Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities

Abstract

We investigate the metastable behavior of reversible Markov chains on\npossibly countable infinite state spaces. Based on a new definition of\nmetastable Markov processes, we compute precisely the mean transition time\nbetween metastable sets. Under additional size and regularity properties of\nmetastable sets, we establish asymptotic sharp estimates on the Poincar\\'e and\nlogarithmic Sobolev constant. The main ingredient in the proof is a capacitary\ninequality along the lines of V. Maz'ya that relates regularity properties of\nharmonic functions and capacities. We exemplify the usefulness of this new\ndefinition in the context of the random field Curie-Weiss model, where\nmetastability and the additional regularity assumptions are verifiable.\n