Making Markov chains less lazy

The mixing time of an ergodic, reversible Markov chain can be bounded in terms of the eigenvalues of the chain: specifically, the second-largest eigenvalue and the smallest eigenvalue. It has become standard to focus only on the secondlargest eigenvalue, by making the Markov chain “lazy”. (A lazy chain does nothing at each step with probability at least 1 2 , and has only nonnegative eigenvalues.) An alternative approach to bounding the smallest eigenvalue was given by Diaconis and Stroock [5, Proposition 2] and Diaconis and Saloff-Coste [4, p.702]. We give examples to show that using this approach it can be quite easy to obtain a bound on the smallest eigenvalue of a combinatorial Markov chain which is several orders of magnitude below the best-known bound on the second-largest eigenvalue.