On a long range particle system with unbounded flip rates

Abstract

We consider an interacting particle system on f0; 1g Z with non-local, unbounded ip rates. Zeroes ip to one at a rate that depends on the number of ones to the right until we see a zero (the ip rate equals times one plus this number). The ip rate of the ones equals . We give motivation for models like this in general, and this one in particular. The system turns out not to be Feller, and we construct it using monotonicity. We show thatfor < the system has a unique non-trivial stationary distribution, which is ergodic, stationary, and has a density ofones of . For the limit is degenerate at f1g Z . Our main tool is an explicit formula for the density of ones at any given moment.