Let T be a supercritical Galton–Watson tree with a bounded offspring distribution that has mean μ> 1 , conditioned to survive. Let φT be a random embedding of T into Zd according to a simple random walk step distribution. Let Tp be percolation on T with parameter p, and let pc= μ- 1 be the critical percolation parameter. We consider a random walk (Xn)n≥1 on Tp and investigate the behavior of the embedded process φTp(Xn) as n→ ∞ and simultaneously, Tp becomes critical, that is, p= pn↘ pc. We show that when we scale time by n/(pn-pc)3 and space by (pn-pc)/n, the process (φTp(Xn))n≥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.
- Branching random walk
- Random walk indexed by a tree
- Scaling limit