Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where its 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 achive excellent agreement with known exact numbers of small triangulations as well as an improvement of analytical calculated asymptotics. The entropy density is C = 2.196(3) consistent with rigorous upper and lower bounds. The presented numerical scheme can easily be applied to other counting and optimization problems.Introduction. -Triangulations of spaces are relevant for a broad range of physical phenomena. They serve as discretisation of all kinds of surfaces, hypersurfaces and volumes [1], yielding applications of computational geometry in physics, material science, medical image processing or even in computer graphics and visualisation [2][3][4][5][6]. Many physical systems can be described by random surface models [7] -based on random triangulations. For instance, biological membranes and vesicles can be modelled using triangulated surfaces with curvature-dependent Hamiltonians [8][9][10][11][12][13].Triangulations are also used as a random graph model for real world networks: Random Apollonian networks [14][15][16], which are the dual graphs of classical Apollonian packed granular matter and therewith triangulations, show both small-world and scale-free behaviour, as many real world networks. The properties of triangulations of closed surfaces with arbitrary genus are of much interest, since each graph can be embedded into a closed surface with high enough genus [17,18].The (Causal) Dynamical Triangulation approach even tries to describe quantum gravity from scratch with an ensemble of random space-time triangulations as their central entity [19]. Based on a discrete version of general relativity, where spacetime is approximated by triangles or higher-dimensional analogues, the curvatures become determined purly by the topological structure of the underlying triangulation, e.g. the number of triangles. The resulting action of the theory can be used to extract a

Real-world networks, e.g., the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages, using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real-world graphs; for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees ⩾ k 5. Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.

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, what might guide a mathematicians proof for the exact asymptotics.

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Random unimodular lattice triangulations have been recently used as an embedded random graph model, which exhibit a crossover behaviour between an ordered, large-world and a disordered, small-world behaviour. Using the ergodic Pachner flips that transform such triangulations into another and an energy functional that corresponds to the degree distribution variance, Markov chain Monte-Carlo simulations can be applied to study these graphs. Here, we consider the spectra of the adjacency and the Laplacian matrix as well as the algebraic connectivity and the spectral radius. Power law dependencies on the system size can clearly be identified and compared to analytical solutions for periodic ground states. For random triangulations we find a qualitative agreement of the spectral properties with well-known random graph models. In the microcanonical ensemble analytical approximations agree with numerical simulations. In the canonical ensemble a crossover behaviour can be found for the algebraic connectivity and the spectral radius, thus combining large-world and small-world behavior in one model. The considered spectral properties can be applied to transport problems on triangulation graphs and the crossover behaviour allows a tuning of important transport quantities.