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

TL;DR

The paper investigates tilings of a bounded plane region using maximal one-dimensional $K$-mers, allowing a range of lengths under a maximality constraint. By encoding tile orientations with Ising-like row configurations and formulating an energy based on nearest-neighbor interactions with parameters $f_L$, $f_U$, $g_L$, and $g_U$, the authors apply a transfer-matrix approach on a toroidal $m\times n$ lattice to compute thermodynamic observables. They identify a parameter-dependent phase boundary marked by a peak in the heat capacity $C_V$ and a growing correlation length $\xi$, indicating ordering at a critical temperature $T_c$ that shifts with parameter choices; the low-temperature regime exhibits distinct ordered and disordered (spin-glass-like) behaviors, including residual entropy in certain limits. The work demonstrates that the transfer-matrix framework can accommodate concomitant use of multiple $K$-mers under maximality and suggests future work with Monte Carlo methods to study larger systems and broader energy landscapes.

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.

Figures (10)

• Figure 1: Example of a tiling for $m=n=3$. The coordinates are chosen so that the origin is located at the bottom left corner. Cell state $-1$ is represented by horizontal stripes, cell state $+1$ by vertical stripes. The row and column indices increase from left to right and from bottom to top, respectively. Owing to the periodic boundary conditions, two horizontal and two vertical dimers are present in this configuration. For details, see the text.
• Figure 2: Left axis: Heat capacity for different system sizes, obtained with $f_L=1, f_U=2, g_L=1,$ and $g_U=2$. Right axis: The inverse of the correlation length for the same system sizes and parameters. Continuous lines represent the heat capacity while dashed lines depict the inverse of the correlation length. For details, see the text.
• Figure 3: Same as Fig. \ref{['fig:cv12_12']}, now for $f_L=1, f_U=4,g_L=1$, and $g_U=4$.
• Figure 4: Same as Fig. \ref{['fig:cv12_12']}, now for $f_L=1, f_U=2,g_L=1$, and $g_U=4$.
• Figure 5: Inverse of the correlation length as a function of temperature for $n=m=16$ and different values of $f_U$ and $g_U$ (see legends). In both cases, $f_L=g_L=1$. For details, see the text.