Metastability of reversible finite state Markov processes

Beltrán, Johel; Landim, Cláudio

doi:10.1016/j.spa.2011.03.008

Cited by 33 publications

(69 citation statements)

References 24 publications

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“…In the present paper we study the low-temperature behavior of the Potts model using the pathwise approach (see [58] for a sistematic overview and further references) and its more recent extensions [24,53,56], but also other techniques have been successfully used in the literature to study tunneling and metastability phenomena, e.g. the potential theoretical approach (introduced in [17], for an overview see the recent book [15]) and the martingale approach [7,8,9].…”

Section: Related Results and Discussionmentioning

confidence: 99%

Tunneling behavior of Ising and Potts models in the low-temperature regime

Nardi

Zocca

2019

Stochastic Processes and their Applications

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We consider the ferromagnetic q-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature β. Our analysis concerns the low-temperature regime β → ∞, in which this multi-spin system has q stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the q-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as β → ∞ in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result. In the special case q = 2, our results characterize the tunneling behavior of the Ising model on grid graphs.

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Section: Related Results and Discussionmentioning

confidence: 99%

Tunneling behavior of Ising and Potts models in the low-temperature regime

Nardi

Zocca

2019

Stochastic Processes and their Applications

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“…x , are cycles while the set of the following partitions are cycles of cycles. These results have been proved in [3] for finite state reversible Markovian dynamics. We remove in this article the assumption of reversibility and we simplify some proofs.…”

Section: Introductionmentioning

confidence: 80%

“…We present in this section some estimates for the capacity of reversible, finite state Markov chains obtained in in [3]. There are useful below since se proved in Corollary 4.4 that the capacity between two disjoint subsets A, B of E is of the same order as the capacity with respect to the reversible chain.…”

Section: Reversible Chains and Capacitiesmentioning

confidence: 99%

“…Not long ago, Beltrán and one of the authors introduced a general approach to derive the metastable behavior of continuous-time Markov chains, particularly convenient in the presence of several valleys with the same depth [2,4,5]. In the context of finite state Markov chains [3], it permits to identify the slow variables and to reduce the model and the state space.…”

Section: Introductionmentioning

confidence: 99%

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Metastability of finite state Markov chains: a recursive procedure to identify slow variables for model reduction

Landim¹,

Xu²

2016

ALEA

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“…Very recently, Beltran and Landim [5] describe for the continuous time Markov process of the Ising model all metastable behaviors, defining time scales at which they occur, the metastable set associated to each time scale, and the asymptotic dynamics which specifies at which rate the process jumps from one metastable state to another.…”

Section: Introductionmentioning

confidence: 99%

Metastability of Logit Dynamics for Coordination Games

Auletta

Ferraioli

Pasquale

et al. 2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. This process defines an ergodic Markov chain, over the set of strategy profiles of the game, whose unique stationary distribution is the long-term equilibrium concept for the game. However, when the mixing time of the chain is large (e.g., exponential in the number of players), the stationary distribution loses its appeal as equilibrium concept, and the transient phase of the Markov chain becomes important. In several cases it happens that on a time-scale shorter than mixing time the chain is "quasistationary", meaning that it stays close to some small set of the state space, while in a time-scale multiple of the mixing time it jumps from one quasi-stationary configuration to another; this phenomenon is usually called "metastability".In this paper we give a quantitative definition of "metastable probability distributions" for a Markov chain and we study the metastability of the Logit dynamics for some classes of coordination games. In particular, we study no-risk-dominant coordination games on the clique (which is equivalent to the well-known Glauber dynamics for the Ising model) and coordination games on a ring (both the risk-dominant and norisk-dominant case). We also describe a simple "artificial" game that highlights the distinctive features of our metastability notion based on distributions.

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Metastability of reversible finite state Markov processes

Abstract: We prove the metastable behavior of reversible Markov processes on finite state spaces under minimal conditions on the jump rates. To illustrate the result we deduce the metastable behavior of the Ising model with a small magnetic field at very low temperature.

Cited by 33 publications

References 24 publications

Tunneling behavior of Ising and Potts models in the low-temperature regime

Tunneling behavior of Ising and Potts models in the low-temperature regime

Metastability of finite state Markov chains: a recursive procedure to identify slow variables for model reduction

Metastability of Logit Dynamics for Coordination Games

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