Critical percolation in high dimensions

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4–13 dimensions. For d<6 they are preliminary, for d>~6 they are between 20 and 10⁴ 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 ≈30 and more. Based on these data we propose a scaling law for finite cluster size corrections.