CONTINUITY AND CONVERGENCE OF THE PERCOLATION FUNCTION IN CONTINUUM PERCOLATION

We consider a percolation model on the d-dimensional Euclidean space ( d greater than or equal to 2) which consists of spheres centred at the points of a Poisson point process of intensity lambda. The radii of t he spheres are random and are chosen independently and identically acc ording to a distribution of a positive random variable. We show that t he percolation function is continuous everywhere except perhaps at the critical point. Further, we show that the percolation functions conve rge to the appropriate percolation function except at the critical poi nt when the radius random variables are uniformly bounded and converge weakly to another bounded random variable.