  We define a discrete-time Markov chain for abstract polymer models and show
that under sufficient decay of the polymer weights, this chain mixes rapidly.
We apply this Markov chain to polymer models derived from the hard-core and
ferromagnetic Potts models on bounded-degree (bipartite) expander graphs. In
this setting, Jenssen, Keevash and Perkins (2019) recently gave an FPTAS and an
efficient sampling algorithm at sufficiently high fugacity and low temperature
respectively. Their method is based on using the cluster expansion to obtain a
complex zero-free region for the partition function of a polymer model, and
then approximating this partition function using the polynomial interpolation
method of Barvinok.
  Our approach via the polymer model Markov chain circumvents the zero-free
analysis and the generalization to complex parameters, and leads to a sampling
algorithm with a fast running time of $O(n \log n)$ for the Potts model and
$O(n^2 \log n)$ for the hard-core model, in contrast to typical running times
of $n^{O(\log \Delta)}$ for algorithms based on Barvinok's polynomial
interpolation method on graphs of maximum degree $\Delta$. We finally combine
our results for the hard-core and ferromagnetic Potts models with standard
Markov chain comparison tools to obtain polynomial mixing time for the usual
spin Glauber dynamics restricted to even and odd or `red' dominant portions of
the respective state spaces.
