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

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.