### Gibbs Measures of Disordered Lattice Systems with Unbounded Spins

Kondratiev Y, Kozitsky Y, Pasurek T (2012)
Markov Processes And Related Fields 18(3): 553-582.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

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Einrichtung
Abstract / Bemerkung
The Gibbs measures of a spin system on Z(d) with pair interactions J(xy)sigma(x)sigma(y) are studied. Here <{x, y > is an element of E, i.e. x and y are neighbors in Z(d). The intensities J(xy) and the spins sigma(x), sigma(y) are arbitrary real. To control their growth we introduce appropriate sets Jq subset of R-E and S-p subset of R-Zd and prove that for every J = (J(xy)) is an element of J(q): (a) the set of Gibbs measures G(p)(J) = (mu : solves DLR, mu(S-p) = 1} is non-void and weakly compact; (b) each mu is an element of G(p)(J) obeys an integrability estimate, the same for all mu. Next we equip J(q) with a norm, with the Borel sigma-field B(J(q)), and with a complete probability measure nu. We show that the set-valued map J(q) epsilon J -> G(p)(J) is measurable and hence there exist measurable selections Jq is an element of J -> mu(J) is an element of G(p)(J), which are random Gibbs measures. We prove that the empirical distributions N-1 Sigma(n)(n=1) pi Delta(n) (.vertical bar J, xi), obtained from the local conditional Gibbs measures pi(Delta n) (.vertical bar J, xi) and from exhausting sequences of Delta(n) subset of Z(d), have v-a.s, weak limits as N -> +infinity, which are random Gibbs measures. Similarly, we prove the existence of the v-a.s. weak limits of the empirical metastates N-1 Sigma(N)(n=1) delta(pi Delta n) (.vertical bar j, xi), which are Aizenman-Wehr metastates. Finally, we prove the existence of the limiting thermodynamic pressure under some further conditions on v. The proof is based on a generalization of the Contucci-Lebowitz. inequality which we obtain for our model.
Stichworte
chaotic size dependence; unbounded random interaction; measure; random Gibbs; Aizenman-Wehr metastate; Newman-Stein empirical metastate; Komlos; theorem; quenched pressure; set-valued map; measurable selection
Erscheinungsjahr
2012
Zeitschriftentitel
Markov Processes And Related Fields
Band
18
Ausgabe
3
Seite(n)
553-582
ISSN
1024-2953
Page URI
https://pub.uni-bielefeld.de/record/2544310

### Zitieren

Kondratiev Y, Kozitsky Y, Pasurek T. Gibbs Measures of Disordered Lattice Systems with Unbounded Spins. Markov Processes And Related Fields. 2012;18(3):553-582.
Kondratiev, Y., Kozitsky, Y., & Pasurek, T. (2012). Gibbs Measures of Disordered Lattice Systems with Unbounded Spins. Markov Processes And Related Fields, 18(3), 553-582.
Kondratiev, Y., Kozitsky, Y., and Pasurek, T. (2012). Gibbs Measures of Disordered Lattice Systems with Unbounded Spins. Markov Processes And Related Fields 18, 553-582.
Kondratiev, Y., Kozitsky, Y., & Pasurek, T., 2012. Gibbs Measures of Disordered Lattice Systems with Unbounded Spins. Markov Processes And Related Fields, 18(3), p 553-582.
Y. Kondratiev, Y. Kozitsky, and T. Pasurek, “Gibbs Measures of Disordered Lattice Systems with Unbounded Spins”, Markov Processes And Related Fields, vol. 18, 2012, pp. 553-582.
Kondratiev, Y., Kozitsky, Y., Pasurek, T.: Gibbs Measures of Disordered Lattice Systems with Unbounded Spins. Markov Processes And Related Fields. 18, 553-582 (2012).
Kondratiev, Yuri, Kozitsky, Yuri, and Pasurek, Tatiana. “Gibbs Measures of Disordered Lattice Systems with Unbounded Spins”. Markov Processes And Related Fields 18.3 (2012): 553-582.

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