# Entropy of unimodular lattice triangulations

@article{Knauf2014EntropyOU, title={Entropy of unimodular lattice triangulations}, author={Johannes F. Knauf and Benedikt Kr{\"u}ger and Klaus Mecke}, journal={Europhysics Letters}, year={2014}, volume={109} }

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…

## 8 Citations

### Spectral Properties of Unimodular Lattice Triangulations

- MathematicsJournal 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…

### Genus dependence of the number of (non-)orientable surface triangulations

- Mathematics
- 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,…

### Unimodular lattice triangulations as small-world and scale-free random graphs

- Computer Science
- 2015

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.

### Flip Paths Between Lattice Triangulations

- MathematicsArXiv
- 2020

The first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such thatthere is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations.

### A Lyapunov function for Glauber dynamics on lattice triangulations

- MathematicsArXiv
- 2015

A height function is constructed on lattice triangulations and it is proved that, in the whole subcritical regime, the function behaves as a Lyapunov function with respect to Glauber dynamics; that is, thefunction is a supermartingale.

### Spectral Properties of Unimodular Lattice Triangulations

- Mathematics
- 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…

### A Lyapunov function for Glauber dynamics on lattice triangulations

- Materials ScienceProbability Theory and Related Fields
- 2016

We study random triangulations of the integer points [0,n]2∩Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

## References

SHOWING 1-10 OF 18 REFERENCES

### Triangulations: Structures for Algorithms and Applications

- Mathematics
- 2010

Triangulations appear everywhere, from volume computations and meshing to algebra and topology. This book studies the subdivisions and triangulations of polyhedral regions and point sets and presents…

### Building complex networks with Platonic solids.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

Analytical expressions for the degree distribution, the clustering coefficient, and the mean degree of nearest neighbors are derived showing that these networks have power-law degree distributions with tunable exponents associated with the building polyhedron, and they possess a hierarchical organization that is determined by planarity.

### Second- and first-order phase transitions in causal dynamical triangulations

- Physics
- 2012

Causal Dynamical Triangulations (CDT) is a proposal for a theory of quantum gravity, which implements a path-integral quantization of gravity as the continuum limit of a sum over piecewise flat…

### Reconstructing the universe

- Physics
- 2005

We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of…

### Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs.

- Computer SciencePhysical review letters
- 2005

A new family of networks are introduced that are simultaneously scale-free, small-world, Euclidean, space filling, and with matching graphs, that could be applied to the geometry of fully fragmented porous media, hierarchical road systems, and area-covering electrical supply networks.

### Residual entropy of ordinary ice from multicanonical simulations

- Computer Science, Physics
- 2007

Two simple models with nearest neighbor interactions on 3D hexagonal lattices are introduced and the correction to the residual entropy derived by Linus Pauling in 1935 is found to be within less than 0.1% of an analytical approximation by Nagle, which is an improvement of Pauling's result.

### Wang-Landau algorithm: a theoretical analysis of the saturation of the error.

- PhysicsThe Journal of chemical physics
- 2007

A theoretical analysis of the convergence of the Wang-Landau algorithm, introduced years ago to calculate the density of states in statistical models, concludes that the source of the saturation of the error is due to the decreasing variations of the refinement parameter.

### Maximal planar networks with large clustering coefficient and power-law degree distribution.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2005

It is proved that RANs are maximal planar networks, which are of particular practicability for layout of printed circuits and so on, and the diseases spread slower in R ANs than BA networks in the early stage of the susceptible-infected process, indicating that the large clustering coefficient may slow the spreading velocity.

### Grand canonical Monte Carlo simulations of elastic membranes with fluidity

- Physics, Mathematics
- 2003

### Flow-induced clustering and alignment of vesicles and red blood cells in microcapillaries

- EngineeringProceedings of the National Academy of Sciences
- 2009

This model combines a particle-based mesoscale simulation technique for the fluid hydrodynamics with a triangulated-membrane model and shows that already at very low HT, the deformability of RBCs implies a flow-induced cluster formation above a threshold flow velocity.