Modified logarithmic Sobolev inequalities for some models of random walk

Logarithmic Sobolev inequalities are a well-studied technique for estimating rates of convergence of Markov chains to their stationary distributions. In contrast to continuous state spaces, discrete settings admit several distinct log Sobolev inequalities, one of which is the subject of this paper. Here we derive modified log Sobolev inequalities for some models of random walk, including the random transposition shuffle and the top-random transposition shuffle on Sn, and the walk generated by 3-cycles on An. As an application, we derive concentration inequalities for these models.