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Random Tilings: Concepts and Examples
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.Comment: 32 pages, 42 eps-figures, Latex2e updated version, minor grammatical
change