Zeros for Partition Functions and Graph-counting Polynomials.
Talkat Deterministic Counting, Probability, and Zeros of Partition Functions (Simons Institute workshop, 2019) · 2019-03-18..03-22
summary
Extends Asano’s lemma to multi-affine polynomials and uses repeated Asano contraction of vertex polynomials to produce edge-variable subgraph-counting polynomials; proves monomer–dimer (matching) polynomials have zeros in the negative real axis and unbranched-subgraph polynomials avoid the right half-plane, by mapping allowed zero-free regions (half-planes → parabolas) under contractions. Shows factorization into linear/quadratic positive-coefficient factors and uses that location of zeros plus independence heuristics to derive an asymptotic Gaussian law for coefficients on large lattices.
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