Abstract
We consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in the low temperature limit. We generalize previous model-independent results in several directions. First, we give an estimate of the mixing time of the dynamics in terms of the maximal stability level. Second, assuming the dynamics is reversible, we give an estimate of the associated spectral gap. Third, we give precise asymptotics for the expected transition time from any metastable state to the stable state using potential-theoretic techniques. We do this in a general reversible setting where two or more metastable states are allowed and some of them may even be degenerate. This generalizes previous results that hold for a series of only two metastable states. We then focus on a specific Probabilistic Cellular Automata (PCA) with configuration space {mathcal {X}}={-1,+1}^varLambda where varLambda subset {mathbb {Z}}^2 is a finite box with periodic boundary conditions. We apply our model-independent results to find sharp estimates for the expected transition time from any metastable state in {underline{-1}, {underline{c}}^o,{underline{c}}^e} to the stable state underline{+1}. Here {underline{c}}^o,{underline{c}}^e denote the odd and the even chessboard respectively. To do this, 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.
Highlights
Metastability is a phenomenon that occurs when a physical system is close to a first order phase transition
Metastability occurs in several physical situations and this has led to the formulation of numerous models for metastable behavior
In order to find sharp estimates of the transition time from −1 to +1 for the Probabilistic Cellular Automata (PCA) model in Sect. 3.1, we extend the model-independent theorems given in [25], which hold for a series of two metastable states
Summary
Metastability is a phenomenon that occurs when a physical system is close to a first order phase transition. The authors use Sobolev inequalities to study the simulated annealing algorithm and they demonstrate that this approach gives detailed information about the rate at which the process is tending to its ground state Thanks to this result, the mixing time is estimated for Metropolis dynamics in [46, Proposition 3.24]. In [10] the authors focus on the connection between metastability and spectral theory for the so-called generic Markov chains under the assumption of non-degeneracy They use spectral information to derive sharp estimates on the transition times. In order to study the PCA, we need to extend their estimates of the spectral gap to the case of degenerate in energy metastable states and to a model with the Hamiltonian that depends on the asymptotic.
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