Universal crossing probabilities and incipient spanning clusters in directed percolation

Details Universal crossing probabilities and incipient spanning clusters in directed percolation Universal crossing probabilities and incipient spanning clusters in directed percolation

2003-02-10

arxiv.org/abs/cond-mat/0302181v2

Eur. Phys. J. B33 (2003) 331-338

Physics

Condensed Matter

Statistical Mechanics

8 pages, 12 figures; minor corrections

Scientific paper

10.1140/epjb/e2003-00173-8

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.

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