Universal crossing probabilities and incipient spanning clusters in directed percolation

TL;DR

This paper investigates universal crossing probabilities and the emergence of spanning clusters in directed percolation on a square lattice, emphasizing shape dependence, initial state effects, and anisotropic finite-size scaling through Monte Carlo simulations.

Contribution

It introduces a generalized anisotropic finite-size scaling approach and extends Aizenman's argument to directed percolation, providing new insights into universal crossing probabilities.

Findings

Crossing probabilities depend on an effective aspect ratio in anisotropic systems.

The initial state significantly influences the universal behavior of crossing probabilities.

Numerical results support the conjecture relating incipient spanning clusters to anisotropic scaling.

Abstract

Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. Since the system is strongly anisotropic, the shape-dependence enters through the effective aspect ratio r_eff=cL^z/t, where c is a non-universal constant and z the anisotropy exponent. A particular attention is paid to the influence of the initial state on the universal behaviour of the crossing probability. Using anisotropic finite-size scaling and generalizing a simple argument given by Aizenman for isotropic percolation, we obtain the behaviour of the probability to find n incipient spanning clusters on a finite system at time t. The numerical results are in good agreement with the conjecture.