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.
@article{1034968224,
author = {Diaconis, P. and Saloff-Coste, L.},
title = {Logarithmic Sobolev inequalities for finite Markov chains},
journal = {Ann. Appl. Probab.},
volume = {6},
number = {1},
year = {1996},
pages = { 695-750},
language = {en},
url = {http://dml.mathdoc.fr/item/1034968224}
}
Diaconis, P.; Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab., Tome 6 (1996) no. 1, pp. 695-750. http://gdmltest.u-ga.fr/item/1034968224/