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

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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(n¹/α); and if the offspring distribution has an exponentially decaying tail, then Vₙ=Oₚ(n) in dimensions 3 and higher, and Vₙ=Oₚ((n)²) in dimension 2. Furthermore, if the offspring distribution is non-degenerate then P(Vₙ≥ δ 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 (that is, at the location of a randomly chosen particle of the nth generation) is of order Oₚ(n), and the number of occupied sites is Oₚ(n/ n).