Random Tilings: Concepts and Examples

TL;DR

This paper introduces a generalized concept of random tilings applicable beyond traditional height representations, explores the entropy related to tile densities, and provides examples and counterexamples to existing hypotheses.

Contribution

It generalizes the first random tiling hypothesis and offers new insights into the entropy function, including counterexamples to the second hypothesis.

Findings

Proves a generalized connection between maximum entropy and symmetry.

Provides explicit examples from exactly solvable models.

Identifies counterexamples to the second random tiling hypothesis.

Abstract

We introduce a concept for random tilings which, comprising the conventional one, is also applicable to tiling ensembles without height representation. In particular, we focus on the random tiling entropy as a function of the tile densities. In this context, and under rather mild assumptions, we prove a generalization of the first random tiling hypothesis which connects the maximum of the entropy with the symmetry of the ensemble. Explicit examples are obtained through the re-interpretation of several exactly solvable models. This also leads to a counterexample to the analogue of the second random tiling hypothesis about the form of the entropy function near its maximum.