### Abstract

Let T be a supercritical Galton–Watson tree with a bounded offspring distribution that has mean μ> 1 , conditioned to survive. Let φ_{T} be a random embedding of T into Z^{d} according to a simple random walk step distribution. Let T_{p} be percolation on T with parameter p, and let p_{c}= μ^{- 1} be the critical percolation parameter. We consider a random walk (Xn)n≥1 on T_{p} and investigate the behavior of the embedded process φTp(Xn) as n→ ∞ and simultaneously, T_{p} becomes critical, that is, p= p_{n}↘ p_{c}. 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.

Original language | English |
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Number of pages | 53 |

Journal | Probability Theory and Related Fields |

DOIs | |

Publication status | E-pub ahead of print - 25 Sep 2019 |

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### Keywords

- Branching random walk
- Percolation
- Random walk indexed by a tree
- Scaling limit
- Supercriticality