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 ${\phi }_{\mathcal{T}}$ 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 ${\phi }_{{\mathcal{T}}_p}(X_n)$ 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 $({\phi }_{{\mathcal{T}}_p}(X_n){)}_{n\ge 1}$ 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.

47 pages, 1 figure