Critical branching random walk in an IID environment

TL;DR

This paper uses high-performance simulations to investigate the survival probabilities of critical branching random walks in IID environments, revealing self-averaging behavior and specific asymptotic survival probabilities.

Contribution

It provides the first simulation-based evidence for self-averaging in critical branching random walks with random environments, highlighting the asymptotic behavior of survival probabilities.

Findings

Survival probability asymptotics are the same in annealed and quenched cases.

Asymptotic survival probability is 2/(qn) where q=1-p.

Tail behavior matches non-spatial critical branching with probabilistic branching.

Abstract

Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p>0, there is a cookie, completely suppressing the branching of any particle located there. Abstract. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is \frac{2}{qn}, where q:=1-p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probability q for every particle at every iteration.