## The Annals of Applied Probability

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

#### Abstract

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. Maz’ya 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.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 6 (2019), 3438-3488.

Dates
Revised: January 2019
First available in Project Euclid: 7 January 2020

https://projecteuclid.org/euclid.aoap/1578366319

Digital Object Identifier
doi:10.1214/19-AAP1484

Mathematical Reviews number (MathSciNet)
MR4047985

#### Citation

Schlichting, André; Slowik, Martin. Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities. Ann. Appl. Probab. 29 (2019), no. 6, 3438--3488. doi:10.1214/19-AAP1484. https://projecteuclid.org/euclid.aoap/1578366319

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