Metastability of reversible condensed zero range processes on a finite set

Let be the jump rates of an irreducible random walk on a finite set S , reversible with respect to some probability measure m . For α > 1, let be given by g (0) = 0, g (1) = 1, g ( k ) = ( k / k − 1) α , k ≥ 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 , all particles but a finite number accumulate on one single site. We show in this article that in the time scale N 1+ α 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.