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 zero-range 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 zero-range process. We first give a lower bound on the spectral gap for the zero-range 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 mean-zero condition on f(eta) is required for the integral above to converge when eta(t) corresponds to exclusion and zero-range dynamics.
- Publication:
-
Ph.D. Thesis
- Pub Date:
- 1995
- Bibcode:
- 1995PhDT.......185S
- Keywords:
-
- EXCLUSION;
- ZERO-RANGE;
- Mathematics; Physics: General