Random trimer tilings

TL;DR

This paper investigates the statistical mechanics of random trimer tilings on a square lattice, deriving entropy estimates, analyzing conformal invariance, and exploring height correlations through numerical and Monte Carlo methods.

Contribution

It introduces a conserved functional to block-diagonalize the transfer matrix, estimates the entropy per site, and provides numerical evidence for conformal invariance in the continuum limit.

Findings

Entropy per site estimated as 0.158520 +- 0.000015

Height-height correlations grow logarithmically with distance

Orientation correlations decay as a power law

Abstract

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 ground-state block is shown to grow as (3 / 2^{1/3})^m. We numerically diagonalize this block for m <= 27, obtaining the estimate S = 0.158520 +- 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 dimers 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.