[Paper Review] The range of tree-indexed random walk

Jean‐François Le Gall, Shen Lin|arXiv (Cornell University)|Jul 19, 2013

Stochastic processes and statistical mechanics22 references3 citations

TL;DR

This paper establishes asymptotic scaling laws for the range of a tree-indexed random walk on ℤ^d, using Kingman’s subadditive ergodic theorem and properties of the integrated super-Brownian excursion (ISE). It shows that in dimensions d ≥ 5, the range grows linearly with n; in d = 4, it grows as n / log n; and in d ≤ 3, it scales as n^{d/4} with a limiting distribution involving the Lebesgue measure of ISE support.

ABSTRACT

We provide asymptotics for the range R(n) of a random walk on the d-dimensional lattice indexed by a random tree with n vertices. Using Kingman's subadditive ergodic theorem, we prove under general assumptions that R(n)/n converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension 4 and in the case of a symmetric random walk with exponential moments, we prove that R(n) grows like n/(log n). We apply our results to asymptotics for the range of branching random walk when the initial size of the population tends to infinity.

Motivation & Objective

• To derive almost sure and in probability asymptotics for the range of a random walk indexed by a random tree with n vertices.
• To identify the critical dimension d=4 for recurrence behavior in tree-indexed random walks, analogous to d=2 for ordinary random walks.
• To establish connections between the range of tree-indexed random walks and the integrated super-Brownian excursion (ISE) in low dimensions.
• To extend results on ordinary random walk ranges to the tree-indexed setting, particularly under symmetry and moment conditions on the jump distribution.
• To apply the findings to branching random walks with large initial populations, deriving scaling limits for their range.

Proposed method

• Applying Kingman’s subadditive ergodic theorem to establish almost sure convergence of R_n / n to a constant in d ≥ 5.
• Using the discrete snake representation and convergence to integrated super-Brownian excursion (ISE) to analyze the range in low dimensions.
• Employing moment bounds and coupling arguments to control intersection probabilities between independent subtrees, especially in the critical d=4 case.
• Leveraging the convergence of rescaled branching random walks to superprocesses and the scaling properties of ISE to derive limiting distributions.
• Using the fact that the number of vertices in a critical Galton-Watson tree with geometric offspring distribution scales as p², and applying this to the branching random walk framework.
• Applying the Cauchy-Schwarz inequality and Green’s function estimates to bound the expected number of common sites visited by two independent tree-indexed walks.

Experimental results

Research questions

• RQ1 How does the range of a tree-indexed random walk on ℤ^d scale with the size n of the tree in different dimensions?
• RQ2 What is the critical dimension for recurrence in tree-indexed random walks, and how does it compare to ordinary random walks?
• RQ3 In dimension d=4, does the range grow as n / log n, and what is the precise constant in the asymptotic scaling?
• RQ4 How do the asymptotics for the range of tree-indexed walks relate to the integrated super-Brownian excursion (ISE) in low dimensions?
• RQ5 Can the results for tree-indexed walks be extended to branching random walks with large initial populations?

Key findings

• For d ≥ 5, the normalized range R_n / n converges in probability to a positive constant c_θ depending on the jump distribution θ.
• In dimension d = 4, the range satisfies (log n)/n × R_n → 8π²σ⁴ in probability, where σ² = (det M_θ)^{1/4} and M_θ is the covariance matrix of θ.
• For d ≤ 3, the rescaled range n^{-d/4} R_n converges in distribution to c_θ × λ_d(supp(ℐ)), where ℐ is the ISE measure and λ_d is Lebesgue measure.
• The constant c_θ = 2^{d/4} (det M_θ)^{1/2} depends on the jump distribution and captures the scaling of the range in low dimensions.
• In the critical case d=4, the range of a branching random walk with initial size p grows as (log p)/p² × R(𝒵^{(p)}) → 4π²σ⁴ × J in distribution, where J is the first hitting time of 1 by standard Brownian motion.
• The expected number of common sites visited by two independent tree-indexed walks decays as o(p²) in the critical regime, enabling convergence results via moment bounds.