Universal crossing probabilities and incipient spanning clusters in directed percolation |
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Authors: | L Turban |
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Institution: | (1) Laboratoire de Physique des Matériaux, Université Henri Poincaré (Nancy I), BP 239, 54506 Vandœuvre lès Nancy Cedex, France, FR |
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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 r
eff = ct/L
z, 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.
Received 10 February 2003 Published online 20 June 2003
RID="a"
ID="a"e-mail: turban@lpm.u-nancy.fr
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ID="b"UMR CNRS 7556 |
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Keywords: | 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 |
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