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Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities. (English) Zbl 1432.60070
Summary: We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. G. Maz’ya [Sobolev spaces. With applications to elliptic partial differential equations. Transl. from the Russian by T. O. Shaposhnikova. Berlin: Springer (2011; Zbl 1217.46002)] that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new definition in the context of the random field Curie-Weiss model, where metastability and the additional regularity assumptions are verifiable.
MSC:
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
60J45 Probabilistic potential theory
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
60E15 Inequalities; stochastic orderings
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