Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous

Abstract

We study the onset of percolation of overlapping discrete hyperspheres on hypercubic lattices in dimension D = 3, 4, 5. Taking the continuum limit of the thresholds for discrete hyperspheres we obtain the values of percolation thresholds for continuous hyperspheres. In D = 3 we improved the value of the correlation length exponent: ν = 0.8762(7). In D = 4 and 5 we obtained the continuous percolation thresholds of hyperspheres with much better quality than previously known (the uncertainties reduced by the factor of 230 and 10 respectively). We discuss the hypothesis of constant exponent governing the rate of convergence of discrete models to the continuous one for hyperspheres.