Abstract
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 ${\sim} 1/\log(L M)$ to the finite-size scaling.
Highlights
The geometrical description of critical phenomena has renewed interest in the theoretical study of phase transitions
The Schramm-Loewner evolution (SLE) [1, 2, 3, 4, 5] describes the random fractals arising in 2D critical phenomena as a growth process defined by a stochastic evolution of conformal maps: the SLE generates a random curve on a planar domain from a 1D
Numerical Results let us numerically calculate the crossing probability of the fourstate Potts model (4) on a rectangle on the square lattice
Summary
The geometrical description of critical phenomena has renewed interest in the theoretical study of phase transitions. The crossing probability for the interfaces characterized by the SLE with κ ≤ 4, which includes the spin cluster boundaries of the Ising model (κ = 3; c = 1/2), may be evaluated by considering the multiple (three) SLE curves as constructed in [13]. There still exists, a non-trivial problem: what kinds of cluster boundaries in the scaling limit of the lattice model do SLE curves correspond to?
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