Branching Random Walks in Time Inhomogeneous Environments

TL;DR

This paper analyzes the maximum displacement of binary branching random walks with Gaussian increments in time inhomogeneous environments, revealing how variance profiles influence asymptotic behavior and phase transitions.

Contribution

It provides the first detailed asymptotic analysis of maximal displacement in time-inhomogeneous branching random walks with Gaussian increments, highlighting the impact of variance profiles.

Findings

Asymptotics of maximum displacement are derived up to an $O_P(1)$ error.

Variance profile significantly affects the velocity and logarithmic correction terms.

A phase transition in the correction term depending on the variance profile is identified.

Abstract

We study the maximal displacement of branching random walks in a class of time inhomogeneous environments. Specifically, binary branching random walks with Gaussian increments will be considered, where the variances of the increments change over time macroscopically. We find the asymptotics of the maximum up to an $O_P(1)$ (stochastically bounded) error, and focus on the following phenomena: the profile of the variance matters, both to the leading (velocity) term and to the logarithmic correction term, and the latter exhibits a phase transition.