Left Tail of the Subcritical Derivative Martingale in a Branching Wiener Process

TL;DR

This paper provides precise two-sided estimates for the tail probability of the derivative martingale limit in a branching Wiener process, confirming a recent conjecture for log-correlated Gaussian fields on trees.

Contribution

It establishes sharp tail estimates for the derivative martingale in a branching Wiener process, extending results to the entire subcritical regime and confirming a conjecture for log-correlated Gaussian fields.

Findings

Sharp two-sided tail estimates derived

Confirmation of conjecture for log-correlated Gaussian fields

Results applicable throughout the subcritical regime

Abstract

We establish a rather sharp two-side estimate for the tail probability of the derivative martingale limit in a branching random walk throughout the entire subcritical regime, confirming a conjecture by Lacoin, Rhodes, and Vargas (\emph{Duke Math. J.} 171(3):483--545, 2022.) for the special case of log-correlated Gaussian fields on trees.