 07181 Paul Federbush
 Tilings with very Elastic Tiles
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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 nontrivial. This seems
like a very basic result in the theory of pressure of tilings.
Though at the same time, perhaps nonglamorous, 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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