1-stable fluctuations in branching Brownian motion at critical temperature I: the derivative martingale

Abstract

Let $(Z_t)_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, i.e.\@ the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [\textit{Ann. Probab.}, 15(3):1052--1061, 1987] says that this martingale converges almost surely to a limit $Z_\infty$, positive on the event of survival. In this paper, our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier [\textit{Phys. Rev. E}, 90:042143, 2014]: \[ \sqrt{t} \left( Z_\infty - Z_t + \frac{\log t}{\sqrt{2\pi t}} Z_\infty \right) \xrightarrow[t\to\infty]{} S_{Z_\infty}, \quad \text{in law}, \] where $S$ is a spectrally positive 1-stable L\'evy process independent of $Z_\infty$. In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of $Z_\infty$ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way, we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale. All proofs in this paper are given under the hypothesis $E[L(\log L)^3] < \infty$, where the random variable $L$ follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.