Condensed Matter Physics (Jan 2010)
Gibbs states of lattice spin systems with unbounded disorder
Abstract
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.
