2007/07/25 by Steven P. Lalley, Steven Lalley, Lalley, Steven +2 · 1 citation
Mathematics · #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60K35
paper · pdf · doi:10.48550/arxiv.0707.3829
openalex publication_date 2007/07/25 · arxiv created 2010/04/07 · arxiv updated 2010/04/08 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
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(log n) in dimensions 3 and higher, and Vn=Op((log n)2) in dimension 2. Furthermore, if the offspring distribution is non-degenerate 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 (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).