Critical percolation in high dimensions

Abstract

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4–13 dimensions. For $d<\mathrm{}6\mathrm{}$ they are preliminary, for $d>~6$ they are between 20 and ${10}^4$ times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing that allowed us to simulate clusters of millions of sites on computers with less than 500 Mbytes memory; (ii) a histogram method that allowed us to obtain information for several p values from a single simulation; and (iii) a variance reduction technique that is especially efficient at high dimensions where it reduces error bars by a factor of up to $\approx 30$ and more. Based on these data we propose a scaling law for finite cluster size corrections.