Percolation Thresholds and Fisher Exponents in Hypercubic Lattices

TL;DR

This paper uses invasion percolation to accurately determine percolation thresholds and Fisher exponents on hypercubic lattices for dimensions 4 through 13, confirming theoretical predictions about critical behavior.

Contribution

It provides highly precise numerical estimates of percolation thresholds and Fisher exponents across multiple dimensions, supporting mean-field theory for d >= 6.

Findings

Percolation thresholds computed for d=4 to 13.

Fisher exponent tau confirmed as 5/2 for d >= 6.

Logarithmic corrections observed at d=6.

Abstract

We use invasion percolation to compute highly-accurate numerical values for bond and site percolation thresholds p_c on the hypercubic lattice Z^d for d = 4,,,,,13. We also compute the Fisher exponent tau governing the cluster size distribution at criticality. Our results support the claim that the mean-field value tau = 5/2 holds for d >= 6, with logarithmic corrections to power-law scaling at d=6.