Entropy of unimodular lattice triangulations

Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where their entropy is crucial for many physical properties. Due to their inherent complex topological structure even the number of possible triangulations is unknown for large systems. We...

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Bibliographic Details
Published in:Europhysics letters 2015-02, Vol.109 (4), p.40011-p1-40011-p6
Main Authors: Knauf, Johannes F., Krüger, Benedikt, Mecke, Klaus
Format: Article
Language:English
Subjects:
02.10.Ox
05.10.Ln
64.60.aq
Algorithms
Asymptotic properties
Combinatorial analysis
Density
Entropy
Enumeration
Lattices
Lower bounds
Mathematical analysis
Physical properties
Quantum gravity
Sampling
Triangulation
Two dimensional analysis
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Summary:Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where their entropy is crucial for many physical properties. Due to their inherent complex topological structure even the number of possible triangulations is unknown for large systems. We present a novel algorithm for an approximate enumeration which is based on calculations of the density of states using the Wang-Landau flat histogram sampling. For triangulations on two-dimensional integer lattices we achieve excellent agreement with known exact numbers of small triangulations as well as an improvement of analytical calculated asymptotics. The entropy density is consistent with rigorous upper and lower bounds. The presented numerical scheme can easily be applied to other counting and optimization problems.
ISSN:0295-5075
1286-4854
DOI:10.1209/0295-5075/109/40011