Numerical Study on a Crossing Probability for the Four-State Potts Model: Logarithmic Correction to the Finite-Size Scaling

Abstract

A crossing probability for the critical four-state Potts model on an $L\times M$ rectangle on a square lattice is numerically studied. The crossing probability here denotes the probability that spin clusters cross from one side of the boundary to the other. First, by employing a Monte Carlo method, we calculate the fractal dimension of a spin cluster interface with a fluctuating boundary condition. By comparison of the fractal dimension with that of the Schramm-Loewner evolution (SLE), we numerically confirm that the interface can be described by the SLE with $\kappa =4$, as predicted in the scaling limit. Then, we compute the crossing probability of this spin cluster interface for various system sizes and aspect ratios. Furthermore, comparing with the analytical results for the scaling limit, which have been previously obtained by a combination of the SLE and conformal field theory, we numerically find that the crossing probability exhibits a logarithmic correction $\sim 1/\log (LM)$ to the finite-size scaling.

arXiv comment11 pages

References in corpus (20)

• Critical Percolation in Finite Geometries
• SLE for theoretical physicists
• The dimension of the SLE curves
• 2D growth processes: SLE and Loewner chains
• A Guide to Stochastic Loewner Evolution and its Applications
• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• Logarithmic Corrections and Finite-Size Scaling in the Two-Dimensional 4-State Potts Model
• Commutation relations for SLE
• SLE_k growth processes and conformal field theories
• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• On Multiple Schramm-Loewner Evolutions
• SLE in the three-state Potts model - a numerical study
• A solution space for a system of null-state partial differential equations 3
• A solution space for a system of null-state partial differential equations 2
• Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
• A solution space for a system of null-state partial differential equations 1
• A formula for crossing probabilities of critical systems inside polygons
• A solution space for a system of null-state partial differential equations 4
• Exact logarithmic four-point functions in the critical two-dimensional Ising model
• Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance

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