Random walk on barely supercritical branching random walk

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 $n\rightarrow \infty$ 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.