Random trimer tilings

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 block corresponding to the largest eigenvalue 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 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.