Mertens, Stephen

We present an algorithm to compute the exact probability Rn(p) for a site percolation cluster to span an n × n square lattice at occupancy p. The algorithm has time and space complexity O(λn) with λ ≈ 2.6. It allows us to compute Rn(p) up to n = 24. We use the data to compute estimates for the percolation threshold pc that are several orders of magnitude more precise than estimates based on Monte-Carlo simulations.