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Gibbsian description of mean-field models. (English) Zbl 1151.82338

Sidoravicius, Vladas (ed.) et al., In and out of equilibrium 2. Papers celebrating the 10th edition of the Brazilian school of probability (EBP), Rio de Janiero, Brazil, July 30 to August 4, 2006. Basel: Birkhäuser (ISBN 978-3-7643-8785-3/hbk). Progress in Probability 60, 463-480 (2008).
Summary: We introduce a new framework to describe mean-field models in the spirit of the DLR description of probability measures on infinite prodnet probability spaces used for lattice spin systems. The approach, originally introduced by C. Kuelske in 2003, is inspired by the generalized Gibbsian formalism recently developed in the context of the Dobrushin program of restoration of Gibbsianness, and enables the recovery of many of its features in the mean-field context. It is based on a careful study of the continuity properties of the limiting conditional probabilities of the finite-volume mean-field measures as a function of, empirical averages, when the limiting procedure is properly done to avoid trivialities. This contribution is an extended version of a poster presented at the Xth Brazilian school of probability and has mainly a review character.
For the entire collection see [Zbl 1141.82002].

MSC:

82B23 Exactly solvable models; Bethe ansatz
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G09 Exchangeability for stochastic processes
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