Metastability and low lying spectra in reversible Markov chains
type: Item
instance of: scholarly article
title: Metastability and low lying spectra in reversible Markov chains
zbMATH Open document ID: 1010.60088
author: Anton Bovier
author: Michael Eckhoff
author: V. Gayrard
author: M. Klein
published in: Communications in Mathematical Physics
publication date: 2002-12-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0007160
review text: The authors continue their investigation of metastability [Probab. Theory Relat. Fields 119, No. 1, 99-161 (2001)]. In the present paper they study the connection between metastability and spectral theory in a more general setting. They consider a reversible Markov chain with transition matrix \(P_N\) with a finite state space \(\Gamma_N\), whose cardinality tends to infinity as \(N\to\infty\). They define a set of metastable points each of them representing a metastable state. This notion is based on the idea that the time it takes to visit this set from anywhere is much smaller compared to the transition times between different metastable points. The set depends on the corresponding timescales. Under an additional non-degeneracy assumption the authors establish a precise relation between the low-lying eigenvalues of the operator \(1-P_N\) and the mean metastable transition times with sharp estimates of the convergence of the distribution of these times to the exponential distribution.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C05
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 1836923
zbMATH Keywords: reversible Markov chains
zbMATH Keywords: metastability
zbMATH Keywords: low-lying spectrum
MaRDI profile type: Publication
OpenAlex ID: W2090987881
arXiv ID: math/0007160
DOI: 10.1007/S002200200609
description: scientific article