We consider critical reversible nearest particle systems. We assume that the associated renewal measure has large moments as well as some regularity conditions. It is shown that such processes, started from a nontrivial ergodic translation invariant distribution, converge in distribution to the upper invariant measure.
"The Convergence Result for Critical Reversible Nearest Particle Systems." Ann. Probab. 30 (1) 1 - 61, January 2002. https://doi.org/10.1214/aop/1020107760