  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.
