Metastability and low lying spectra in reversible Markov chains

TL;DR

This paper analyzes reversible Markov chains to identify metastable states, linking their mean exit times to eigenvalues of the transition matrix and demonstrating the exponential nature of metastable exit time distributions.

Contribution

It introduces a framework for defining metastable points and states in reversible Markov chains, relating eigenvalues to mean exit times and providing precise spectral and probabilistic characterizations.

Findings

Metastable states correspond to eigenvalues close to 1 of the transition matrix.

Mean exit times from metastable states can be computed explicitly.

Exit time distributions are sharply close to exponential.

Abstract

We study a large class of reversible Markov chains with discrete state space and transition matrix $P_N$. We define the notion of a set of {\it metastable points} as a subset of the state space $\G_N$ such that (i) this set is reached from any point $x\in \G_N$ without return to x with probability at least $b_N$, while (ii) for any two point x,y in the metastable set, the probability $T^{-1}_{x,y}$ to reach y from x without return to x is smaller than $a_N^{-1}\ll b_N$. Under some additional non-degeneracy assumption, we show that in such a situation: \item{(i)} To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. \item{(ii)} To each metastable point corresponds one simple eigenvalue of $1-P_N$ which is essentially equal to the inverse mean exit time from this state. The corresponding eigenfunctions are close to the indicator function of the support of the metastable state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.