Efficient Monte Carlo algorithm and high-precision results for percolation

We present a new Monte Carlo algorithm for studying site or bond percolatio n on any lattice. The algorithm allows us to calculate quantities such as t he cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run w hich takes an amount of time scaling linearly with the number of sites on t he lattice. We use our algorithm to determine that the percolation transiti on occurs at p(c) = 0.592 746 21(13) for site percolation on the square lat tice and to provide clear numerical confirmation of the conjectured 4/3-pow er stretched-exponential tails in the spanning probability functions.