Tilings With Very Elastic Dimers

TL;DR

This paper proves that the pressure of weighted dimer tilings on a lattice converges uniformly to a simple limit as the weighting function becomes smoother, under certain basic conditions, advancing understanding of tiling entropy.

Contribution

It establishes a uniform convergence result for the pressure of weighted dimers as the weighting function's smoothness increases, removing previous fall-off restrictions.

Findings

Pressure converges uniformly to the limit p_0 as smoothness increases.

The result holds without fall-off conditions, assuming dimers connect black and white vertices.

The theorem simplifies understanding of entropy in lattice tilings.

Abstract

We consider tiles (dimers) each of which covers two vertices of a rectangular lattice. There is a normalized translation invariant weighting on the shape of the tiles. We study the pressure, p, or entropy, (one over the volume times the logarithm of the partition function). We let p_0 (easy to compute) be the pressure in the limit of absolute smoothness (the weighting function is constant). We prove that as the smoothness of the weighting function, suitably defined, increases, p converges to p_0, uniformly in the volume. It is the uniformity statement that makes the result non-trivial. In an earlier paper the author proved this, but with an additional requirement of a certain fall-off on the weighting function. Herein fall-off is not demanded, but there is the technical requirement that each dimer connect a black vertex with a white vertex, vertices colored as on a checker board. This seems like a very basic result in the theory of pressure (entropy) of tilings.