Map enumeration from a dynamical perspective

TL;DR

This paper integrates dynamical systems and asymptotic analysis to develop a framework for enumerating maps on surfaces of any genus, providing explicit formulas for 4-valent maps and general methods for other valences.

Contribution

It introduces a novel approach combining Painlevé equations and orthogonal polynomial recurrences to compute map enumeration generating functions for arbitrary genus.

Findings

Explicit formulas for 4-valent map counts.

A general method for maps with even or mixed valence.

Connections established between map enumeration and dynamical systems theory.

Abstract

This contribution summarizes recent work of the authors that combines methods from dynamical systems theory (discrete Painlev\'e equations) and asymptotic analysis of orthogonal polynomial recurrences, to address long-standing questions in map enumeration. Given a genus $g$, we present a framework that provides the generating function for the number of maps that can be realized on a surface of that genus. In the case of 4-valent maps, our methodology leads to explicit expressions for map counts. For general even or mixed valence, the number of vertices of the map specifies the relevant order of the derivatives of the generating function that needs to be considered. Beyond summarizing our own results, we provide context for the program highlighted in this article through a brief review of the literature describing advances in map enumeration. In addition, we discuss open problems and challenges related to this fascinating area of research that stands at the intersection of statistical physics, random matrices, orthogonal polynomials, and discrete dynamical systems theory.