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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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