Possible crossover of a nonuniversal quantity at the upper critical dimension.
We report on a possible crossover of a nonuniversal quantity at the upper critical dimensionality in the field of percolation. Plotting recent estimates for site percolation thresholds of hypercubes in dimension 6 < or =d < or =13 against corresponding predictions from the Galam-Mauger (GM) formula p(c) = p(0) [(d-1) (q-1)](-a) d(b) for percolation thresholds, a significant departure of p(c) is observed for d > or =6 . This result is reminiscent of the crossover undergone by universal quantities in critical phenomena. For bond percolation, evidence of such a crossover of dimensionality would require an improvement of the GM formula to reach a relative error of typically 0.2%, while it is currently at 0.9% for hypercubes.
Galam
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Laboratoire des Milieux Désordonnés et Hétérogènes, Case 86, 4 place Jussieu, F-75252 Paris Cedex 05, France.
Mauger
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