From discrete to continuous percolation in dimensions 3 to 7. (24th October 2016) - Koza, Zbigniew; Poła, Jakub
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 θ = 3 / 2 . This allows us to estimate the continuous percolation thresholds in a model of aligned hypercubes in dimensions d = 3, …, 7 with accuracy far better than that attained using any other method before. We also report improved values of the correlation length critical exponent ν in dimensions d   =  4, 5 and the values of several universal wrapping probabilities for d = 4, …, 7 .