Random Tilings: Concepts and Examples

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.

32 pages, 42 eps-figures, Latex2e updated version, minor grammatical changes

References in corpus (4)

• Local statistics of lattice dimers
• A constrained Potts antiferromagnet model with an interface representation
• The Entropy of Square-Free Words
• Bethe Ansatz solution of a decagonal rectangle triangle random tiling

Cited by in corpus (16)

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• Quasicrystals in a Monodisperse System
• Diffraction of random tilings: some rigorous results
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• Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
• Classical Dimers on Penrose Tilings
• A short guide to pure point diffraction in cut-and-project sets
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• q-linear approximants: Scaling functions for polygon models
• An alternative view on random tilings
• Random colourings of aperiodic graphs: Ergodic and spectral properties
• Repetitive Delone Sets and Quasicrystals
• Tiles and colors
• Diffraction of the Dart-Rhombus Random Tiling
• On the Entropy and Letter Frequencies of Ternary Square-Free Words