Lee-Yang phenomena in edge-coloured graph counting

TL;DR

This paper investigates the distribution of zeros in a polynomial related to counting edge-coloured graphs, revealing Lee-Yang phenomena analogous to phase transitions in statistical physics.

Contribution

It introduces a new perspective on the zero distribution of a graph polynomial related to the Ising model, connecting combinatorics with complex analysis and physics.

Findings

Zeros accumulate along semialgebraic limit curves.

Limit curves are derived from anti-Stokes curves of an exponential integral.

The polynomial generalizes the edge-chromatic polynomial and relates to the Ising model.

Abstract

We study the accumulation of zeros of a polynomial arising from the enumeration of edge-coloured graphs along certain limit curves. The polynomial is a variant of an edge-chromatic polynomial, which specialises to the partition function of the ferromagnetic Ising model on a random regular graph. We call this accumulation behaviour a Lee-Yang phenomenon in analogy with the Lee-Yang theorem. The limiting loci are semialgebraic and arise from anti-Stokes curves of an exponential integral.