Dresden 2014 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 22: Statistical Physics (general)

DY 22.12: Talk

Wednesday, April 2, 2014, 18:00–18:15, ZEU 160

Approximative counting of Manifold Triangulations — •Benedikt Krüger and Klaus Mecke — Institut für Theoretische Physik, FAU Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen

Each topological manifold in 2d and 3d permits a finite number of non-equivalent discretisations into combinatorial manifolds or triangulations with given number of vertices or maximal simplices. This number of distinct triangulations is important for questions arising in topology, geometry and physics. e.g. in quantum gravity [1].

Until now the best method for the counting of combinatorial manifolds was the isomorphism free enumeration of all possible triangulations which succeeded for vertex numbers below about 15 [2]. We use Monte-Carlo algorithms for estimating the number of triangulations of two- and three-dimensional manifolds and show that we are able to increase the known regime of triangulation counts by one magnitude. We give numerical evidence that the number of surface triangulations scales exponentially with the vertex number and that the rate of growth depends linearly on the genus of the surface.

[1] J. Ambjørn, J. Jurkiewicz, and R. Loll, Phys. Rev. D 72, 064014 (2005)

[2] T. Sulanke and F. H. Lutz, Eur. J. Comb. 30, 1965 (2009)