SIZE OF THE CLUSTERS UNDER LOW DENSITY ZERO-RANGE INVARIANT MEASURES

Regarding all particles at a fixed site as a cluster, the size of the largest cluster under the zero range invariant measures is well studied by Jeon et al.[5] for the case of density one. Here, the density of the finite zero-range process is given by the ratio between the number m of particles and the number n of sites. In this paper, we study the lower density case, i.e., the case m = o(n). Especially, when m ~ $n^{\beta}$,0 < ${\beta}$ < 1, we show that there is an interesting cutoff point around $\beta$ = 1/2.