Limit Theorems and Estimates for Interacting Particle Systems
Abstract
In this thesis we consider the exclusion and zero range processes, two of the most widely studied interacting particle systems. Informally, the exclusion process is a collection of particles performing random walk on the lattice Z^{d}; each particle moves independently except in that jumps to occupied vertices are suppressed. The zerorange processes also consist of collections of particles executing random walk. The interaction between particles is specified in the following way: If there are k particles at a vertex iin Z^{d}, then infinitesimally the rate at which one of them departs is given as c(k) where c : N to R_sp{+ }{1} is a specified rate function identifying the process. Our main goal is to establish a functional central limit theorem, or invariance principle, for these systems. One of the basic tools developed, of independent interest, is an estimate for the spectral gap of the zerorange process. We first give a lower bound on the spectral gap for the zerorange process. Under some conditions on the rate function, we show that the gap shrinks on the order O(n^{2}), independent of the density, for the dynamics localized on a cube of size n^{d}.. This estimate is then applied to derive easily verifiable conditions under which a functional central limit theorem holds for additive functionals of certain conservative particle systems. Let eta(t) be the configuration of the process at time t and let f(eta) be a function on the state space. The question is: For which functions f does lambda^{1/2} int_sp {0}{lambda t} f(eta(s))ds converge to a Brownian motion? We determine what conditions beyond a meanzero condition on f(eta) is required for the integral above to converge when eta(t) corresponds to exclusion and zerorange dynamics.
 Publication:

Ph.D. Thesis
 Pub Date:
 1995
 Bibcode:
 1995PhDT.......185S
 Keywords:

 EXCLUSION;
 ZERORANGE;
 Mathematics; Physics: General