Abstract
In this paper we introduce a new renormalization group method for the study of the long-time behavior of Markov chains with finite state space and with transition probabilities exponentially small in an external parameterβ. A general approach of metastability problems emerges from this analysis and is discussed in detail in the case of a two-dimensional Ising system at low temperature,
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. I. Freidlin and A. D. Wentzell,Random Perturbations of Dynamical Systems (Springer-Verlag, Berlin, 1984).
R. Z. Has'minskii,Stochastic Stability of Differential Equations (Sijthoff-Noordhoff, 1980).
E. J. Neves and R. H. Schonmann, Behaviour of droplets for a class of Glauber dynamics at very low temperatures,Commun. Math. Phys. 137:209 (1991).
E. J. Neves and R. H. Schonmann, Critical droplets and metastability for a Glauber dynamics at very low temperatures,Prob. Theory Related Fields 91:331 (1992).
S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by simulated annealing,Science 220:671 (1983).
R. Kotecky and E. Olivieri, Droplet dynamics for asymmetric Ising model, Preprint (1992).
R. Kotecky and E. Olivieri, Shapes of growing droplets-A model of escape from a metastable phase, in preparation.
F. Martinelli, E. Oliveri, and E. Scoppola, On the Swendsen and Wang dynamics II: Critical droplets and homogeneous nucleation at low temperature for the 2 dimensional Ising model,J. Stat. Phys. 62:1135 (1991).
F. Martinelli, E. Olivieri, and E. Scoppola, Metastability and exponential approach to equilibrium for low temperature stochastic Ising models,J. Stat. Phys. 61:1105 (1990).
F. Martinelli, E. Olivieri, and E. Scoppola, Small random perturbations of finite and infinite dimensional dynamical systems: Unpredictability of exit times.J. Stat. Phys. 55:477 (1989).
D. Revuz,Markov Chains (North-Holland, Amsterdam, 1975), Chapter 1, Section 3.
E. Scoppola, Metastability and nucleation for 2-dimensional Ising systems,Physica A (Proceedings Statphys 18-Berlin 1992), to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Scoppola, E. Renormalization group for Markov chains and application to metastability. J Stat Phys 73, 83–121 (1993). https://doi.org/10.1007/BF01052752
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01052752