Occupation statistics of critical branching random walks in two or higher dimensions

Abstract

Consider a critical nearest-neighbor branching random walk on the d-dimensional integer lattice initiated by a single particle at the origin. Let G_n be the event that the branching random walk survives to generation n. We obtain the following limit theorems, conditional on the event G_n, for a variety of occupation statistics: (1) Let V_n be the maximal number of particles at a single site at time n. If the offspring distribution has finite αth moment for some integer α≥2, then, in dimensions 3 and higher, V_n=O_p(n^1∕α). If the offspring distribution has an exponentially decaying tail, then V_n=O_p(log n) in dimensions 3 and higher and V_n=O_p((log n)²) in dimension 2. Furthermore, if the offspring distribution is nondegenerate, then P(V_n≥δlog n|G_n)→1 for some δ>0. (2) Let M_n(j) be the number of multiplicity-j sites in the nth generation, that is, sites occupied by exactly j particles. In dimensions 3 and higher, the random variables M_n(j)∕n converge jointly to multiples of an exponential random variable. (3) In dimension 2, the number of particles at a “typical” site (i.e., at the location of a randomly chosen particle of the nth generation) is of order O_p(log n) and the number of occupied sites is O_p(n∕log n). We also show that, in dimension 2, there is particle clustering around a typical site.