Gibbs states of lattice spin systems with unbounded disorder

Abstract

The Gibbs states of a spin system on the lattice Z d with pair interactions Jxyσ(x)σ(y) are studied. Here x, y ∈ E, i.e. x and y are neighbors in Z d . The intensities Jxy and the spins σ(x), σ(y) are arbitrarily real. To control their growth we introduce appropriate sets Jq ⊂ R E and Sp ⊂ R Z d and show that, for every J = (Jxy) ∈ Jq: (a) the set of Gibbs states Gp(J) = {µ : solves DLR, µ(Sp) = 1} is non-void and weakly compact; (b) each µ ∈ Gp(J) obeys an integrability estimate, the same for all µ. Next we study the case where Jq is equipped with a norm, with the Borel σ-field B(Jq), and with a complete probability measure ν. We show that the set-valued map Jq ∋ J → Gp(J) has measurable selections Jq ∋ J → µ(J) ∈ Gp(J), which are random Gibbs measures. We demonstrate that the empirical distributions N −1 N n=1 π ∆n (·|J, ξ), obtained from the local conditional Gibbs measures π ∆n (·|J, ξ) and from exhausting sequences of ∆n ⊂ Z d , have ν-a.s. weak limits as N → +∞, which are random Gibbs measures. Similarly, we show the existence of the ν-a.s. weak limits of the empirical metastates N −1 N n=1 δ π ∆n (·|J,ξ) , which are Aizenman-Wehr metastates. Finally, we demonstrate that the limiting thermodynamic pressure exists under some further conditions on ν.