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Metastable states, quasi-stationary distributions and soft measures

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  • Bianchi, Alessandra
  • Gaudillière, Alexandre

Abstract

We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypotheses for Markov chains on a finite configuration space in some asymptotic regime. By comparing restricted ensembles and quasi-stationary measures, and introducing soft measures as an interpolation between the two, we prove an asymptotic exponential exit law and, on a generally different time scale, an asymptotic exponential transition law. By using potential-theoretic tools, and introducing “(κ,λ)-capacities”, we give sharp estimates on relaxation time, as well as mean exit time and transition time. We also establish local thermalization on shorter time scales.

Suggested Citation

  • Bianchi, Alessandra & Gaudillière, Alexandre, 2016. "Metastable states, quasi-stationary distributions and soft measures," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1622-1680.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:6:p:1622-1680
    DOI: 10.1016/j.spa.2015.11.015
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    References listed on IDEAS

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    1. Gaudillière, A. & den Hollander, F. & Nardi, F.R. & Olivieri, E. & Scoppola, E., 2009. "Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 737-774, March.
    2. Beltrán, J. & Landim, C., 2011. "Metastability of reversible finite state Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1633-1677, August.
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    Cited by:

    1. Baldassarri, Simone & Gallo, Anna & Jacquier, Vanessa & Zocca, Alessandro, 2023. "Ising model on clustered networks: A model for opinion dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 623(C).
    2. Baldassarri, Simone & Nardi, Francesca R., 2022. "Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 107-144.
    3. Bet, Gianmarco & Gallo, Anna & Nardi, F.R., 2024. "Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics," Stochastic Processes and their Applications, Elsevier, vol. 172(C).

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