Gibbs Measures of Disordered Lattice Systems with Unbounded Spins

The Gibbs measures of a spin system on Z(d) with pair interactions J(xy)sigma(x)sigma(y) are studied. Here <{x, y > is an element of 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 Jq subset of R-E and S-p subset of R-Zd and prove that for every J = (J(xy)) is an element of 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 is an element of G(p)(J) obeys an integrability estimate, the same for all mu. Next we equip J(q) 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(q) epsilon J -> G(p)(J) is measurable and hence there exist measurable selections Jq is an element of J -> mu(J) is an element of G(p)(J), which are random Gibbs measures. We prove that the empirical distributions N-1 Sigma(n)(n=1) pi Delta(n) (.vertical bar J, xi), obtained from the local conditional Gibbs measures pi(Delta n) (.vertical bar J, xi) and from exhausting sequences of Delta(n) subset of Z(d), have v-a.s, weak limits as N -> +infinity, which are random Gibbs measures. Similarly, we prove the existence of the v-a.s. weak limits of the empirical metastates N-1 Sigma(N)(n=1) delta(pi Delta n) (.vertical bar j, xi), which are Aizenman-Wehr metastates. Finally, we prove the existence of the limiting thermodynamic pressure under some further conditions on v. The proof is based on a generalization of the Contucci-Lebowitz. inequality which we obtain for our model.