\subset R^E$ and $S_p\subset R^{Z^d}$ and prove that for every $J = (J_{xy}) \in J_q$: (a) the set of Gibbs measures $G_p(J)= \{\mu: solves DLR, \mu(S_p)=1\}$ is non-void and weakly compact; (b) each $\mu\inG_p(J)$ obeys an integrability estimate, the same for all $\mu$. Next we study the case where $J_q$ is equipped with a norm, with the Borel $\sigma$-field $B(J_q)$, and with a complete probability measure $\nu$. We show that the set-valued map $J \mapsto G_p(J)$ is measurable and hence there exist measurable selections $J_q \ni J \mapsto \mu(J) \in G_p(J)$, which are random Gibbs measures. We prove that the empirical distributions $N^{-1} \sum_{n=1}^N \pi_{\Delta_n} (\cdot| J, \xi)$, obtained from the local conditional Gibbs measures $\pi_{\Delta_n} (\cdot| J, \xi)$ and from exhausting sequences of $\Delta_n \subset Z^d$, have $\nu$-a.s. weak limits as $N\rightarrow +\infty$, which are random Gibbs measures. Similarly, we prove the existence of the $\nu$-a.s. weak limits of the empirical metastates $N^{-1} \sum_{n=1}^N \delta_{\pi_{\Delta_n} (\cdot| J,\xi)}$, which are Aizenman-Wehr metastates. Finally, we prove the existence of the limiting thermodynamic pressure under some further conditions on $\nu$.