Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 1-53 |
| Number of pages | 53 |
| Journal | Probability Theory and Related Fields |
| Volume | 177 |
| Issue number | 1-2 |
| Early online date | 25 Sept 2019 |
| DOIs | |
| Publication status | Published - Jun 2020 |
Funding
The work of J.N. was supported by the Deutsche Forschungsgemeinschaft (DFG) through Grant NA 1372/1. R.H. was supported by NWO through VICI-grant 639.033.806. The work of RvdH and TH is also supported by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003. We thank the anonymous referee for a careful reading of the paper and several helpful comments. 1 Note that this is a different process than if we were to consider a simple random walk on the subgraph of Z d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^d$$\end{document} traced out by the branching random walk, as is for instance the topic of [ 7 ] (for a different kind of tree): random walk on the trace is, in our setting, a vacuous complication, because T \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}$$\end{document} is supercritical, and thus grows at an exponential rate, while Z d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^d$$\end{document} has polynomial growth, so that φ T ( T ) = Z d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{{\mathcal {T}}}({\mathcal {T}}) = {\mathbb {Z}}^d$$\end{document} P p \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}_p$$\end{document} -almost surely.
Keywords
- Branching random walk
- Percolation
- Random walk indexed by a tree
- Scaling limit
- Supercriticality