Non-recursive Counts of Graphs on Surfaces

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

doi:10.48550/arxiv.2210.00671

Cited by 1 publication

(1 citation statement)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When ν = 2, the expressions for z g and e g presented in this article may be used to derive closed-form expressions for N g,ν (j) for all genera for which z g (z 0 ) is known. This work is beyond the scope of the present article and will be described separately [ELT22b]. In summary, the present article illustrates how results from Riemann-Hilbert analysis and either Plancherel-Rotach asymptotics or centre manifold theory may be combined to provide a solution to a longstanding combinatorial problem in map enumeration.…”

Section: Closed-form Expressions For the Map Countsmentioning

confidence: 97%

Multiple scale asymptotics of map enumeration

2023

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: Closed-form Expressions For the Map Countsmentioning

confidence: 97%