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\(L_ 2\) rates of convergence for attractive reversible nearest particle systems: The critical case. (English) Zbl 0737.60092
Reversible nearest particle systems are a class of one-dimensional interacting particle systems whose transition rates are determined by a probability density \(\beta(n)\) with finite mean on the positive integers. In an earlier paper the author proved that the system converges exponentially rapidly in \(L_ 2(v)\) if and only if the system is supercritical, where \(v\) is the reversible measure for the system. In the present paper the author considers the critical case, and gives moment conditions on \(\beta(n)\) which are necessary and sufficient for convergence of the processes in \(L_ 2(v)\) at a specified algebraic rate. Conditions for algebraic \(L_ 2\) convergence of general Markov processes are developed as a tool.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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