Metastability and Low Lying Spectra in Reversible Markov Chains

We study a large class of reversible Markov chains with discrete state space and transition matrixPN . We define the notion of a set of metastable points as a subset of the state space N such that (i) this set is reached from any point x ∈ N without return to x with probability at least bN , while (ii) for any two points 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−1 N bN . Under some additional non-degeneracy assumption, we show that in such a situation: (i) To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. (ii) To each metastable point corresponds one simple eigenvalue of 1 − PN 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.