Exponential $L_2$ Convergence of Attractive Reversible Nearest Particle Systems

Abstract

Nearest particle systems are continuous-time Markov processes on $\{0, 1\}^Z$ in which particles die at rate 1 and are born at rates which depend on their distances to the nearest particles to the right and left. There is a natural parametrization of these systems with respect to which they exhibit a phase transition. When the process is attractive and reversible, the critical value $\lambda_c$ above which a nontrivial invariant measure exists can be computed exactly. This invariant measure is the distribution $v$ of a stationary discrete time renewal process. Under a mild regularity assumption, we prove that the following three statements are equivalent: (a) The nearest particle system converges to equilibrium exponentially rapidly in $L_2(v)$. (b) The density of the interarrival times in the renewal process has exponentially decaying tails. (c) The nearest particle system is supercritical in the sense that $\lambda > \lambda_c$. Under an additional second-moment assumption, we prove that the critical exponent associated with the exponential convergence is 2. The proof of exponential convergence is based on an unusual comparison of the nearest particle system with an infinite system of independent birth and death chains. To carry out this comparison, a new representation is developed for a stationary renewal process with a log-convex renewal sequence in terms of a sequence of i.i.d. random variables.