On the Aizenman exponent in critical percolation

L.N. Shchur and T.A. Rostunov

Письма в ЖЭТФ 76 (2002) 553-558



The probabilities of clusters spanning a hypercube of dimensions two to seven along one axis of a percolation system under criticality were investigated numerically. We used a modified Hoshen--Kopelman algorithm combined with Grassberger's "go with the winner" strategy for the site percolation. We carried out a finite-size analysis of the data and found that the probabilities confirm Aizenman's proposal of the multiplicity exponent for dimensions three to five. A crossover to the mean-field behavior around the upper critical dimension is also discussed. 

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