The Annals of Probability

$L_2$ Rates of Convergence for Attractive Reversible Nearest Particle Systems: The Critical Case

Thomas M. Liggett

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Reversible nearest particle systems are certain one-dimensional interacting particle systems whose transition rates are determined by a probability density $\beta(n)$ with finite mean on the positive integers. The reversible measure for such a system is the distribution $\nu$ of the stationary renewal process corresponding to this density. In an earlier paper, we proved under certain regularity conditions that the system converges exponentially rapidly in $L_2(\nu)$ if and only if the system is supercritical. This in turn is equivalent to $\beta(n)$ having exponential tails. In this paper, we consider the critical case, and give moment conditions on $\beta(n)$ which are separately necessary and sufficient for the convergence of the process in $L_2(\nu)$ at a specified algebraic rate. In order to do so, we develop conditions on the generator of a general Markov process which correspond to algebraic $L_2$ convergence of the process. The use of these conditions is also illustrated in the context of birth and death chains on the positive integers.

Article information

Ann. Probab., Volume 19, Number 3 (1991), 935-959.

First available in Project Euclid: 19 April 2007

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Interacting particle systems nearest particle systems algebraic rates of convergence for semigroups renewal theory


Liggett, Thomas M. $L_2$ Rates of Convergence for Attractive Reversible Nearest Particle Systems: The Critical Case. Ann. Probab. 19 (1991), no. 3, 935--959. doi:10.1214/aop/1176990330.

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