Possible crossover of a non universal quantity at the upper critical dimension

Abstract

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 p_c=p_0 [ (d-1)(q-1) ]^{-a} d^b for percolation thresholds, a significant departure of p_c 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.