We consider the problem of metastability for stochastic reversible dynamics with exponentially small transition probabilities. We generalize previous results in several directions. We give an estimate of the spectral gap of the transition matrix and of the mixing time of the associated dynamics in terms of the maximal stability level. These model-independent results hold in particular for a large class of Probabilistic Cellular Automata (PCA), which we then focus on. We consider the PCA in a finite volume, at small and fixed magnetic field, and in the limit of vanishing temperature. This model is peculiar because of the presence of three metastable states, two of which are degenerate with respect to their energy. We identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability. Further, we also identify the metastable states according to the potential-theoretic approach to metastability, and this allows us to give precise asymptotics for the expected transition time from any such metastable state to the stable state.

Effect of energy degeneracy on the transition time for a series of metastable states: application to Probabilistic Cellular Automata / Gianmarco Bet; Vanessa Jacquier; Francesca R. Nardi. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - ELETTRONICO. - (2021), pp. 1-44. [10.1007/s10955-021-02788-0]

Effect of energy degeneracy on the transition time for a series of metastable states: application to Probabilistic Cellular Automata

Gianmarco Bet
Membro del Collaboration Group
;
Vanessa Jacquier
Membro del Collaboration Group
;
Francesca R. Nardi
Membro del Collaboration Group
2021

Abstract

We consider the problem of metastability for stochastic reversible dynamics with exponentially small transition probabilities. We generalize previous results in several directions. We give an estimate of the spectral gap of the transition matrix and of the mixing time of the associated dynamics in terms of the maximal stability level. These model-independent results hold in particular for a large class of Probabilistic Cellular Automata (PCA), which we then focus on. We consider the PCA in a finite volume, at small and fixed magnetic field, and in the limit of vanishing temperature. This model is peculiar because of the presence of three metastable states, two of which are degenerate with respect to their energy. We identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability. Further, we also identify the metastable states according to the potential-theoretic approach to metastability, and this allows us to give precise asymptotics for the expected transition time from any such metastable state to the stable state.
2021
1
44
Goal 17: Partnerships for the goals
Gianmarco Bet; Vanessa Jacquier; Francesca R. Nardi
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1209239
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