Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes

We present a general method to derive the metastable behavior of weakly mixing Markov chains. This approach is based on properties of the resolvent equations and can be applied to metastable dynamics which do not satisfy the mixing conditions required in Beltr\'an and Landim (2010,2012) or in Landim et. al. (2020). As an application, we study the metastable behavior of critical zero-range processes. 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 the uniform measure. For $\alpha >0$, 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 $g(k) r(x,y)$. For $\alpha \ge 1$, in the stationary state, as the total number of particles, represented by $N$, tends to infinity, all particles but a negligible number accumulate at one single site. This phenomenon is called condensation. Since condensation occurs if and only if $\alpha\ge 1$, we call the case $\alpha =1$ critical. By applying the general method established in the first part of the article to the critical case, we show that the site which concentrates almost all particles evolves in the time-scale $N^2 \log N$ as a random walk on $S$ whose transition rates are proportional to the capacities of the underlying random walk.