Exact Percolation Probability on the Square Lattice

We present an algorithm to compute the exact probability $R_n(p)$ for a site percolation cluster to span an $n\times n$ square lattice at occupancy $p$. The algorithm has time and space complexity $O({\lambda }^n)$ with $\lambda \approx 2.6$\mathrm{.}Itallowsustocompute$R_{n}(p)$upto$n=24$. We use the data to compute estimates for the percolation threshold $p_c$ that are several orders of magnitude more precise than estimates based on Monte-Carlo simulations.