Approximating a critical attractive reversible nearest particle system on a finite set from above is not difficult, but approximating it from below is less trivial, as the empty configuration is invariant. We develop a finite state Markov chain that deals with this issue. The rate of convergence for this chain is discovered through a mixing inequality in Jerrum and Sinclair; an application of that spectral gap bound in this case requires the use of ‘‘randomized paths from state to state.’’ For applications, we prove distributional results for semiinfinite and infinite critical RNPS.
"Finite approximations to the critical reversible nearest particle system." Ann. Probab. 26 (4) 1751 - 1780, October 1998. https://doi.org/10.1214/aop/1022855881