Random walk on barely supercritical branching random walk

TL;DR

This paper studies the behavior of a random walk on a barely supercritical branching random walk, showing it converges to Brownian motion under specific scaling as the environment approaches criticality.

Contribution

It introduces a new scaling limit for random walks on near-critical branching random walks, bridging static and critical random environment behaviors.

Findings

Convergence to Brownian motion under specific scaling

Identification of a transition between static and critical environment behaviors

Analysis of the process as the percolation parameter approaches criticality

Abstract

Let $\mathcal{T}$ be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean $\mu >1$, conditioned to survive. Let $\varphi_{\mathcal{T}}$ be a random embedding of $\mathcal{T}$ into $\mathbb{Z}^d$ according to a simple random walk step distribution. Let $\mathcal{T}_p$ be percolation on $\mathcal{T}$ with parameter $p$, and let $p_c = \mu^{-1}$ be the critical percolation parameter. We consider a random walk $(X_n)_{n \ge 1}$ on $\mathcal{T}_p$ and investigate the behavior of the embedded process $\varphi_{\mathcal{T}_p}(X_n)$ as $n\to \infty$ and simultaneously, $\mathcal{T}_p$ becomes critical, that is, $p=p_n\searrow p_c$. We show that when we scale time by $n/(p_n-p_c)^3$ and space by $\sqrt{(p_n-p_c)/n}$, the process $(\varphi_{\mathcal{T}_p}(X_n))_{n \ge 1}$ converges to a $d$-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.