2008•arXiv (Cornell University)Open access

Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights

Bill Jackson, Aldo Procacci, Alan D. Sokal

Abstract

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) Z_G(q,w) of a graph G with general complex edge weights w = {w_e}. This generalizes a result of Sokal (cond-mat/9904146) that applies only within the complex antiferromagnetic regime |1+w_e| \le 1. Our proof uses the polymer-gas representation of the multivariate Tutte polynomial together with the Penrose identity.

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Key concepts: Potts model, Tutte polynomial, Chromatic polynomial, Partition function (quantum field theory), Combinatorics, Mathematics, Partition (number theory), Complex plane