Genus dependence of the number of (non-)orientable surface triangulations

Benedikt Krüger and Klaus Mecke
Phys. Rev. D 93, 085018 – Published 13 April 2016

Abstract

Topological triangulations of orientable and nonorientable surfaces with arbitrary genus have important applications in quantum geometry, graph theory and statistical physics. However, until now, only the asymptotics for 2-spheres have been known analytically, and exact counts of triangulations are only available for both small genera and triangulations. We apply the Wang-Landau algorithm to calculate the number N(m,h) of triangulations for several orders of magnitude in system size m and type h (equals genus in orientable triangulations). We verify that the limit of the entropy density of triangulations is independent of genus and orientability and are able to determine the next-to-leading-order and the next-to-next-to-leading-order terms. We conjecture for the number of surface triangulations the asymptotic behavior N(m,h)(170.4±15.1)hm2(h1)/5(25627)m/2,which might guide a mathematician’s proof for the exact asymptotics.

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  • Received 27 July 2015

DOI:https://doi.org/10.1103/PhysRevD.93.085018

© 2016 American Physical Society

Physics Subject Headings (PhySH)

Particles & Fields

Authors & Affiliations

Benedikt Krüger* and Klaus Mecke

  • FAU Erlangen-Nuremberg, Institute for Theoretical Physics, Staudtstraße 7, 91058 Erlangen, Germany

  • *benedikt.krueger@fau.de

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Issue

Vol. 93, Iss. 8 — 15 April 2016

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