Phase separation in tilings of a bounded region of the plane

Eduardo J. Aguilar, Valmir C. Barbosa, Raul Donangelo, Sergio R. Souza

Published 2023-08-07Version 1

Given a finite set of two-dimensional tile types, the field concerned with covering the plane with tiles of these types only has a long history, having enjoyed great prominence in the last six to seven decades, not only as a topic of recreational mathematics but mainly as a topic of scientific interest. Much of this interest has revolved around fundamental geometrical problems such as minimizing the variety of tile types to be used, and also around important applications in areas such as crystallography and others concerned with various atomic- and molecular-scale phenomena. All applications are of course confined to finite regions, but in many cases they refer back directly to progress in tiling the whole plane. Tilings of bounded regions of the plane have also been actively studied, but in general the additional complications imposed by the boundary conditions tend to constrain progress to mostly indirect results, such as recurrence relations. Here we study the tiling of rectangular regions of the plane by squares, dominoes, and straight tetraminoes. For this set of tile types, not even recurrence relations seem to be available. Our approach is to seek to characterize this system through some of the fundamental quantities of statistical physics. We do this on two parallel tracks, one fully analytical for a special case, the other based on the Wang-Landau method for state-density estimation. Given a simple Hamiltonian based solely on tile contacts, we have found either approach to lead to illuminating depictions of entropy, temperature, and above all phase separation. The notion of phase separation, in this context, refers to keeping track of how many tiles of each type are used in each of the many possibilities. We have found that this helps bind together different aspects of the system in question and conjecture that future applications will benefit from the possibilities it affords.