Metastability of reversible condensed zero range processes on a finite set

TL;DR

This paper studies the metastable behavior of a reversible zero range process on a finite set, showing that as the number of particles grows, the site with almost all particles moves according to a random walk with rates related to the underlying capacity.

Contribution

It demonstrates that in the large particle limit, the condensate's location evolves as a random walk with transition rates tied to the capacities of the base process.

Findings

Particles condense on a single site as N increases

The condensate's movement follows a random walk on S

Transition rates are proportional to capacities

Abstract

Let $r: S\times S\to \bb R_+$ 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 $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = (k/k-1)^\alpha$, $k\ge 2$. 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 $N\uparrow\infty$, 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.