On a Long Range Particle System with Unbounded Flip Rates,

#### R. Meester, C. Quant

2003, v.9, №1, 59-84

ABSTRACT

We consider an interacting particle system on $\{0,1\}^{Z}$ with non-local, unbounded flip rates. A zero flips to a one at a rate which depends on the number of ones to the right until we see a zero (in fact, the flip rate equals $\lambda$ times one plus this number). A one flips to a zero at rate $\mu$. We motivate models like this in general, and this one in particular. The system is constructed using monotonicity. We show that for $\lambda < \mu$ the system has a unique non-trivial stationary distribution, which is ergodic, stationary, and has a density of ones of $\lambda/\mu$. For $\lambda\ge\mu$ the limit is degenerate at $\{1\}^{Z}$. Our main tool is an explicit formula for the density of ones at any given moment. We also show that the stationary distribution has positive correlations, and is not a product measure.

Keywords: long range particle system,construction,monotonicity,unique stationary distribution,non-locality