Symmetric Markov chains in ℤd: How fast can they move?

Abstract

Consider a reversible Markov chain Xnwhich takes values in a subset of ℤd. If the steps of the chain are uniformly bounded and the invariant measure satisfies a mild regularity condition, Varopoulos, Carne and Kesten have obtained estimates on {Mathematical expression} which exhibit a Gaussian tail in λ but blow up as n→∞. Following Kesten's approach we derive bounds which are uniform in n in some special cases. Our main result, however, is an example which shows that in general the estimates of Varopoulos, Carne and Kesten are essentially best possible. © 1989 Springer-Verlag.