Logarithmic Sobolev Techniques for Random Walks on Graphs

Abstract

Recently, Diaconis and Sarloff-Coste used logarithmic Sobolev inequalities to improve convergence bounds for random walks on graphs. We will give a strengthened version by showing that the random walk on a graph G reaches stationarity (under total variation distance) after about $\frac{1}{{4\alpha }}$ log(|E|/min _x d _x ) steps, where α denotes the log-Sobolev constant, E is the edge set of G and d _x denotes the degree of x. Under the relative pointwise distance (which is a slightly stronger notion), the random walk converges in about $\frac{1}{{2\alpha }}$ log(|E|/min _x d _x ) steps.