A sharp log-Sobolev inequality for the multislice

Salez, Justin

doi:10.5802/ahl.99

A sharp log-Sobolev inequality for the multislice
[Une égalité de log-Sobolev optimale pour les multi-tranches]

Salez, Justin ¹

¹ Université Paris-Dauphine & PSL, CEREMADE - UMR 7534, Place du Maréchal de Lattre de Tassigny, F-75775, Paris Cedex 16, (France)

Annales Henri Lebesgue, Tome 4 (2021), pp. 1143-1161.

Résumé
Abstract

Nous déterminons, à une petite constante multiplicative près, la constante de log-Sobolev de la diffusion de Bernoulli–Laplace multi-urne avec paramètres arbitraires. Ce résultat étend une estimée classique de Lee et Yau (1998) et confirme une conjecture de Filmus, O’Donnell et Wu (2018). Entre autres applications, nous quantifions complètement le phénomène d’expansion des petits ensembles, et obtenons des estimées optimales pour le temps de mélange du processus d’exclusion coloré sur divers graphes.

We determine the log-Sobolev constant of the multi-urn Bernoulli–Laplace diffusion model with arbitrary parameters, up to a small universal multiplicative constant. Our result extends a classical estimate of Lee and Yau (1998) and confirms a conjecture of Filmus, O’Donnell and Wu (2018). Among other applications, we completely quantify the small-set expansion phenomenon on the multislice, and obtain sharp mixing-time estimates for the colored exclusion process on various graphs.

Reçu le : 2020-04-19
Accepté le : 2021-02-05
Publié le : 2021-09-22

DOI : 10.5802/ahl.99

Classification : 60J27, 60J10, 05C81
Mots clés : Log-Sobolev constant, random transpositions, colored exclusion process

Affiliations des auteurs :

Salez, Justin ¹

¹ Université Paris-Dauphine & PSL, CEREMADE - UMR 7534, Place du Maréchal de Lattre de Tassigny, F-75775, Paris Cedex 16, (France)

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     title = {A sharp {log-Sobolev} inequality for the multislice},
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     publisher = {\'ENS Rennes},
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     year = {2021},
     doi = {10.5802/ahl.99},
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     url = {http://archive.numdam.org/articles/10.5802/ahl.99/}
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Salez, Justin. A sharp log-Sobolev inequality for the multislice. Annales Henri Lebesgue, Tome 4 (2021), pp. 1143-1161. doi : 10.5802/ahl.99. http://archive.numdam.org/articles/10.5802/ahl.99/

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