On the Growth of the Extremal and Cluster Level Sets in Branching Brownian Motion

TL;DR

This paper analyzes the growth of extremal and cluster level sets in branching Brownian motion, providing almost sure asymptotics and confirming conjectures from physics literature about their behavior.

Contribution

It establishes almost sure growth rates of extremal and cluster level sets in branching Brownian motion, improving previous probabilistic results and rigorously confirming physics conjectures.

Findings

Mass of upper level sets grows as $C_ullet Z v e^{ oot2 v}$ almost surely.

Logarithm of cluster level set mass grows as $ oot2 v$ with $v^{2/3}$ fluctuations.

Provides explicit law governing fluctuations in the cluster process.

Abstract

We study the limiting extremal and cluster point processes of branching Brownian motion. The former records the heights of all extreme values of the process, while the latter records the relative heights of extreme values in a genealogical neighborhood of order unity around a local maximum thereof. For the extremal point process, we show that the mass of upper level sets $[-v, \infty)$ grows as $C_\star Z v e^{\sqrt{2} v}(1+o(1))$ as $v \to \infty$, almost surely, where $Z$ is the limit of the associated derivative martingale and $C_\star \in (0, \infty)$ is a universal constant. For the cluster point process, we show that the logarithm of the mass of $[-v, \infty)$ grow as $\sqrt{2}v$ minus random fluctuations of order $v^{2/3}$, which are governed by an explicit law in the limit. The first result improves upon the works of Cortines et al. (arXiv:1703.06529) and Mytnik et al. (arXiv:2009.02042) in which asymptotics are shown in probability, while the second makes rigorous the derivation in the physics literature by Mueller et al. (arXiv:1910.06382) and Le et al. (arXiv:2207.07672) and resolves a conjecture thereof.