Advertisement

Probability Theory and Related Fields

, Volume 119, Issue 1, pp 99–161 | Cite as

Metastability in stochastic dynamics of disordered mean-field models

  • Anton Bovier
  • Michael Eckhoff
  • Véronique Gayrard
  • Markus Klein

Abstract.

We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of “admissible transitions”. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor. The distributions of the rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.

Mathematics Subject Classification (2000): 82C44, 60K35 
Key words or phrases: Metastability – Stochastic dynamics – Markov chains – Wentzell-Freidlin theory – Disordered systems – Mean field models – Random field Curie–Weiss model 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Anton Bovier
    • 1
  • Michael Eckhoff
    • 2
  • Véronique Gayrard
    • 3
  • Markus Klein
    • 4
  1. 1.Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany. e-mail: bovier@wias-berlin.deDE
  2. 2.Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. e-mail: meckhoff@math.uni-potsdam.deDE
  3. 3.Centre de Physique Théorique, CNRS, Luminy, Case 907, 13288 Marseille, Cedex 9, France. e-mail: Veronique.Gayrard@cpt.univ-mrs.frFR
  4. 4.Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. e-mail: mklein@math.uni-potsdam.deDE

Personalised recommendations