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2017 Survival asymptotics for branching random walks in IID environments
János Engländer, Yuval Peres
Electron. Commun. Probab. 22(none): 1-12 (2017). DOI: 10.1214/17-ECP60

Abstract

We first study a model, introduced recently in [4], of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no ‘obstacle’ placed there. The obstacles appear at each site with probability $p\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical.

Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\mathbb P_p$-measure, as $n\to \infty $, $P^{\omega }(S_n)\sim 2/(qn)$ in the critical case, while this probability is asymptotically stretched exponential in the subcritical case.

Hence, the model exhibits ‘self-averaging’ in the critical case but not in the subcritical one. I.e., in the first case, the asymptotic tail behavior is the same as in a ‘toy model’ where space is removed, while in the second, the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies.

A spine decomposition of the branching process along with known results on random walks are utilized.

Citation

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János Engländer. Yuval Peres. "Survival asymptotics for branching random walks in IID environments." Electron. Commun. Probab. 22 1 - 12, 2017. https://doi.org/10.1214/17-ECP60

Information

Received: 6 May 2017; Accepted: 15 May 2017; Published: 2017
First available in Project Euclid: 25 May 2017

zbMATH: 1364.60103
MathSciNet: MR3663100
Digital Object Identifier: 10.1214/17-ECP60

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JOURNAL ARTICLE
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