Ja n 20 05 Possible crossover of a non universal quantity at the upper critical dimension

We report on a possible crossover of a non universal quantity at the upper critical dimensionality in the field of percolation. Plotting recent estimates for site percolation thresholds of hypercubes in dimension 6 ≤ d ≤ 13 against corresponding predictions from the GM formula pc = p0[(d − 1)(q − 1)] d for percolation thresholds, a significant departure of pc is observed for d ≥ 6. This result is reminiscent of the crossover undergone by universal quantities in critical phenomena. For bond percolation, the 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. PA Classification Numbers: 64.60.Ak, 05.70.Jk