WIAS Preprint No. 601, (2000)

Metastability and low lying spectra in reversible Markov chains



Authors

  • Bovier, Anton
  • Eckhoff, Michael
  • Gayrard, Veronique
  • Klein, Markus

2010 Mathematics Subject Classification

  • 60J10 60K35

Keywords

  • Markov chains, metastability, eigenvalue problems, exponential distribution

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 $xin G_N$ 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 $a_N^-1ll b_N$. 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. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.

Appeared in

  • Comm. Math. Phys. 228 (2002), pp. 219-255

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