Entropy of unimodular lattice triangulations - Mecke, Klaus; Krüger, Benedikt; Reinhard, Johannes (EPL - Europhysics Letters, 2015)
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 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. Copyright (C) EPLA, 2015

Genus dependence of the number of (non-)orientable surface triangulations - Krüger, Benedikt; Mecke, Klaus (Physical Review D, 2016)
which might guide a mathematician's proof for the exact asymptotics.

Spectral Properties of Unimodular Lattice Triangulations - Krüger, Benedikt; Schmidt, Ella; Mecke, Klaus (Journal of Statistical Physics, 2016)
Random unimodular lattice triangulations have been recently used as an embedded random graph model, which exhibit a crossover behavior between an ordered, large-world and a disordered, small-world behavior. 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 behavior 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 behavior allows a tuning of important transport quantities.
Keywords: Triangulations; Random graphs; Networks; Spectral graph theory

Unimodular lattice triangulations as small-world and scale-free random graphs - Mecke, Klaus; Krüger, Benedikt; Schmidt, Ella (New Journal of Physics, 2015)
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.
Keywords: unimodular lattice triangulations; networks; maximal planar graphs