The Annals of Probability

A central limit theorem for reversible exclusion and zero-range particle systems

Sunder Sethuraman and Lin Xu

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We give easily verifiable conditions under which a functional central limit theorem holds for additive functionals of symmetric simple exclusion and symmetric zero-range processes. Also a reversible exclusion model with speed change is considered. 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 $\lambda^{-1/2} \int_0^{\lambda t} f(\eta(s)) ds$ converge to a Brownian motion? A general but often intractable answer is given by Kipnis and Varadhan. In this article we determine what conditions beyond a mean-zero condition on $f(\eta)$ are required for the diffusive limit above. Specifically, we characterize the $H^{-1}$ space in an applicable way.

Our method of proof relies primarily on a sharp estimate on the "spectral gap" of the process and weak regularity properties for the invariant measures.

Article information

Ann. Probab., Volume 24, Number 4 (1996), 1842-1870.

First available in Project Euclid: 6 January 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems

Simple exclusion process zero-range process invariance principle central limit theorem


Sethuraman, Sunder; Xu, Lin. A central limit theorem for reversible exclusion and zero-range particle systems. Ann. Probab. 24 (1996), no. 4, 1842--1870. doi:10.1214/aop/1041903208.

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