The Annals of Applied Probability

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

André Schlichting and Martin Slowik

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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

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

Received: May 2017
Revised: January 2019
First available in Project Euclid: 7 January 2020

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Mathematical Reviews number (MathSciNet)

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 49J40: Variational methods including variational inequalities [See also 47J20] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 82C26: Dynamic and nonequilibrium phase transitions (general)

Capacitary inequality harmonic functions logarithmic Sobolev constant mean hitting time metastability Poincaré constant random field Curie–Weiss model


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.

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