Occupation Statistics of Critical Branching Random Walks in Two or Higher Dimensions

Consider a critical nearest neighbor branching random walk on the d-dimensional integer lattice initiated by a single particle at the origin. Let Gn be the event that the branching random walk survives to generation n. We obtain limit theorems conditional on the event Gn for a variety of occupation statistics: (1) Let Vn 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, Vn = Op(n); and if the offspring distribution has an exponentially decaying tail, then Vn = Op(log n) in dimensions 3 and higher, and Vn = Op((log n)) in dimension 2. Furthermore, if the offspring distribution is non-degenerate then P (Vn ≥ δ log n|Gn) → 1 for some δ > 0. (2) Let Mn(j) be the number of multiplicityj sites in the nth generation, that is, sites occupied by exactly j particles. In dimensions 3 and higher, the random variables Mn(j)/n converge jointly to multiples of an exponential random variable. (3) In dimension 2, the number of particles at a “typical” site (that is, at the location of a randomly chosen particle of the nth generation) is of order Op(log n), and the number of occupied sites is Op(n/ log n).