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VOLUME 76 | ISSUE 7 | PAGE 553
On the Aizenman exponent in critical percolation
L. N. Shchur, T. Rostunov
Landau Institute for Theoretical Physics RAS, 142432 Chernogolovka, Russia

PACS: 02.70.-c, 05.50.+q, 64.60.Ak, 75.10.-b
Abstract
The probabilities that clusters span 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 performed a finite-size analysis of the data and found that the probabilities confirm Aizenman's proposal for 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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