  We show that the existence of a ``good'' coupling w.r.t. Hamming distance for
any local Markov chain on a discrete product space implies rapid mixing of the
Glauber dynamics in a blackbox fashion. More specifically, we only require the
expected distance between successive iterates under the coupling to be
summable, as opposed to being one-step contractive in the worst case. Combined
with recent local-to-global arguments \cite{CLV21}, we establish asymptotically
optimal lower bounds on the standard and modified log-Sobolev constants for the
Glauber dynamics for sampling from spin systems on bounded-degree graphs when a
curvature condition \cite{Oll09} is satisfied. To achieve this, we use Stein's
method for Markov chains \cite{BN19, RR19} to show that a ``good'' coupling for
a local Markov chain yields strong bounds on the spectral independence of the
distribution in the sense of \cite{ALO20}.
  Our primary application is to sampling proper list-colorings on
bounded-degree graphs. In particular, combining the coupling for the flip
dynamics given by \cite{Vig00, CDMPP19} with our techniques, we show optimal
$O(n\log n)$ mixing for the Glauber dynamics for sampling proper list-colorings
on any bounded-degree graph with maximum degree $\Delta$ whenever the size of
the color lists are at least $\left(\frac{11}{6} - \epsilon\right)\Delta$,
where $\epsilon \approx 10^{-5}$ is small constant. While $O(n^{2})$ mixing was
already known before, our approach additionally yields Chernoff-type
concentration bounds for Hamming Lipschitz functions in this regime, which was
not known before. Our approach is markedly different from prior works
establishing spectral independence for spin systems using spatial mixing
\cite{ALO20, CLV20, CGSV20, FGYZ20}, which crucially is still open in this
regime for proper list-colorings.
