Multiple Scale Asymptotics of Map Enumeration

TL;DR

This paper develops a unified asymptotic framework for enumerating maps on high-genus surfaces, connecting Riemann-Hilbert analysis and dynamical systems to derive new generating function expressions.

Contribution

It introduces a systematic method to relate generating functions for map enumeration across different surface genera using asymptotic expansions.

Findings

Recovered known enumeration results for surfaces of genus 0 to 7

Derived new expressions for generating functions of maps

Extended methodology to regular maps of arbitrary even valence

Abstract

We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two asymptotic expansions obtained from two different fields of mathematics: the Riemann-Hilbert analysis of orthogonal polynomials and the theory of discrete dynamical systems. By equating the coefficients of these expansions in a common region of uniform validity in their parameters, we recover known results and provide new expressions for generating functions associated with graphical enumeration on surfaces of genera 0 through 7. Although the body of the article focuses on 4-valent maps, the methodology presented here extends to regular maps of arbitrary even valence and to some cases of odd valence, as detailed in the appendices.