Metastability of reversible condensed zero range processes on a finite set

Let $r:S\times S\to \text{bb}{R}_{+}$ be the jump rates of an irreducible random walk on a finite set $S$, reversible with respect to some probability measure $m$. For $\alpha >1$, let $g:\text{bb}N\to \text{bb}{R}_{+}$ be given by $g(0)=0$, $g(1)=1$, $g(k)=(k\mathrm{/}k-1{)}^{\alpha }$, $k\ge 2$. Consider a zero range process on $S$ in which a particle jumps from a site $x$, occupied by $k$ particles, to a site $y$ at rate $g(k)r(x,y)$. Let $N$ stand for the total number of particles. In the stationary state, as $N\uparrow \mathrm{\infty }$, all particles but a finite number accumulate on one single site. We show in this article that in the time scale $N^{1+\alpha }$ the site which concentrates almost all particles evolves as a random walk on $S$ whose transition rates are proportional to the capacities of the underlying random walk.