Random tilings of high symmetry: I. Mean-field theory

TL;DR

This paper develops a mean-field theory for high-symmetry random tilings, providing predictions for entropy, correlations, and elasticity, primarily in two dimensions, with brief considerations of higher dimensions.

Contribution

It introduces a novel mean-field approach based on an iterative tiling construction to analyze high-symmetry random tilings.

Findings

Mean-field theory accurately predicts configurational entropy in 2D tilings.

Correlation functions and phason elasticity are analyzed within the model.

Brief discussion on tilings in dimensions higher than two.

Abstract

We study random tiling models in the limit of high rotational symmetry. In this limit a mean-field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions. We base our mean-field theory on an iterative tiling construction inspired by the work of de Bruijn. In addition to the entropy, we consider correlation functions, phason elasticity and the thermodynamic limit. Tilings of dimension other than two are considered briefly.