07-181 Paul Federbush

Tilings with very Elastic Tiles (40K, LaTeX) Jul 13, 07

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Abstract. We consider tiles of some fixed size, with an associated weighting on the shapes of tile, of total mass 1. We study the pressure, $p$, of tilings with those tiles; the pressure, one over the volume times the logarithm of the partition function. (The quantity we define as ``pressure" could, perhaps equally harmoniously with physics notation, be called ``entropy per volume", neither nomenclature is ``correct".) We let $\hat p^0$ (easy to compute) be the pressure in the limit of absolute smoothness (the weighting function is constant). Then as smoothness, suitably defined, increases, $p$ converges to $\hat p^0$, uniformly in the volume. It is the uniformity requirement that makes the result non-trivial. This seems like a very basic result in the theory of pressure of tilings. Though at the same time, perhaps non-glamorous, being bereft of geometry and not very difficult. The problem arose for us out of study of a problem in mathematical physics, associated to a model of ferromagnetism.

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