Multiple Scale Asymptotics of Map Enumeration
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.
Cite
@article{arxiv.2210.00668,
title = {Multiple Scale Asymptotics of Map Enumeration},
author = {Nicholas Ercolani and Joceline Lega and Brandon Tippings},
journal= {arXiv preprint arXiv:2210.00668},
year = {2023}
}