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.