Random walk on barely supercritical branching random walk

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.