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 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.

12 pages, 11 figures

References in corpus (11)

• Coulomb and Liquid Dimer Models in Three Dimensions
• Classical dimers on the triangular lattice
• Interacting classical dimers on the square lattice
• Logarithmic Conformal Field Theory and Boundary Effects in the Dimer Model
• Unconventional continuous phase transition in a three dimensional dimer model
• Slow Relaxation in a Model with Many Conservation Laws : Deposition and Evaporation of Trimers on a Line
• Triangular Trimers on the Triangular Lattice: an Exact Solution
• Orientational correlations and the effect of spatial gradients in the equilibrium steady state of hard rods in 2D : A study using deposition-evaporation kinetics
• Continuous melting of compact polymers
• Conformal field theory of the Flory model of polymer melting
• The Irreducible String and an Infinity of Additional Constants of Motion in a Deposition-Evaporation Model on a Line

Cited by in corpus (25)

• On the orientational ordering of long rods on a lattice
• The nematic-disordered phase transition in systems of long rigid rods on two dimensional lattices
• Thermodynamics of massless particles in curved spacetime
• Stability of columnar order in assemblies of hard rectangles or squares
• Entropy of fully packed hard rigid rods on $d$-dimensional hyper-cubic lattices
• Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics
• Integrability and conformal data of the dimer model
• Non-Local Finite-Size Effects in the Dimer Model
• The phase transition from nematic to high-density disordered phase in a system of hard rods on a lattice
• Entropy of rigid k-mers on a square lattice
• Refined conformal spectra in the dimer model
• Resonating Valence Bond States with Trimer Motifs
• The three-state Potts antiferromagnet on plane quadrangulations
• Phase diagram of a bidispersed hard rod lattice gas in two dimensions
• Absence of logarithmic divergence of the entanglement entropies at the phase transitions of a 2D classical hard rod model
• The columnar-disorder phase boundary in a mixture of hard squares and dimers
• Exact finite-size corrections in the dimer model on a planar square lattice
• Trimer classical spin liquid from interacting fractional charges
• Critical intermediate phase and phase transitions in a triangular-lattice three-spin interaction model: Level-spectroscopy approach
• Polyomino Models of Surface Supramolecular Assembly: Design Constraints and Structural Selectivity
• Configuration space partitioning in tilings of a bounded region of the plane
• Errata and Addenda to Mathematical Constants
• Semi-flexible trimers on the square lattice in the full lattice limit
• Entropy of fully-packed rigid rods on generalized Husimi trees: a route to the square lattice limit
• Tilings of a bounded region of the plane by maximal one-dimensional tiles