Survival asymptotics for branching random walks in IID environments

János Engländer,Yuval Peres

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.

Highlights

The environment is determined by placing obstacles on each site, with probability
Turning back to our spatial model, simulations suggested the self averaging property of the model: the asymptotics for the annealed and the quenched case are the same. This asymptotics is the same as the one in (1.4), where p = 1 − q is the probability that a site has a obstacle
The proof only uses the fact that if φ is the generating function of the offspring distribution, φ(z) ≥ z on [0, 1]. This remains the case for subcritical branching too, since φ(1) = 1, φ (1) < 1 and φ is convex from above on the interval

Summary

Introduction

We will consider the same model when critical branching is replaced by a subcritical one, with mean μ < 1 In this latter case we will make the following standard assumption. Turning back to our spatial model (with critical branching), simulations suggested (see [4]) the self averaging property of the model: the asymptotics for the annealed and the quenched case are the same. This asymptotics is the same as the one in (1.4), where p = 1 − q is the probability that a site has a obstacle. Those simulation results suggest that spatial survival strategies do exists, which are not detectable at the logarithmic scale but are visible at the second-order level

Some preliminary results

Further preparation

Proof of Theorem 2 — critical case

Proof of Theorem 2 — subcritical case