Excess number of percolation clusters on the surface of a sphere

Monte Carlo simulations were performed in order to determine the excess number of clusters b and the average density of clusters nc for the two-dimensional “Swiss cheese” continuum percolation model on a planar L×L system and on the surface of a sphere. The excess number of clusters for the L×L system was confirmed to be a universal quantity with a value b=0.8841 as previously predicted and verified only for lattice percolation. The excess number of clusters on the surface of a sphere was found to have the value b=1.215(1) for discs with the same coverage as the flat critical system. Finally, the average critical density of clusters was calculated for continuum systems as nc=0.04075(5).