  An important paradigm in the understanding of mixing times of Glauber
dynamics for spin systems is the correspondence between spatial mixing
properties of the models and bounds on the mixing time of the dynamics. This
includes, in particular, the classical notions of weak and strong spatial
mixing, which have been used to show the best known mixing time bounds in the
high-temperature regime for the Glauber dynamics for the Ising and Potts
models.
  Glauber dynamics for the random-cluster model does not naturally fit into
this spin systems framework because its transition rules are not local. In this
paper, we present various implications between weak spatial mixing, strong
spatial mixing, and the newer notion of spatial mixing within a phase, and
mixing time bounds for the random-cluster dynamics in finite subsets of
$\mathbb Z^d$ for general $d\ge 2$. These imply a host of new results,
including optimal $O(N\log N)$ mixing for the random cluster dynamics on torii
and boxes on $N$ vertices in $\mathbb Z^d$ at all high temperatures and at
sufficiently low temperatures, and for large values of $q$ quasi-polynomial (or
quasi-linear when $d=2$) mixing time bounds from random phase initializations
on torii at the critical point (where by contrast the mixing time from
worst-case initializations is exponentially large). In the same parameter
regimes, these results translate to fast sampling algorithms for the Potts
model on $\mathbb Z^d$ for general $d$.
