## Juggler's exclusion process

*Journal of Applied Probability*49(1):266–279, 2012.- doi:10.1239/jap/1331216846
- arXiv:1104.3397

### Abstract

Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.

**Keywords**: exclusion process, juggling pattern, set-valued Markov process,
ergodicity, positive recurrence, set-avoiding memoryless distribution,
noncolliding random walk, Gibbs measure, maximum entropy