Entropy of unimodular Lattice Triangulations

TL;DR

This paper introduces a new algorithm using Wang-Landau sampling to approximate the entropy of 2D lattice triangulations, providing accurate estimates and improving asymptotic calculations.

Contribution

A novel Wang-Landau based algorithm for approximate enumeration of lattice triangulations, enhancing accuracy and applicability in combinatorics and physics.

Findings

Excellent agreement with known small triangulation counts

Improved asymptotic entropy estimates

Entropy density C=2.196(3) consistent with bounds

Abstract

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.