Multiple scale asymptotics of map enumeration

Nicholas Ercolani, Joceline Lega, Brandon Tippings

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1663-1698
Number of pages36
Issue number3
StatePublished - Mar 1 2023


  • asymptotic analysis
  • graphical enumeration
  • integrable systems

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics


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