2017/03/28 by János Engländer, Janos Englander, Yuval Peres +2
Computer Science · Mathematics · Physics and Astronomy · #60J80 #Cellular Automata and Applications #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60J80
paper · pdf · doi:10.48550/arxiv.1703.09731
2 figures
arxiv created 2017/03/28 · openalex publication_date 2017/03/28 · arxiv updated 2017/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We first study a model, introduced recently in \citeES, 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 \mathbb Pp-measure, as n→∞, (i) Critical case: Pω(Sn)∼(2)/(qn); (ii) Subcritical case: Pω(Sn)= exp[( -Cd,q⋅ \fracn(log n)2/d )(1+o(1))], where Cd,q>0 does not depend on the branching law. Hence, the model exhibits `self-averaging' in the critical case but not in the subcritical one. I.e., in (i) the asymptotic tail behavior is the same as in a "toy model" where space is removed, while in (ii) the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. We utilize a spine decomposition of the branching process as well as some known results on random walks.