  This paper formalizes connections between stability of polynomials and
convergence rates of Markov Chain Monte Carlo (MCMC) algorithms. We prove that
if a (multivariate) partition function is nonzero in a region around a real
point $\lambda$ then spectral independence holds at $\lambda$. As a
consequence, for Holant-type problems (e.g., spin systems) on bounded-degree
graphs, we obtain optimal $O(n\log n)$ mixing time bounds for the single-site
update Markov chain known as the Glauber dynamics. Our result significantly
improves the running time guarantees obtained via the polynomial interpolation
method of Barvinok (2017), refined by Patel and Regts (2017).
  There are a variety of applications of our results. In this paper, we focus
on Holant-type (i.e., edge-coloring) problems, including weighted edge covers
and weighted even subgraphs. For the weighted edge cover problem (and several
natural generalizations) we obtain an $O(n\log{n})$ sampling algorithm on
bounded-degree graphs. The even subgraphs problem corresponds to the
high-temperature expansion of the ferromagnetic Ising model. We obtain an
$O(n\log{n})$ sampling algorithm for the ferromagnetic Ising model with a
nonzero external field on bounded-degree graphs, which improves upon the
classical result of Jerrum and Sinclair (1993) for this class of graphs. We
obtain further applications to antiferromagnetic two-spin models on line
graphs, weighted graph homomorphisms, tensor networks, and more.
