Dichromatic polynomials and Potts models summed over rooted maps

TL;DR

This paper explores the sum of dichromatic polynomials over rooted planar maps, linking it to the q-state Potts model, revealing a phase transition at a high critical q value around 72.

Contribution

It derives new results on the enumeration of rooted maps and their connection to Potts models, extending known theories to higher q values.

Findings

The Potts model exhibits a first-order transition at q > 72.

New enumeration formulas for rooted planar maps are presented.

The critical q value for phase transition is significantly larger than in regular lattices.

Abstract

We consider the sum of dichromatic polynomials over non-separable rooted planar maps, an interesting special case of which is the enumeration of such maps. We present some known results and derive new ones. The general problem is equivalent to the $q$-state Potts model randomized over such maps. Like the regular ferromagnetic lattice models, it has a first-order transition when $q$ is greater than a critical value $q_c$, but $q_c$ is much larger - about 72 instead of 4.