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Metastability in stochastic dynamics of disordered mean-field models
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  • Published: January 2001

Metastability in stochastic dynamics of disordered mean-field models

  • Anton Bovier1,
  • Michael Eckhoff2,
  • Véronique Gayrard3 &
  • …
  • Markus Klein4 

Probability Theory and Related Fields volume 119, pages 99–161 (2001)Cite this article

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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.

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Authors and Affiliations

  1. Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany. e-mail: bovier@wias-berlin.de, , , , , , DE

    Anton Bovier

  2. Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. e-mail: meckhoff@math.uni-potsdam.de, , , , , , DE

    Michael Eckhoff

  3. Centre de Physique Théorique, CNRS, Luminy, Case 907, 13288 Marseille, Cedex 9, France. e-mail: Veronique.Gayrard@cpt.univ-mrs.fr, , , , , , FR

    Véronique Gayrard

  4. Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. e-mail: mklein@math.uni-potsdam.de, , , , , , DE

    Markus Klein

Authors
  1. Anton Bovier
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  2. Michael Eckhoff
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  3. Véronique Gayrard
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  4. Markus Klein
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Additional information

Received: 26 November 1998 / Revised version: 21 March 2000 / Published online: 14 December 2000

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Bovier, A., Eckhoff, M., Gayrard, V. et al. Metastability in stochastic dynamics of disordered mean-field models. Probab Theory Relat Fields 119, 99–161 (2001). https://doi.org/10.1007/PL00012740

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  • Issue Date: January 2001

  • DOI: https://doi.org/10.1007/PL00012740

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  • 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
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