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A Probabilistic Proof for Stable Fluctuations in the Extremal Process of Branching Brownian Motion

Lisa Hartung, Oren Louidor, Tianqi Wu

Abstract

We give a probabilistic proof for the emergence of the Stable-$1$ Law for the random fluctuations of the mass of the extremal process of branching Brownian Motion away from its tip. This result was already shown by Mytnik et al. albeit using PDE techniques. As a consequence, we demystify the origin of these fluctuations and the meaning of the deterministic centering function required.

A Probabilistic Proof for Stable Fluctuations in the Extremal Process of Branching Brownian Motion

Abstract

We give a probabilistic proof for the emergence of the Stable- Law for the random fluctuations of the mass of the extremal process of branching Brownian Motion away from its tip. This result was already shown by Mytnik et al. albeit using PDE techniques. As a consequence, we demystify the origin of these fluctuations and the meaning of the deterministic centering function required.
Paper Structure (29 sections, 30 theorems, 210 equations)

This paper contains 29 sections, 30 theorems, 210 equations.

Table of Contents

  1. Introduction and results
  2. Setup and background
  3. Contribution
  4. Heuristic for 1-stable fluctuations in $\mathcal{E}_\circ([-u, \infty))$
  5. Other log-correlated fields and universality
  6. Organization of the paper
  7. Preliminaries
  8. Additional notation
  9. Weak representation of the cluster law
  10. Spinal decomposition
  11. Decorated random walk representation
  12. Ballot estimates for DRW
  13. Joint convergence of critical extreme level-sets and maximum
  14. Stable limit of compensated process
  15. The intensity of unusually large clusters
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $x^+, x^-: \mathbb{R}_+ \to \mathbb{R}$ be functions which tend sub-linearly but arbitrarily slow to $\pm \infty$ respectively and abbreviate $u_* \equiv -\frac{1}{\sqrt{2}} \log u$. Then and where $C_\star$ is as in e:29 and $C_\circ \in \mathbb{R}$ is a universal constant.

Theorems & Definitions (52)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1: Many-To-One
  • Lemma 2.2: Lemma 5.1 in CHL19
  • Lemma 2.3: Lemma 3.1 in CHL19
  • Lemma 2.4: Lemma 3.2 in CHL19
  • Lemma 2.5: Lemma 3.3 in CHL19
  • Lemma 2.6: Essentially Lemma 3.4 in CHL19
  • ...and 42 more