Central Limit Theorem for Branching Random Walks in Random Environment

TL;DR

This paper proves a central limit theorem for the population density in high-dimensional branching random walks within random environments, revealing phase transitions and providing bounds on population concentration.

Contribution

It establishes a CLT for branching random walks in random environments with new bounds and insights into phase transitions related to directed polymers.

Findings

Proves a CLT for population density in dimensions d ≥ 3.

Provides upper bounds for the density of the most populated site.

Discusses phase transition phenomena in the model.

Abstract

We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d \ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase transition of this model in connection with directed polymers in random environment.