The Annals of Probability

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

Steven P. Lalley and Xinghua Zheng

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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 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(Vnδlog 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.

Article information

Ann. Probab., Volume 39, Number 1 (2011), 327-368.

First available in Project Euclid: 3 December 2010

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G60: Random fields 60F05: Central limit and other weak theorems

Critical branching random walks limit theorems occupation statistics


Lalley, Steven P.; Zheng, Xinghua. Occupation statistics of critical branching random walks in two or higher dimensions. Ann. Probab. 39 (2011), no. 1, 327--368. doi:10.1214/10-AOP551.

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