Gibbs measures of disordered lattice systems with unbounded spins

The Gibbs measures of a spin system on $Z^d$ with unbounded pair interactions $J_{xy} σ(x) σ(y)$ 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 $σ(x) , σ(y)$ 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 $G_p(J)= \{μ: solves DLR, μ(S_p)=1\}$ is non-void and weakly compact; (b) each $μ\inG_p(J)$ obeys an integrability estimate, the same for all $μ$. Next we study the case where $J_q$ is equipped with a norm, with the Borel $σ$-field $B(J_q)$, and with a complete probability measure $ν$. We show that the set-valued map $J \mapsto G_p(J)$ is measurable and hence there exist measurable selections $J_q \ni J \mapsto μ(J) \in G_p(J)$, which are random Gibbs measures. We prove that the empirical distributions $N^{-1} \sum_{n=1}^N π_{Δ_n} (\cdot| J, ξ)$, obtained from the local conditional Gibbs measures $π_{Δ_n} (\cdot| J, ξ)$ and from exhausting sequences of $Δ_n \subset Z^d$, have $ν$-a.s. weak limits as $N\rightarrow +\infty$, which are random Gibbs measures. Similarly, we prove the existence of the $ν$-a.s. weak limits of the empirical metastates $N^{-1} \sum_{n=1}^N δ_{π_{Δ_n} (\cdot| J,ξ)}$, which are Aizenman-Wehr metastates. Finally, we prove the existence of the limiting thermodynamic pressure under some further conditions on $ν$.

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