Let r : S × S → R + 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 g: N → R + 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 N ↑ ∞ 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. © 2011 Springer-Verlag.