Exponential \(L_ 2\) convergence of attractive reversible nearest particle systems (Q1822845)

Nearest particle systems on \({\mathbb{Z}}\) which are attractive and reversible are studied. The main result shows that the convergence of the system to the equilibrium \(\nu\) in \(L_ 2(\nu)\) is exponential (even if phase transition occurs), and the upper and lower bounds for the exponent are given (up to the critical value of parameter). The method used for the bound which proves the exponential convergence is a comparison of the corresponding exponent to the exponent for a Markov chain on \(({\mathbb{Z}}^+)^{{\mathbb{Z}}}\) which is the family of independent birth and death chains in \({\mathbb{Z}}^+\). The comparison is based on a representation of the invariant measure \(\nu\), which is the distribution of a renewal process, as an image of a distribution of a sequence of independent identically distributed random variables in \({\mathbb{Z}}^+\).