On the asymptotic geometrical behaviour of percolation processes

Abstract

In this paper the global behaviour of percolation processes on the d-dimensional square lattice is studied. Using techniques of Richardson (1973) we prove, under weak moment assumptions on the time coordinate distribution, the following result. There exists a norm N(·) on Rd such that, for all 0 < ε < 1, we have that almost surely for all sufficiently large t the N-ball of radius (1 – ε)t is contained in η, (the set of all sites occupied by time t) and η, is contained in the N-ball of radius (1 + ε)t. Richardson (1973) derived the corresponding ‘in probability' result for a class of spread processes on Rd, satisfying certain conditions.