Non-recursive counts of graphs on surfaces

Ercolani, Nicholas M.; Lega, J.; Tippings, Brandon

doi:10.54550/eca2023v3s3r20

Cited by 2 publications

(17 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where the bidiagonal matrices M (j) depend on the coefficients α j,ℓ and β j,ℓ . In addition, the vector V (j) has only a finite number of non-zero entries, consistent with the expression given in (2.5) [ELT23b].…”

Section: Dynamical Perspectivessupporting

confidence: 79%

“…This section explains how ideas from dynamical systems theory, both linear and nonlinear, lead to explicit expressions for the generating functions z g (t, α) and e g (t, α) and, in the 4-valent case, to explicit formulas for map counts. We focus on the broad ideas and refer the interested reader to [ELT22,ELT23a,ELT23b] for proofs and further details.…”

Section: Dynamical Perspectivesmentioning

confidence: 99%

“…where the bracket [i] refers to the i th entry of the corresponding vector. Whereas the semi-infinite matrices M (n) and M (j) do not commute when n ̸ = j, it turns out that the matrices A (n) and A (j) always do [ELT23b]! Writing…”

Section: Dynamical Perspectivesmentioning

confidence: 99%

“…Recently, we showed [ELT23b] that such a formulation provides an explicit expression for the counts in terms of hypergeometric functions. Specifically, because the e g are analytic in a neighborhood of t = 0, one may write…”

Section: Introduction: Map Enumeration Random Matrices and Orthogonal...mentioning

confidence: 99%

“…The remainder of this article is organized as follows. Section 2 describes how adding a dynamical perspective to the results of [EM03, EMP08, Er11, Er14] leads to explicit formulations for the generating functions z g and e g in terms of z 0 [Tip20, ELT22, ELT23a], as well as to non-recursive expressions for four-valent map counts [ELT23b]. Section 3 explains how all of these advances may be combined into a general framework for map enumeration [Tip20,ELT23a].…”

Section: Introduction: Map Enumeration Random Matrices and Orthogonal...mentioning

confidence: 99%

See 4 more Smart Citations

Multiple scale asymptotics of map enumeration

2023

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Dynamical Perspectivessupporting

confidence: 79%

Section: Dynamical Perspectivesmentioning

confidence: 99%

Section: Dynamical Perspectivesmentioning

confidence: 99%

Section: Introduction: Map Enumeration Random Matrices and Orthogonal...mentioning

confidence: 99%

Section: Introduction: Map Enumeration Random Matrices and Orthogonal...mentioning

confidence: 99%

See 3 more Smart Citations

Multiple scale asymptotics of map enumeration

2023

Self Cite

View full text Add to dashboard Cite

show abstract

Map enumeration from a dynamical perspective

Ercolani,

Lega,

Tippings

2024

Contemporary Mathematics

View full text Add to dashboard Cite

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 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.

show abstract

Non-recursive counts of graphs on surfaces

Cited by 2 publications

References 0 publications

Multiple scale asymptotics of map enumeration

Multiple scale asymptotics of map enumeration

Map enumeration from a dynamical perspective

Contact Info

Product

Resources

About