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

TL;DR

This paper investigates a potential crossover in a non-universal percolation quantity at the upper critical dimension, highlighting deviations from the GM formula in high-dimensional hypercubes.

Contribution

It provides evidence of a crossover in non-universal quantities at the upper critical dimension in percolation, suggesting limitations of the GM formula for high-dimensional hypercubes.

Findings

Significant departure of percolation thresholds for d>6.

Crossover behavior reminiscent of universal quantities in critical phenomena.

Current GM formula accuracy insufficient for precise bond percolation predictions.

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.