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Map enumeration from a dynamical perspective

Nicholas Ercolani, Joceline Lega, Brandon Tippings

TL;DR

The work develops a rigorous, dynamical-systems–inspired framework for map enumeration on genus-g surfaces by coupling discrete Painlevé dynamics with orthogonal-polynomial asymptotics. Central to the approach are four pillars: genus expansion for 2-legged maps, center-manifold expansions around Fixed Points, a rational form for z_g(z0), and a common region of validity that ties the two expansions together to yield explicit generating functions e_g(z0) and counts for 4-valent maps. The methodology produces explicit expressions and non-recursive counts in the 4-valent case and extends, in progress, to mixed-valence (3 and 4) maps and the singular 3-valent limit, with several conjectures and open problems guiding future work. This program connects map enumeration to random matrices, orthogonal polynomials, and discrete dynamical systems, and reveals deep links to special functions and moduli-space invariants such as Bernoulli numbers and orbifold Euler characteristics.

Abstract

This contribution summarizes recent work of the authors that combines methods from dynamical systems theory (discrete Painlevé 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.

Map enumeration from a dynamical perspective

TL;DR

The work develops a rigorous, dynamical-systems–inspired framework for map enumeration on genus-g surfaces by coupling discrete Painlevé dynamics with orthogonal-polynomial asymptotics. Central to the approach are four pillars: genus expansion for 2-legged maps, center-manifold expansions around Fixed Points, a rational form for z_g(z0), and a common region of validity that ties the two expansions together to yield explicit generating functions e_g(z0) and counts for 4-valent maps. The methodology produces explicit expressions and non-recursive counts in the 4-valent case and extends, in progress, to mixed-valence (3 and 4) maps and the singular 3-valent limit, with several conjectures and open problems guiding future work. This program connects map enumeration to random matrices, orthogonal polynomials, and discrete dynamical systems, and reveals deep links to special functions and moduli-space invariants such as Bernoulli numbers and orbifold Euler characteristics.

Abstract

This contribution summarizes recent work of the authors that combines methods from dynamical systems theory (discrete Painlevé equations) and asymptotic analysis of orthogonal polynomial recurrences, to address long-standing questions in map enumeration. Given a genus , 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.
Paper Structure (19 sections, 85 equations, 3 figures, 2 tables)

This paper contains 19 sections, 85 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction: map enumeration, random matrices, and orthogonal polynomials
  2. Dynamical perspectives
  3. Orthogonal polynomials, Painlevé equation, and center manifold
  4. Vector recurrence
  5. The 4-valent case
  6. A new methodology for map enumeration
  7. Preliminary results on the mixed 3- and 4-valent case
  8. Derivation of the mixed valence discrete dynamical systems
  9. Center manifold
  10. Conclusions: open problems and challenges
  11. 3-valent case
  12. Rational forms
  13. Center manifold expansion
  14. Conjectures and open questions
  15. An explicit identity involving the Gauss hypergeometric function
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Labeled maps with 1 vertex on a surface of genus 0.
  • Figure 2: The methodology proposed in bib:elt22bib:elt23abib:elt23b to obtain counts of 4-valent maps.
  • Figure 3: Numerical simulation of the Freud orbit (dots) for the mixed valence case and plot of the center manifold of $P_\infty$ (solid curve) expanded to fourth order in $u$. The dynamics is shown in $(s,f,u,z,w)$ coordinates. The dots are colored according to the value of the iteration index $n \in \{2, 3, \cdots, 700\}$, with blue (resp. red) corresponding to low (resp. high) indices. The Freud orbit converges to $P_\infty$, which is on the invariant hyperplane $u=0$ in grey).

Theorems & Definitions (2)

  • Conjecture 5.1
  • Conjecture 5.2