Zero-free regions of partition functions with applications to algorithms and graph limits

TL;DR

This paper identifies classes of graph partition functions that are zero-free on bounded degree graphs, enabling efficient approximation algorithms and establishing continuity in graph limits.

Contribution

It introduces a new class of zero-free partition functions and provides quasi-polynomial algorithms for their approximation and related graph polynomials.

Findings

Partition functions are zero-free on bounded degree graphs.

Quasi-polynomial time approximation schemes are developed.

Continuity of normalized partition functions in the Benjamini-Schramm topology.

Abstract

Based on a technique of Barvinok and Barvinok and Sober\'on we identify a class of edge-coloring models whose partition functions do not evaluate to zero on bounded degree graphs. Subsequently we give a quasi-polynomial time approximation scheme for computing these partition functions. As another application we show that the normalised partition functions of these models are continuous with respect the Benjamini-Schramm topology on bounded degree graphs. We moreover give quasi-polynomial time approximation schemes for evaluating a large class of graph polynomials, including the Tutte polynomial, on bounded degree graphs.