TY - JOUR
T1 - Upgrading MLSI to LSI for reversible Markov chains
AU - Salez, Justin
AU - Tikhomirov, Konstantin
AU - Youssef, Pierre
N1 - Funding Information:
The three authors warmly thank Jonathan Hermon for pointing out the open problem [17, Question 7.1] . J.S. was partially supported by Institut Universitaire de France , and K.T. was partially supported by the Sloan Research Fellowship and by the NSF grant DMS 2054666 .
Publisher Copyright:
© 2023 Elsevier Inc.
PY - 2023/11/1
Y1 - 2023/11/1
N2 - 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.
AB - 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.
KW - Functional inequalities
KW - Logarithmic Sobolev inequalities
KW - Mixing times
KW - Modified log-Sobolev inequalities
KW - Reversible Markov chains
UR - http://www.scopus.com/inward/record.url?scp=85165469968&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85165469968&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2023.110076
DO - 10.1016/j.jfa.2023.110076
M3 - Article
AN - SCOPUS:85165469968
SN - 0022-1236
VL - 285
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 9
M1 - 110076
ER -