April 2023 Modified log-Sobolev inequalities for strong-Rayleigh measures
Jonathan Hermon, Justin Salez
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Ann. Appl. Probab. 33(2): 1501-1514 (April 2023). DOI: 10.1214/22-AAP1847


We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice {0,1}n, when the invariant law π satisfies a form of negative dependence known as the stochastic covering property. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as the uniform distribution over the set of bases of any balanced matroid. In the special case where π is k-homogeneous, our results imply the celebrated concentration inequality for Lipschitz functions due to Pemantle and Peres (Combin. Probab. Comput. 23 (2014) 140–160). As another application, we deduce that the natural Monte-Carlo Markov chain used to sample from π has mixing time at most knloglog1π(x) when initialized in state x. To the best of our knowledge, this is the first work relating negative dependence and modified log-Sobolev inequalities.


The authors would like to thank Radosław Adamczak and Prasad Tetali for useful discussions and for bringing some relevant references to our attention. They also thank two anonymous referees for their numerous remarks.


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Jonathan Hermon. Justin Salez. "Modified log-Sobolev inequalities for strong-Rayleigh measures." Ann. Appl. Probab. 33 (2) 1501 - 1514, April 2023. https://doi.org/10.1214/22-AAP1847


Received: 1 January 2021; Revised: 1 January 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1515.60259
MathSciNet: MR4564432
Digital Object Identifier: 10.1214/22-AAP1847

Primary: 60J10 , 60J45

Keywords: Modified log-Sobolev inequalities , Negative dependence , stochastic covering

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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