Occupation Statistics of Critical Branching Random Walks

We consider a critical nearest neighbor branching random walk on the d−dimensional integer lattice. Denote by Vm the maximal number of particles at a single site at time m, and by Gm the event that the branching random walk survives to generation m. We show that if the offspring distribution has finite n-th moment, then in dimensions d ≥ 3, conditional on Gm, Vm = Op(m 1 n ); and if the offspring distribution has exponentially decaying tail, then, conditional on Gm, (a) Vm = Op(logm) in dimensions d ≥ 3, and (b) Vm = Op((logm)2) in dimension d = 2. On the other hand, we show that if the offspring distribution is non-degenerate then P (Vm ≥ δ logm|Gm) → 1 for some δ > 0. Therefore, in dimensions d ≥ 3, if the offspring distribution has exponentially decaying tail then conditional on Gm, the distribution of Vm/logm must converge to a nontrivial limit as m → ∞. Furthermore, we show that, conditional on Gm, in dimensions d ≥ 3, the number of multiplicity-j sites, j ≥ 1, and the number of occupied sites, normalized by m, converge jointly to multiples of an exponential random variable; in dimension d = 2, however, the number of particles on a ‘typical’ site is Op(logm), and the number of occupied sites is Op(m/ logm).