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

André Schlichting, Martin Slowik

Published 2017-05-15Version 1

We investigate the metastable behaviour of reversible Markov chains on countable infinite state spaces. Based on a definition of metastable sets, we compute precisely the mean exit time from a metastable set. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincar\'e and logarithmic Sobolev constant. The main ingredient are capacitary inequalities along the lines of V. Maz'ya relating regularity properties of harmonic functions and capacities. We exemplify the usefulness of our approach in the context of the random field Curie-Weiss model.