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


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.


Download Citation

János Engländer. Yuval Peres. "Survival asymptotics for branching random walks in IID environments." Electron. Commun. Probab. 22 1 - 12, 2017.


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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