Gibbs states of lattice spin systems with unbounded disorder
Gibbs states of a spin system on the lattice 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 arbitrarily real. To control their growth we introduce appropriate sets I-q subset of R-E and S-p subset of R-zd and show that, for every J = (J(xy)) is an element of I-q : (a) the set of Gibbs states 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 study the case where I-q is equipped with a norm, with the Borel B(I-q), and with a complete probability measure v. We show that the set-valued map I-q (sic) J bar right arrow G(p) (J) has measurable selections I-q (sic) J bar right arrow mu(J) is an element of G(p)(J), which are random Gibbs measures. We demonstrate 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 show 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 demonstrate that the limiting thermodynamic pressure exists under some further conditions on v.
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INST CONDENSED MATTER PHYSICS NATL ACAD SCIENCES UKRAINE