From discrete to continuous percolation in dimensions 3 to 7

Abstract

We propose a method of studying the continuous percolation of aligned objects as a limit of a corresponding discrete model. We show that the convergence of a discrete model to its continuous limit is controlled by a power-law dependency with a universal exponent $\theta = 3/2$. This allows us to estimate the continuous percolation thresholds in a model of aligned hypercubes in dimensions $d = 3,\ldots,7$ with accuracy far better than that attained using any other method before. We also report improved values of the correlation length critical exponent $\nu$ in dimensions $d = 4,5$ and the values of several universal wrapping probabilities for $d=4,\ldots,7$.