Condensed Matter Physics, 2010, vol. 13, No. 4, p. 43601:1-12

Title: Gibbs states of lattice spin systems with unbounded disorder
  Yu. Kondratiev (Fakultät für Mathematik, Universität Bielefeld, D-33615 Bielefeld, Germany) ,
  Yu. Kozitsky (Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland) ,
  T. Pasurek (Fakultät für Mathematik, Universität Bielefeld, D-33615 Bielefeld, Germany)

The Gibbs states of a spin system on the lattice Zd with pair interactions Jxyσ(x) σ(y) are studied. Here <x,y> ∈ 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 JqRE and SpRZd and show that, for every J = (Jxy)∈Jq: (a) the set of Gibbs states Gp(J) = {μ: solves DLR, &mu(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 ΔnZd, 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&Sigman=1NδπΔn(·|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

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