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Genus dependence of the number of (non-)orientable surface triangulations
Computational Physics · Submitted 17 Mar 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We conjecture for the number of surface triangulations the asymptotic behavior \begin{equation*} N(m,h) \rightarrow (170.4 \pm 15.1)^h m^{-2(h - 1)/5} \left( \frac{256}{27} \right)^{m / 2}\;, \end{equation*} what might guide a mathematicians proof for the exact asymptotics.
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The abstract, sentence by sentence
Topological triangulations of orientable and non-orientable surfaces with arbitrary genus have important applications in quantum geometry, graph theory and statistical physics. However, until now only the asymptotics for 2-spheres are known analytically, and exact counts of triangulations are only available for both small genus and small triangulations. We apply the Wang-Landau algorithm to calculate the number $N(m,h)$ of triangulations for several order of magnitudes in system size $m$ and genus $h$. 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 and the next-to-next-to-leading order terms. We conjecture for the number of surface triangulations the asymptotic behavior \begin{equation*} N(m,h) \rightarrow (170.4 \pm 15.1)^h m^{-2(h - 1)/5} \left( \frac{256}{27} \right)^{m / 2}\;, \end{equation*} what might guide a mathematicians proof for the exact asymptotics.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Topological triangulations of orientable and non-orientable surfaces with arbitrary genus have important applications in quantum geometry, graph theory and statistical physics. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | However, until now only the asymptotics for 2-spheres are known analytically, and exact counts of triangulations are only available for both small genus and small triangulations. | gap 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s2 | We apply the Wang-Landau algorithm to calculate the number $N(m,h)$ of triangulations for several order of magnitudes in system size $m$ and genus $h$. | method 0.97 | 0.97 | 0.04 | asserted 1.00 |
| s3 | 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 and the next-to-next-to-leading order terms. | result 1.00 | 0.96 | 0.44 | asserted 1.00 |
| s4 | We conjecture for the number of surface triangulations the asymptotic behavior \begin{equation*} N(m,h) \rightarrow (170.4 \pm 15.1)^h m^{-2(h - 1)/5} \left( \frac{256}{27} \right)^{m / 2}\;, \end{equation*} what might guide a mathematicians proof for the exact asymptotics. | result 0.87 | 0.75 | 0.87 | speculative 1.00 |
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Original abstract
Topological triangulations of orientable and non-orientable surfaces with arbitrary genus have important applications in quantum geometry, graph theory and statistical physics. However, until now only the asymptotics for 2-spheres are known analytically, and exact counts of triangulations are only available for both small genus and small triangulations. We apply the Wang-Landau algorithm to calculate the number $N(m,h)$ of triangulations for several order of magnitudes in system size $m$ and genus $h$. 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 and the next-to-next-to-leading order terms. We conjecture for the number of surface triangulations the asymptotic behavior \begin{equation*} N(m,h) \rightarrow (170.4 \pm 15.1)^h m^{-2(h - 1)/5} \left( \frac{256}{27} \right)^{m / 2}\;, \end{equation*} what might guide a mathematicians proof for the exact asymptotics.
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Furthermore they are also a fundamental object within the group field theory approach to quantum gravity [ref] , which can be seen to relate the previously mentioned approaches.
The microscopic dynamics of quantum space as a group field theory · Introduction · 0.93
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