1-stable fluctuations in branching Brownian motion at critical temperature II: general functionals

TL;DR

The paper addresses fluctuations of BBM front quantities at critical temperature by extending 1-stable fluctuation theory to a broad class of functionals Z_t(F). Using a barrier/decomposition approach and multiscale bootstrap, it shows that, conditionally on the derivative martingale limit Z_ limit, the fluctuations converge to a 1-stable distribution driven by a stable noise M_{Z_ limit}, with a logarithmic correction. For the additive martingale W_t (F(x)=1/x), the result yields a universal Cauchy-type limit: sqrt t ( sqrt t W_t - sqrt{2/π} Z_ limit ) → Z_ limit C, aligning with physics conjectures. The paper also establishes a functional convergence and identifies the particles responsible for fluctuations, via a barrier-based multiscale analysis that isolates the front contributions and their effect on Z_t(F). These findings illuminate the universal 1-stable nature of front fluctuations in BBM and provide precise probabilistic descriptions of extremal-like front dynamics relevant to FKPP-type universality.

Abstract

Let $μ_t$ denote the critical derivative Gibbs measure of branching Brownian motion at time $t$. It has been proved by Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470--502) and Maillard and Zeitouni (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 3, 1144--1160) that $μ_t$ converges weakly to the random measure $Z_\infty \sqrt{2/π} x^2 e^{-x^2/2} \boldsymbol 1_{x >0} d x$, where $Z_\infty$ is the limit of the derivative martingale. In this paper, we are interested in the fluctuations that occur in this convergence and prove for a large class of functions $F$ that \begin{align*} \sqrt{t} \left( \int_{\mathbb R} F d μ_t - Z_\infty \int_0^\infty F(x) \sqrt{\frac{2}π} x^2 e^{-x^2/2} d x - \frac{c(F) \log t}{\sqrt{t}} Z_\infty \right) \to S(F), \end{align*} in law, as $t\to\infty$, where $c(F)$ is a constant depending on $F$ and, given $Z_\infty$, $S(F)$ has an explicit 1-stable distribution. Moreover, we extend this result to a functional convergence, and we identify precisely the particles responsible for the fluctuations. In particular, this proves the following result for the critical additive martingale $(W_t)_{t\geq 0}$: \[ \sqrt{t} \left( \sqrt{t} W_t - \sqrt{\frac{2}π} Z_\infty \right) \xrightarrow[t\to\infty]{} C Z_\infty, \quad \text{in law}, \] where here $C$ is a Cauchy variable independent of $Z_\infty$, confirming a conjecture by Mueller and Munier (Phys. Rev. E 90 (2014), 042143) in the physics literature.

Figures (2)

• Figure 1: Representation of a BBM with binary branching. The killing barrier that defines $\mathcal{L}^{t_0,\gamma}$ is the thick blue line. At time $t$, only the black particles contribute to $\widetilde{Z}^{t_0,\gamma}_t (F,\gamma)$. For each $u$ in the stopping line, the blue particles descending from $u$ at time $t$ contribute to $\Omega^{(u)}_t$.
• Figure 2: Dependencies between the intermediate results leading to the proof of Theorem \ref{['theorem-complete']}. Arrows indicate that a result is used in the proof of another result. The proofs of the results in the rectangular, orange boxes implement the strategy outlined in Section \ref{['subsection:scheme_barrier']}, with $t_0$ and $\gamma$ as indicated, where $\beta_t \to \infty$ with $\beta_t = o(\log t)$. To avoid overloading the diagram, Lemma \ref{['lem:tightness']} is not included because this basic result, as well as the inequalities in its proof, are used widely throughout the paper. Similarly, results in Appendix \ref{['section:technical-results']} are not included.

Theorems & Definitions (13)

• Theorem 1.1: maillardpain2019
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Corollary 1.10