  We give simply exponential lower bounds on the probabilities of a given
strongly Rayleigh distribution, depending only on its expectation. This
resolves a weak version of a problem left open by Karlin-Klein-Oveis Gharan in
their recent breakthrough work on metric TSP, and this resolution leads to a
minor improvement of their approximation factor for metric TSP. Our results
also allow for a more streamlined analysis of the algorithm.
  To achieve these new bounds, we build upon the work of Gurvits-Leake on the
use of the productization technique for bounding the capacity of a real stable
polynomial. This technique allows one to reduce certain inequalities for real
stable polynomials to products of affine linear forms, which have an underlying
matrix structure. In this paper, we push this technique further by
characterizing the worst-case polynomials via bipartitioned forests. This rigid
combinatorial structure yields a clean induction argument, which implies our
stronger bounds.
  In general, we believe the results of this paper will lead to further
improvement and simplification of the analysis of various combinatorial and
probabilistic bounds and algorithms.
