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 firstname.lastname@example.org
Published online: 20 June 2003
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.
PACS: 64.60.Ak – Renormalization-group, fractal, and percolation studies of phase transitions / 05.50.+q – Lattice theory and statistics (Ising, Potts, etc.) / 02.50.-r – Probability theory, stochastic processes, and statistics
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2003