Title: Gibbs states of lattice spin systems with unbounded disorder
Author(s):

The Gibbs states of a spin system on the lattice Z^d with pair interactions J_xyσ(x) σ(y) are studied. Here <x,y> ∈ E, i.e. x and y are neighbors in Z^d. The intensities J_xy and the spins σ(x), σ(y) are arbitrarily real. To control their growth we introduce appropriate sets J_q⊂R^E and S_p⊂R^Zd and show that, for every J = (J_xy)∈J_q: (a) the set of Gibbs states G_p(J) = {μ: solves DLR, &mu(S_p) = 1} is non-void and weakly compact; (b) each μ∈G_p(J) obeys an integrability estimate, the same for all μ. Next we study the case where J_q is equipped with a norm, with the Borel σ-field B(J_q), and with a complete probability measure ν. We show that the set-valued map J_q∋J → G_p(J) has measurable selections J_q∋J → μ(J) ∈G_p(J), which are random Gibbs measures. We demonstrate that the empirical distributions N^-1Σ_n=1^Nπ_Δn(·|J,ξ), obtained from the local conditional Gibbs measures π_Δn(·|J,ξ) and from exhausting sequences of Δ_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&Sigma_n=1^Nδ_{πΔn(·|J,ξ)}, which are Aizenman-Wehr metastates. Finally, we demonstrate that the limiting thermodynamic pressure exists under some further conditions on ν.

Key words: Aizenman-Wehr metastate, Newman-Stein empirical metastate, chaotic size dependence, Komlós theorem, quenched pressure, spin glass
PACS: 61.43.Fs, 64.60.De, 64.70.kj