  We consider spin systems on the integer lattice graph $\mathbb{Z}^d$ with
nearest-neighbor interactions. We develop a combinatorial framework for
establishing that exponential decay with distance of spin correlations,
specifically the strong spatial mixing condition (SSM), implies rapid mixing of
a large class of Markov chains. As a first application of our method we prove
that SSM implies $O(\log n)$ mixing of systematic scan dynamics (under mild
conditions) on an $n$-vertex $d$-dimensional cube of the integer lattice graph
$\mathbb{Z}^d$. Systematic scan dynamics are widely employed in practice but
have proved hard to analyze. A second application of our technology concerns
the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We
show that SSM implies an $O(1)$ bound for the relaxation time (i.e., the
inverse spectral gap). As a by-product of this implication we observe that the
relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$
is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for
all $q \ge 2$. We also use our combinatorial framework to give a simple
coupling proof of the classical result that SSM entails optimal mixing time of
the Glauber dynamics. Although our results in the paper focus on
$d$-dimensional cubes in $\mathbb{Z}^d$, they generalize straightforwardly to
arbitrary regions of $\mathbb{Z}^d$ and to graphs with subexponential growth.
