Modified Logarithmic Sobolev Inequalities in Discrete Settings

Sergey G. Bobkov () and Prasad Tetali ()
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Sergey G. Bobkov: University of Minnesota
Prasad Tetali: School of Mathematics and College of Computing, Georgia Tech

Journal of Theoretical Probability, 2006, vol. 19, issue 2, 289-336

Abstract: Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare’ (spectral gap) inequality. We show that, in contrast with the spectral gap, for bounded degree expander graphs, various log-Sobolev constants go to zero with the size of the graph. We also derive a hypercontractivity formulation equivalent to our main modified log-Sobolev inequality. Along the way we survey various recent results that have been obtained in this topic by other researchers.

Keywords: Spectral gap; entropy decay; logarithmic Sobolev Inequalities; 60Jxx; 46Exx (search for similar items in EconPapers)
Date: 2006
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-006-0016-3

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