# Counting Cubic Maps with Large Genus

Z. Gao, M. Kang

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandBegutachtung

## Abstract

We derive an asymptotic expression for the number of cubic maps on orientable surfaces when the genus is proportional to the number of vertices. Let Σ_g denote the orientable surface of genus g and θ=g/n∈ (0,1/2). Given g,n∈ ℕ with g→ ∞ and n/2-g→ ∞ as n→ ∞, the number C_{n,g} of cubic maps on Σ_g with 2n vertices satisfies C_{n,g} ∼ (g!)² α(θ) β(θ)ⁿ γ(θ)^{2g}, as g→ ∞, where α(θ),β(θ),γ(θ) are differentiable functions in (0,1/2). This also leads to the asymptotic number of triangulations (as the dual of cubic maps) with large genus. When g/n lies in a closed subinterval of (0,1/2), the asymptotic formula can be obtained using a local limit theorem. The saddle-point method is applied when g/n→ 0 or g/n→ 1/2.
Originalsprache englisch 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms Michael Drmota, Clemens Heuberger Saarbrücken/Wadern Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH 13 9783959771474 18688969 https://doi.org/10.4230/LIPIcs.AofA.2020.13 Veröffentlicht - 2020 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms - https://www.math.aau.at/AofA2020/, Virtuell, ÖsterreichDauer: 15 Jun 2020 → 19 Jun 2020

### Publikationsreihe

Name Leibniz International Proceedings in Informatics, LIPIcs 159

### Konferenz

Konferenz 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms AofA2020 Österreich Virtuell 15/06/20 → 19/06/20

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## Fields of Expertise

• Information, Communication & Computing

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