[Paper Review] Occupation Statistics of Critical Branching Random Walks

Steven P. Lalley, Xinghua Zheng|arXiv (Cornell University)|Jul 25, 2007

Forest Biomass Utilization and Management6 citations

TL;DR

This paper analyzes occupation statistics in critical nearest-neighbor branching random walks on the d-dimensional integer lattice, conditioning on survival to generation n. It establishes limit theorems showing that in dimensions 3 and higher, the maximum number of particles at a single site grows as Op(n^{1/α}) for finite αth moments or Op(log n) for light-tailed offspring distributions, while site multiplicity and site occupancy statistics converge to exponential and logarithmic scaling limits, respectively.

ABSTRACT

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(n 1/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 ≥ δ log 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).

Motivation & Objective

• To understand the distribution of particle counts at individual sites in critical branching random walks on Z^d.
• To analyze how the maximum number of particles at any single site evolves over time, conditioned on survival to generation n.
• To characterize the joint distribution of sites with exactly j particles (multiplicity-j sites) across generations.
• To determine the typical number of particles per site and the number of occupied sites in dimension 2 under conditioning.

Proposed method

• Conditioning on the survival event Gn, where the branching process survives to generation n, to study rare but relevant sample paths.
• Applying moment moment methods and large deviation heuristics to analyze the maximum particle count Vn at any site.
• Using branching process theory and generating functions to derive asymptotic distributions for Mn(j), the number of sites with exactly j particles.
• Employing coupling and stochastic comparison techniques to bound Vn under different offspring tail behaviors.
• Analyzing the typical site occupancy via size-biased sampling of particles in the nth generation.
• Deriving scaling limits for Mn(j)/n in dimensions d ≥ 3, showing convergence to multiples of exponential random variables.

Experimental results

Research questions

• RQ1 How does the maximum number of particles at a single site scale with n in critical branching random walks on Z^d for d ≥ 3?
• RQ2 What is the asymptotic distribution of sites with exactly j particles (Mn(j)) in dimensions d ≥ 3, conditioned on survival to generation n?
• RQ3 How does the typical number of particles per site behave in dimension 2 under conditioning on survival to generation n?
• RQ4 What is the asymptotic order of the number of occupied sites in dimension 2, given survival to generation n?
• RQ5 How do the scaling laws for particle counts and site occupancy depend on the tail behavior of the offspring distribution?

Key findings

• In dimensions 3 and higher, if the offspring distribution has finite αth moment for α ≥ 2, then Vn = Op(n^{1/α}) under conditioning on survival to generation n.
• 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.
• For any non-degenerate offspring distribution, P(Vn ≥ δ log n | Gn) → 1 for some δ > 0, indicating that logarithmic growth is almost sure under conditioning.
• In dimensions 3 and higher, the normalized counts Mn(j)/n converge jointly to multiples of an exponential random variable, establishing a joint limit law for site multiplicities.
• In dimension 2, the number of particles at a typical site (chosen uniformly from the nth generation) is Op(log n), reflecting higher site concentration.
• In dimension 2, the number of occupied sites is Op(n / log n), indicating sparser spatial spread compared to higher dimensions.