Document Type

Article

Publication Date

2017

Publication Title

ELECTRONIC COMMUNICATIONS in PROBABILITY

ISSN

1083-589X

Volume

22

Issue

29

First Page

1

Last Page

12

DOI

http://dx.doi.org/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 ∈ [0, 1) independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let Sn be the event of survival up to time n. We show that on a set of full Ppmeasure, as n → ∞, P ω (Sn) ∼ 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.

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