Gibbs states of lattice spin systems with unbounded disorder

The Gibbs states of a spin system on the lattice Zd with pair interactions Jxyσ(x) σ(y) are studied. Here ∈ E, i.e. x and y are neighbors in Zd. The intensities Jxy and the spins σ(x), σ(y) are arbitrarily real. To control their growth we introduce appropriate sets Jq RE and Sp RZd 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-1Σn=1NπΔn(·|J,ξ), obtained from the local conditional Gibbs measures πΔn(·|J,ξ) and from exhausting sequences of Δn Zd, 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-1Σn=1NδπΔn(·|J,ξ), which are Aizenman-Wehr metastates. Finally, we demonstrate that the limiting thermodynamic pressure exists under some further conditions on ν.