Random walk on barely supercritical branching random walk

Remco van der Hofstad, Tim Hulshof, Jan Nagel (Corresponding author)

Research output: Contribution to journalArticleAcademicpeer-review

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 languageEnglish
Number of pages53
JournalProbability Theory and Related Fields
DOIs
Publication statusE-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

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