Open Access
Translator Disclaimer
August 1996 Logarithmic Sobolev inequalities for finite Markov chains
P. Diaconis, L. Saloff-Coste
Ann. Appl. Probab. 6(3): 695-750 (August 1996). DOI: 10.1214/aoap/1034968224


This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a self-contained development. Examples of applications include the study of a Metropolis chain for the binomial distribution, sharp results for natural chains on the box of side n in d dimensions and improved rates for exclusion processes. We also show that for most r-regular graphs the log-Sobolev constant is of smaller order than the spectral gap. The log-Sobolev constant of the asymmetric two-point space is computed exactly as well as the log-Sobolev constant of the complete graph on n points.


Download Citation

P. Diaconis. L. Saloff-Coste. "Logarithmic Sobolev inequalities for finite Markov chains." Ann. Appl. Probab. 6 (3) 695 - 750, August 1996.


Published: August 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0867.60043
MathSciNet: MR1410112
Digital Object Identifier: 10.1214/aoap/1034968224

Primary: 60F05 , 60J15 , 60J27

Keywords: logarithmic Sobolev inequalities , Markov chains

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.6 • No. 3 • August 1996
Back to Top