Universal crossing probability in anisotropic systems

TL;DR

This paper explores universal crossing probabilities in anisotropic critical systems, generalizing known formulas and providing exact and numerical results for different anisotropic models, revealing a universal function dependent on an effective aspect ratio.

Contribution

It extends crossing probability formulas to anisotropic systems and derives exact and numerical results demonstrating universality in these models.

Findings

Generalized Cardy's formula for weak anisotropy

Exact results for 1+1D anisotropic systems

Universal crossing probability as a function of effective aspect ratio

Abstract

Scale-invariant universal crossing probabilities are studied for critical anisotropic systems in two dimensions. For weakly anisotropic standard percolation in a rectangular-shaped system, Cardy's exact formula is generalized using a length-rescaling procedure. For strongly anisotropic systems in 1+1 dimensions, exact results are obtained for the random walk with absorbing boundary conditions, which can be considered as a linearized mean-field approximation for directed percolation. The bond and site directed percolation problem is itself studied numerically via Monte Carlo simulations on the diagonal square lattice with either free or periodic boundary conditions. A scale-invariant critical crossing probability is still obtained, which is a universal function of the effective aspect ratio r_eff=c r where r=L/t^z, z is the dynamical exponent and c is a non-universal amplitude.