Genus dependence of the number of (non-)orientable surface triangulations
Benedikt Krüger, Klaus R Mecke · Physical review. D/Physical review. D. · 2016
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 $N(m,h)\ensuremath{\rightarrow}(170.4\ifmmode\pm\else\textpm\fi{}15.1{)}^{h}{m}^{\ensuremath{-}2(h\ensuremath{-}1)/5}{(\frac{256}{27})}^{m/2},$which might guide a mathematician's proof for the exact asymptotics.