# Survival asymptotics for branching random walks in IID environments

## TL;DR

This paper analyzes the long-term survival probabilities of critical and subcritical branching random walks in IID obstacle environments on integer lattices, revealing different asymptotic behaviors and the influence of spatial strategies.

## Contribution

It provides the first detailed asymptotic analysis of survival probabilities in these models, highlighting the phenomenon of self-averaging in the critical case and spatial effects in the subcritical case.

## Key findings

- Critical case: survival probability asymptotically behaves as 2/(qn).
- Subcritical case: survival probability decays exponentially with a specific rate involving n and log n.
- Model exhibits self-averaging in the critical case but not in the subcritical case.

## Abstract

We first study a model, introduced recently in \cite{ES}, 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 $n\to\infty$, (i) Critical case: P^{\omega}(S_n)\sim\frac{2}{qn}; (ii) Subcritical case: P^{\omega}(S_n)= \exp\left[\left( -C_{d,q}\cdot \frac{n}{(\log n)^{2/d}} \right)(1+o(1))\right], where $C_{d,q}>0$ 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.