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Analogues of Non-Gibbsianness in Joint Measures of Disordered Mean Field Models

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Abstract

It is known that the joint measures on the product of spin-space and disorder space are very often non-Gibbsian measures, for lattice systems with quenched disorder, at low temperature. Are there reflections of this non-Gibbsianness in the corresponding mean-field models? We study the continuity properties of the conditional probabilities in finite volume of the following mean field models: (a) joint measures of random field Ising, (b) joint measures of dilute Ising, (c) decimation of ferromagnetic Ising. The conditional probabilities are functions of the empirical mean of the conditionings; so we look at the large volume behavior of these functions to discover non-trivial limiting objects. For (a) we find (1) discontinuous dependence for almost any realization and (2) dependence of the conditional probabilities on the phase. In contrast to that we see continuous behavior for (b) and (c), for almost any realization. This is in complete analogy to the behavior of the corresponding lattice models in high dimensions. It shows that non-Gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean-field models.

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  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117, Berlin, Germany

    Christof Külske

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Külske, C. Analogues of Non-Gibbsianness in Joint Measures of Disordered Mean Field Models. Journal of Statistical Physics 112, 1079–1108 (2003). https://doi.org/10.1023/A:1024615625364

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