COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS

By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same state space. This extends the ideas introduced in Diaconis and Saloff-Coste (1993), where random walks on finite groups were considered. The bounds involve geometric properties such as the diameter and covering number of an associated graph along the lines of Diaconis and Stroock (1991). The main application gives a sharp upper bound on the second eigenvalue of the symmetric exclusion process. Thus, let S0 be a connected undirected graph with n vertices. For simplicity, we assume in this introduction that SW is regular. To start, r unlabelled particles are placed in an initial configuration, 1 < r < n. At each step, a particle is chosen at random; then one of the neighboring sites of this particle is chosen at random. If the neighboring site is unoccupied, the chosen particle is moved there; if the neighboring site is occupied, the system stays as it was. This is a reversible Markov chain on the r-sets of {1, 2, . . ., n} with uniform stationary distribution. Liggett (1985) gives background and motivation (he focuses on infinite systems). Fill (1991) gives bounds on the second eigenvalue of the labeled exclusion process on the finite circle ZZn 1 We study this chain by comparison with a second Markov chain on r-sets that proceeds by picking a particle at random, picking an unoccupied site at random (not necessarily a neighboring site) and moving the particle to the unoccupied site. This is a well studied chain (the Bernoulli-Laplace model for diffusion). Its eigenvalues are known. We show that the comparison techniques apply to give upper bounds on the eigenvalues of the exclusion

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