Abstract
Symmetric nearest-particle systems are certain spin systems on {0, 1}z in which the flip rate is a function of the distances to the nearest particle of different type to the left and right. The process differs from the ordinary nearest-particle system in that the rates are preserved if zeros and ones are interchanged. The only reversible measure for the symmetric nearest-particle system is a “renewaltype” measure (the natural analog to the nonsymmetric case). Also as in the nonsymmetric case, reversibility only occurs when the rates are of a specific form. By imposing additional conditions on the rates it can be shown that the reversible measure is the only translation-invariant, invariant measure which concentrates on configurations having infinitely many zeros and ones to either side of the origin. This can be used to prove that for a large class of translation-invariant initial distributions, weak limits are reversible measures. Then we can conclude that the process is convergent for several examples of initial distributions.
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Handjani, S.J. The reversible measures for symmetric nearest-particle systems. J Stat Phys 80, 1119–1164 (1995). https://doi.org/10.1007/BF02179866
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DOI: https://doi.org/10.1007/BF02179866