Tilings of a bounded region of the plane by maximal one-dimensional tiles

TL;DR

This paper investigates complex tilings of plane regions using variable-length one-dimensional tiles called K-mers, introducing a maximality constraint and an energy-based model that reveals potential phase transitions.

Contribution

It extends previous tiling models by allowing variable K-mers with a maximality constraint, and explores their statistical physics behavior including phase transitions.

Findings

Observation of unexpected behavior in tiling configurations

Evidence suggesting phase transitions as temperature varies

Introduction of an energy function based on cell contacts

Abstract

We study the tiling of a two-dimensional region of the plane by $K$-cell one-dimensional tiles, or $K$-mers. Unlike previous studies, which typically allowed for one single value of $K$ or sometimes a small assortment of fixed values, here a tiling may concomitantly employ $K$-mers comprising any number $K$ of cells, provided a maximality constraint is satisfied. In essence, this constraint requires each of the $K$-mers in use to be as lengthy as possible, given its surroundings in the resulting tiling. Maximality aims to limit the variety of possible tilings while allowing for interesting behavior in terms of the statistical physical observables of interest. In fact, by introducing an energy function based on cell contacts and parameterizing it appropriately, we have been able to observe relatively unexpected behavior, including the suggestion of phase transitions as the system's temperature evolves.