Condensed Matter Physics, 2010, vol. 13, No. 4, p. 43601:112
DOI:10.5488/CMP.13.43601
Title:
Gibbs states of lattice spin systems with unbounded disorder
Author(s):

Yu. Kondratiev
(Fakultät für Mathematik, Universität Bielefeld, D33615 Bielefeld, Germany)
,


Yu. Kozitsky
(Instytut Matematyki, Uniwersytet Marii CurieSkłodowskiej, 20031 Lublin, Poland)
,


T. Pasurek
(Fakultät für Mathematik, Universität Bielefeld, D33615 Bielefeld, Germany)

The Gibbs states of a spin system on the lattice Z^{d} with pair interactions J_{xy}σ(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 nonvoid 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 σfield B(J_{q}), and with a complete probability measure ν. We show that the setvalued 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^{1}Σ_{n=1}^{N}π_{Δn}(·J,ξ), obtained from the local conditional Gibbs measures π_{Δn}(·J,ξ) and from exhausting sequences of Δ_{n}⊂Z^{d}, have νa.s. weak limits as N→+∞, which are random Gibbs measures. Similarly, we show the existence of the νa.s. weak limits of the empirical metastates N^{1}&Sigma_{n=1}^{N}δ_{πΔn(·J,ξ)}, which are AizenmanWehr metastates. Finally, we demonstrate that the limiting thermodynamic pressure exists under some further conditions on ν.
Key words:
AizenmanWehr metastate, NewmanStein empirical metastate, chaotic size dependence, Komlós theorem, quenched pressure, spin glass
PACS:
61.43.Fs, 64.60.De, 64.70.kj
