Two-dimensional random tilings of large codimension: new progress

TL;DR

This paper develops a mean-field theory to analyze the thermodynamic properties of two-dimensional random tilings projected from high-dimensional spaces, focusing on the large codimension limit and comparing with numerical results.

Contribution

It introduces a new mean-field approach for large codimension tilings and investigates the entropy and correlations, advancing understanding of their thermodynamic behavior.

Findings

Thermodynamic properties become boundary-condition independent at large D.

The mean-field theory provides estimates for the limiting entropy.

Numerical comparisons validate the theoretical predictions.

Abstract

Two-dimensional random tilings of rhombi can be seen as projections of two-dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a $D$-dimensional space. We study the limiting case, when the quantity $D$, and therefore the number of different species of tiles, become large. We had previously demonstrated [ICQ6] that, in this limit, the thermodynamic properties of the tiling become independent of the boundary conditions. The exact value of the limiting entropy and finite $D$ corrections remain open questions. Here, we develop a mean-field theory, which uses an iterative description of the tilings based on an analogy with avoiding oriented walks on a random tiling. We compare the quantities so-obtained with numerical calculations. We also discuss the role of spatial correlations.