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 the following 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(n1∕α). If the offspring distribution has an exponentially decaying tail, then Vn=Op(log n) in dimensions 3 and higher and Vn=Op((log n)2) in dimension 2. Furthermore, if the offspring distribution is nondegenerate, then P(Vn≥δlog n|Gn)→1 for some δ>0. (2) Let Mn(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 Mn(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 Op(log n) and the number of occupied sites is Op(n∕log n). We also show that, in dimension 2, there is particle clustering around a typical site.
"Occupation statistics of critical branching random walks in two or higher dimensions." Ann. Probab. 39 (1) 327 - 368, January 2011. https://doi.org/10.1214/10-AOP551