Upgrading MLSI to LSI for reversible Markov chains

Upgrading MLSI to LSI for reversible Markov chains

For reversible Markov chains on finite state spaces, we show that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) at the surprisingly low cost of degrading the associated constant by log⁡(1/p), where p is the minimum non-zero transition probability. We il...

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Veröffentlicht in:Journal of functional analysis 2023-11, Vol.285 (9), p.110076, Article 110076
Hauptverfasser: Salez, Justin, Tikhomirov, Konstantin, Youssef, Pierre
Format: Artikel
Sprache:eng
Schlagworte:
Functional inequalities
Logarithmic Sobolev inequalities
Mathematics
Mixing times
Modified log-Sobolev inequalities
Reversible Markov chains
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creator Salez, Justin
Tikhomirov, Konstantin
Youssef, Pierre
description For reversible Markov chains on finite state spaces, we show that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) at the surprisingly low cost of degrading the associated constant by log⁡(1/p), where p is the minimum non-zero transition probability. We illustrate this by providing the first log-Sobolev estimate for Zero-Range processes on arbitrary graphs. As another application, we determine the modified log-Sobolev constant of the Lamplighter chain on all bounded-degree graphs, and use it to provide negative answers to two open questions by Montenegro and Tetali (2006) [27] and Hermon and Peres (2018) [17]. Our proof builds upon the ‘regularization trick’ recently introduced by the last two authors.
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subjects Functional inequalities
Logarithmic Sobolev inequalities
Mathematics
Mixing times
Modified log-Sobolev inequalities
Reversible Markov chains
title Upgrading MLSI to LSI for reversible Markov chains
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