Gibbs Measures of Disordered Lattice Systems with Unbounded Spins

#### Yu.G. Kondratiev, Y. Kozitsky, T. Pasurek

2012, v.18, №3, 553-582

ABSTRACT

The Gibbs measures of a spin system on $Z^d$ with pair interactions $J_{xy} \sigma(x) \sigma(y)$ are studied. Here $\in E$, 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 $\mathcal{J}_q\subset R^{E}$ and $\mathcal{S}_p\subset R^{Z}^d$ and prove that for every $J = (J_{xy}) \in \mathcal{J}_q$: (a) the set of Gibbs measures $\mathcal{G}_p(J)= \{ \mu:$ solves DLR, $\mu(\mathcal{S}_p)=1\}$ is non-void and weakly compact; (b) each $\mu\in\mathcal{G}_p(J)$ obeys an integrability estimate, the same for all $\mu$. Next we equip $\mathcal{J}_q$ with a norm, with the Borel $\sigma$-field $\mathcal{B}(\mathcal{J}_q)$, and with a complete probability measure $\nu$. We show that the set-valued map $\mathcal{J}_q \ni J \mapsto \mathcal{G}_p(J)$ is measurable and hence there exist measurable selections $\mathcal{J}_q \ni J \mapsto \mu(J) \in \mathcal{G}_p(J)$, which are random Gibbs measures. We prove that the empirical distributions $$N^{-1} \sum_{n=1}^N \pi_{\mathit{\Delta}_n} (\cdot \mid J, \xi),$$ obtained from the local conditional Gibbs measures $\pi_{\mathit{\Delta}_n} (\cdot \mid J, \xi)$ and from exhausting sequences of $\mathit{\Delta}_n \subset \mathbb{Z}^d$, 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 $N^{-1}\sum_{n=1}^N\delta_{\pi_{\mathit{\Delta}_n}(\cdot \mid J,\xi)}$, which are Aizenman - Wehr metastates. Finally, we prove the existence of the limiting thermodynamic pressure under some further conditions on $\nu$. The proof is based on a generalization of the Contucci - Lebowitz inequality which we obtain for our model.

Keywords: Aizenman - Wehr metastate,Newman - Stein empirical metastate,random Gibbs measure,unbounded random interaction,chaotic size dependence,Komlos theorem,quenched pressure,set-valued map,measurable selection