Limit theorems for the empirical distribution of supercritical branching random walks on transitive graphs

TL;DR

This paper develops limit theorems for the empirical distribution of a supercritical branching random walk on transitive graphs. It proves a law of large numbers for the mean displacement, showing (1/n)∑|X_v|/ρ^n → ℓW almost surely, and establishes a Stam-type central limit theorem stating that the mass of particles within a shrinking window around the drift nℓ converges to W times a standard normal distribution. The results rely on spine techniques, many-to-one and many-to-two lemmas, and careful decomposition arguments, without assuming transience or recurrence of the underlying walk. The framework is then specialized to anisotropic random walks on homogeneous trees, yielding explicit LLN and CLT statements with simulations illustrating the phenomena. Together, the findings quantify how the BRW’s empirical distribution concentrates around its deterministic drift, modulated by the population limit W, on broad transitive graphs.

Abstract

We consider supercritical branching random walks on transitive graphs and we prove a law of large numbers for the mean displacement of the ensemble of particles, and a Stam-type central limit theorem for the empirical distributions, thus answering the questions from Kaimanovich-Woess [KW23, Section 6.2].

Figures (5)

• Figure 1: The following diagram visualizes the proof of Lemma \ref{['lem: Lemma many-to-two product']}: The red and blue coloured part is the skeleton, on red vertices the offspring distribution is $\pi^{(2)}$, on blue vertices the offspring distribution is $\pi^{(1)}$, and on grey vertices the offspring distribution is $\pi$. For every vertex $v$ on the skeleton we look at the parent $p(v)$ and add either the first moment $\rho$ or the second moment $\theta$ to $p(v)$, depending on weather $p(v)$ is on the blue or the red part (weather there are one or two spine particles at $p(v)$). When taking the product, on the event $\lbrace \tau = k \rbrace$, the red part gives a contribution of $\theta^{k}\theta^2$ and the blue part a contribution of $\rho^{2(n-k-1)}$, which yields the desired result.
• Figure 2: Normalized by $Z_n$
• Figure 3: Normalized by $\rho^n$
• Figure 4: $T = 15$
• Figure 5: $T = 36$

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Lemma 1: Many-to-one ManyToFew
• proof
• Lemma 2: Many-to-two ManyToFew
• Lemma 3
• proof
• Theorem 3
• Lemma 4
• proof