The capacitance of the cube and square plate by random walk methods

We have calculated improved upper and lower bounds for the capacitance of an isolated cube. These come to C*≡C/4πε0l=0.6619 and 0.6596, respectively, where C is the capacitance and l is the length of the side. The lower bound is sufficiently accurate to rule out a recent theoretical conjecture. We show that the traditional methods of computing the actual capacitance contain systematic errors, and we use the Kelvin inversion, followed by a random walk method with variance reduction, to provide the estimate 0.6606±0.0001, which is consistent with the bounds just given. We give an alternative method for the square plate, with a preliminary result C*≡C/4πε0l=0.36±0.01.