Tilings With Very Elastic Tiles

TL;DR

This paper investigates the pressure of tilings with elastic tiles, showing that as the tiles become smoother, the pressure converges uniformly to a simpler, computable limit, linking tiling theory to physical models.

Contribution

It establishes the uniform convergence of tiling pressure to a limit as tile smoothness increases, a fundamental but previously unproven result in tiling theory.

Findings

Pressure converges uniformly to the smooth limit as smoothness increases.

The limit pressure, , is easier to compute.

The result connects tiling pressure with models in mathematical physics.

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.