Eur. Phys. J. B 33, 331-338 (2003)
https://doi.org/10.1140/epjb/e2003-00173-8

Universal crossing probabilities and incipient spanning clusters in directed percolation

Laboratoire de Physique des Matériaux (UMR CNRS 7556) , Université Henri Poincaré (Nancy I), BP 239, 54506 Vandœuvre lès Nancy Cedex, France

Corresponding author: ^a turban@lpm.u-nancy.fr

Received: 10 February 2003
Published online: 20 June 2003

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. In a dynamical interpretation, the crossing probability is the probability that, on a system with size L, an epidemic spreading without immunization remains active at time t. Since the system is strongly anisotropic, the shape dependence in space-time enters through the effective aspect ratio , 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 also 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.