First order transition for the branching random walk at the critical parameter

TL;DR

This paper investigates a critical branching random walk on the real line, demonstrating a first order phase transition in the partition function at the critical parameter, linking to recent work on normalization in the critical regime.

Contribution

It establishes a first order transition for the partition function at the critical parameter using the law of the trajectory under the polymer measure.

Findings

First order transition identified at the critical parameter.

Connection to recent work on partition function normalization.

Insights into the behavior of the branching random walk at criticality.

Abstract

Considering a critical branching random walk on the real line. From a study of the law of the trajectory of a particle chosen under the polymer measure, we establish a first order transition for the partition function at the critical parameter. This result is strongly related to a recent paper of A\"id\'ekon and Shi in which they solved the problem of the normalisation of the partition function in the critical regime.