Random walk on barely supercritical branching random walk

Autor: Tim Hulshof, Remco van der Hofstad, Jan Nagel
Přispěvatelé: Stochastic Operations Research, Eurandom, Probability, ICMS Core
Rok vydání: 2019
Předmět:
Zdroj: Pure TUe
Probability Theory and Related Fields, 177(1-2), 1-53. Springer
arXiv. Cornell University Library
arXiv
ISSN: 1432-2064
0178-8051
DOI: 10.1007/s00440-019-00942-0
Popis: 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 $n\to \infty$ 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.
Comment: 47 pages, 1 figure
Databáze: OpenAIRE
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