Random tilings of high symmetry: II. Boundary conditions and numerical studies

TL;DR

This paper uses numerical simulations to analyze high-symmetry random tilings with fixed boundary conditions, estimating their configurational entropy and comparing strained and unstrained cases.

Contribution

It provides detailed numerical analysis of boundary effects and entropy in high-symmetry random tilings, confirming mean-field predictions and exploring the thermodynamic limit.

Findings

Entropy approaches 0.568 per vertex in the continuous symmetry limit

Strained and unstrained tilings share the same entropy as predicted by mean-field theory

Fixed boundary entropy equals free boundary entropy in the thermodynamic limit

Abstract

We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in the limit of continuous symmetry. Tilings with phason strain appear to share the same entropy as unstrained tilings, as predicted by mean-field theory. We consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy, although these differ for finite rotational symmetry.