Large Deviations and Metastability

Olivieri, E.; Vares, Maria Eulália

doi:10.1017/cbo9780511543272

Cited by 262 publications

(297 citation statements)

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“…The transition path theory [25,26] investigates the most probable paths that the Markov chain uses when travelling between different metastable states. Recent books on various aspects of metastability include the monograph [19], and the lecture notes [4].…”

Section: Introductionmentioning

confidence: 99%

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Betz

Roux

2016

Stochastic Processes and their Applications

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Abstract. We consider a simple but important class of metastable discrete time Markov chains, which we call perturbed Markov chains. Basically, we assume that the transition matrices depend on a parameter ε, and converge as ε → 0. We further assume that the chain is irreducible for ε > 0 but may have several essential communicating classes when ε = 0. This leads to metastable behavior, possibly on multiple time scales. For each of the relevant time scales, we derive two effective chains. The first one describes the (possibly irreversible) metastable dynamics, while the second one is reversible and describes metastable escape probabilities. Closed probabilistic expressions are given for the asymptotic transition probabilities of these chains, but we also show how to compute them in a fast and numerically stable way. As a consequence, we obtain efficient algorithms for computing the committor function and the limiting stationary distribution.

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Section: Introductionmentioning

confidence: 99%

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Betz

Roux

2016

Stochastic Processes and their Applications

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“…This restriction is in fact not technical: when we look at (3) on a time scale longer than log N , it may happen that the microscopic dynamics differs significantly from its macroscopic limit, due to the influence of the thermal noise and random environment. Various examples in this direction have already been considered in the literature: in case the mean-field PDE admits an isolated stable fixed point, the finite-size system exits from any neighborhood at exponential times in N [13,30] whereas the system escapes at a time or order log N in case of an unstable fixed point [32]. The Kuramoto model (2) corresponds to a situation where the mean-field PDE admits a whole stable manifold of stationary solutions.…”

Section: Perturbations Of Mean Field Particle Systemsmentioning

confidence: 99%

Interacting particle systems

et al. 2017

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Abstract. We present in this paper a number of recent and various works of statistical physics, which involve an interacting particle system. In kinetically constrained models, the particles are placed on Z d , with local constraints on the dynamics, that can slow down the evolution and reproduce the behaviour of glassy systems. The contact process (or Suspected-Infected-Susceptible) is a simple model for the spread of an infection on a (general) graph. In the Kuramoto model, the motion of the particles is determined by a diffusion system with mean-field interaction. Finally, in the modeling of the vertex reinforced jump process (a particle that interacts with itself via its path), we show an unexpected link with an interacting particle system. Résumé. Nous présentons dans ce papier plusieurs travaux récents et divers de physique statistique,qui mettent en jeu un système de particules en interaction. Dans les modèles cinétiquement contraints, les particules sont placées sur Z d avec des contraintes locales sur la dynamique qui peuvent ralentir celle-ci et reproduire le comportement des systèmes vitreux. Le processus de contact (ou SusceptibleInfected-Susceptible) modélise quant à lui la propagation d'une infection sur un graphe quelconque. Dans le modèle de Kuramoto, le mouvement des particules est déterminé par un système de diffusions, avec interaction de type champ moyen. Enfin, pour le processus de sauts renforcé par site, la particule interagit avec elle-même (avec sa trajectoire), la modélisation présentant un lien inattendu avec un système de particules en interaction.

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“…The rigorous description of this phenomenon in the framework of well defined mathematical models is relatively recent, dating back to the pioneering paper [9], and has experienced substantial progress in the last decade. See [7,14,18] for reviews and for a list of the most important papers on this subject.…”

Section: Introductionmentioning

confidence: 99%

“…Think for instance to a stochastic lattice spin system. In this context powerful theories (see [3,[16][17][18]) have been developed. Furthermore, the results in [15] improved those in [16] in the direction of minimizing the number of model-dependent inputs necessary to describe the metastable behavior of the system.…”

Section: Introductionmentioning

confidence: 99%

Sharp Asymptotics for Stochastic Dynamics with Parallel Updating Rule

Nardi

Spitoni

2011

J Stat Phys

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In this paper we study the metastability problem for a stochastic dynamics with a parallel updating rule; in particular we consider a finite volume Probabilistic Cellular Automaton (PCA) in a small external field at low temperature regime. We are interested in the nucleation of the system, i.e., the typical excursion from the metastable phase (the configuration with all minuses) to the stable phase (the configuration with all pluses), triggered by the formation of a critical droplet. The main result of the paper is the sharp estimate of the nucleation time: we show that the nucleation time divided by its average converges to an exponential random variable and that the rate of the exponential random variable is an exponential function of the inverse temperature β times a prefactor that does not scale with β. Our approach combines geometric and potential theoretic arguments.

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Large Deviations and Metastability

Abstract: "Large Deviations and Metastability" Enzo Olivieri, Maria Eulália Vares p. 26, line 5. Replace "x ∈ M " by "x ∈ M such that I(x) < +∞" p. 57, display (1.95). There is a redundant condition. Since ν and η are probability

Cited by 262 publications

References 0 publications

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

Interacting particle systems

Sharp Asymptotics for Stochastic Dynamics with Parallel Updating Rule

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