Branching Random Walks in Space-Time Random Environment: Survival Probability, Global and Local Growth Rates

TL;DR

This paper investigates the survival probability and growth rates of branching random walks in space-time random environments, linking these properties to the free energy of associated directed polymers, and providing criteria for survival.

Contribution

It introduces a criterion for survival probability positivity based on the free energy of the related directed polymer model and characterizes global and local growth rates in BRWRE.

Findings

Survival probability positivity is characterized by the free energy of the associated DPRE.

Global growth rate equals the free energy of the associated DPRE.

Local growth rate is determined by the directional free energy.

Abstract

We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rateis given by the directional free energy.