Metastability and small eigenvalues in Markov chains

In this Letter we announce rigorous results that elucidate the relation bet ween metastable states and low-lying eigenvalues in Markov chains in a much more general setting and with considerably greater precision than has so f ar been available. This includes a sharp uncertainty principle relating all low-lying eigenvalues to mean times of metastable transitions, a relation between the support of eigenfunctions and the attractor of a metastable sta te and sharp estimates of the convergence of the probability distribution o f the metastable transition times to the exponential distribution.