An alternative view on random tilings

TL;DR

This paper explores a new framework for describing quasicrystalline random tilings without relying on height representations, demonstrating its consistency with traditional methods and providing a group theoretic criterion for its validity.

Contribution

It introduces an alternative description of random tilings that does not depend on height functions and validates it through group theory, expanding the theoretical understanding of quasicrystalline tilings.

Findings

The new framework is consistent with conventional descriptions.

Several key examples of quasicrystalline tilings are analyzed.

A group theoretic criterion for the random tiling hypothesis is derived.

Abstract

We apply a framework for the description of random tilings without height representation, which was proposed recently, to the special case of quasicrystalline random tilings. Several important examples are discussed, thereby demonstrating the consistency of this alternative description with the conventional one. We also clarify the latter by deriving a group theoretic criterion for the validity of the first random tiling hypothesis.