Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes
Résumé
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 \cite{BL1, BL2}
or in \cite{LMS2}.
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.