Random trimer tilings

We study tilings of the square lattice by linear trimers. For a cylinder of circumference $m$, we construct a conserved functional of the base of the tilings, and use this to block diagonalize the transfer matrix. The number of blocks increases exponentially with $m$. The dimension of the block corresponding to the largest eigenvalue is shown to grow as $(3∕2^{1∕3}{)}^m$. We numerically diagonalize this block for $m\le 27$, obtaining the estimate $S_{\infty }=0.158520\pm 0.000015$ for the entropy per site in the thermodynamic limit. We present numerical evidence that the continuum limit of the model has conformal invariance. We measure several scaling dimensions, including those corresponding to defects of monomers and $L$-shaped trimers. The trimer tilings of a plane admits a two-dimensional height representation. Monte Carlo simulations of the height variables show that the height-height correlations grows logarithmically at large separation, and the orientation-orientation correlations decay as a power law.

Figure 1

A trimer tiling with periodic boundary conditions in the vertical direction.

Figure 2

Choice of unit vectors.

Figure 3

(a) A trimer tiling of $3\times 5$ lattice with assignment of labels to the edges of the lattice (b) An alternative assignment of labels (c) the height configuration corresponding to label in (a), where $(n_1,n_2)$ denotes the height $\mathbf{h}(n_1,n_2)\equiv n_1{\mathbf{e}}_1+n_2{\mathbf{e}}_2$.

Figure 4

The flip move exchanging three vertical trimers with three horizontal trimers.

Figure 5

(Color online) (a) $3\times 3$ superlattice decomposition of the square lattice into nine sublattices. (b) Height variables corresponding to a tiling of the square lattice form a triangular lattice.

Figure 6

Partial filling on top of a given base (shown hatched).

Figure 7

Labeling of the $R$ matrix.

Figure 8

Convergence of $S_m$ with $1∕m$ for $m=0\left(\mathrm{mod}3\right)$ $(\star )$, $m=1\left(\mathrm{mod}3\right)$ $(●)$, $m=2\left(\mathrm{mod}3\right)$ $(◯)$. Continuous lines are fits from Eq. (28) that converges to $S_{\infty }=0.15852$ for $d=2.774$ and $c$ and $e$ as listed in Table .

Figure 9

(Color online) The correlation length $\xi$ plotted against the cylinder width $m$. The dotted line has the slope $a=0.23$.

Figure 10

Correlation function $H(x,y)$ for $L=60$.

Figure 11

$\mid f(R,\theta )\mid$ as functions of $R$ for $\theta =0$ $(●)$, $\theta =\pi ∕4$ $(◯)$, $\theta =\pi ∕2$ $(\star )$. Straight lines have slope $-1.5$.