2003/03/10 by Peter Grassberger
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · doi:10.1103/physreve.67.036101
openalex publication_date 2003/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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>or =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 approximately 30 and more. Based on these data we propose a scaling law for finite cluster size corrections.