Metastability in stochastic dynamics of disordered mean-field models. (English) Zbl 1012.82015

Bovier, Anton; Eckhoff, Michael; Gayrard, Véronique; Klein, Markus

Probab. Theory Relat. Fields 119, No. 1, 99-161 (2001).

The paper is devoted to developing a systematic approach to obtain sharp estimates on the long time behaviour of certain classes of Markov chains. The main goal of the paper is to give a precise relation between the metastable time scales in the problem of the properties of the rate functions of the corresponding Gibbs measures.
It is shown that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of “admissible transitions”. For admissible transition upper and lower bounds on the expected transition times is given that differ only by a constant factor. It is shown that the distribution of the transition times is asymptotically exponential. The results of the paper for the random field Curie-Weiss model are illustrated.

Reviewer: Utkir Rozikov (Tashkent)

MSC:

82C44	Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60K35	Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

metastability; stochastic dynamics; Markov chains; Wentzell-Freidlin theory; disordered systems; mean field models; random field Curie-Weiss model