  We show fully polynomial time randomized approximation schemes (FPRAS) for
counting matchings of a given size, or more generally sampling/counting
monomer-dimer systems in planar, not-necessarily-bipartite, graphs. While
perfect matchings on planar graphs can be counted exactly in polynomial time,
counting non-perfect matchings was shown by [Jer87] to be #P-hard, who also
raised the question of whether efficient approximate counting is possible. We
answer this affirmatively by showing that the multi-site Glauber dynamics on
the set of monomers in a monomer-dimer system always mixes rapidly, and that
this dynamics can be implemented efficiently on downward-closed families of
graphs where counting perfect matchings is tractable. As further applications
of our results, we show how to sample efficiently using multi-site Glauber
dynamics from partition-constrained strongly Rayleigh distributions, and
nonsymmetric determinantal point processes.
  In order to analyze mixing properties of the multi-site Glauber dynamics, we
establish two notions for generating polynomials of discrete set-valued
distributions: sector-stability and fractional log-concavity. These notions
generalize well-studied properties like real-stability and log-concavity, but
unlike them robustly degrade under useful transformations applied to the
distribution. We relate these notions to pairwise correlations in the
underlying distribution and the notion of spectral independence introduced by
[ALO20], providing a new tool for establishing spectral independence based on
geometry of polynomials. As a byproduct of our techniques, we show that
polynomials avoiding roots in a sector of the complex plane must satisfy what
we call fractional log-concavity; this generalizes a classic result established
by [Gar59] who showed homogeneous polynomials that have no roots in a
half-plane must be log-concave over the positive orthant.
