Large deviations for the maximum of a branching random walk

TL;DR

This paper establishes large deviation principles for the maximum position in a branching random walk, characterizing the rate function through a variational problem and comparing dependent and independent cases.

Contribution

It provides a novel large deviation analysis for the maximum of branching random walks and compares dependent and independent scenarios, revealing differences in their rate functions.

Findings

Rate function characterized via a variational problem.

Maximum of branching random walk dominated by independent case.

Large deviation principles derived for both dependent and independent maxima.

Abstract

We consider real-valued branching random walks and prove a large deviation result for the position of the rightmost particle. The position of the rightmost particle is the maximum of a collection of a random number of dependent random walks. We characterise the rate function as the solution of a variational problem. We consider the same random number of independent random walks, and show that the maximum of the branching random walk is dominated by the maximum of the independent random walks. For the maximum of independent random walks, we derive a large deviation principle as well. It turns out that the rate functions for upper large deviations coincide, but in general the rate functions for lower large deviations do not.