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Branching brownian motion conditioned on large level sets

Xinxin Chen, Heng Ma

TL;DR

This work derives precise large-deviation asymptotics for intermediate level sets in branching Brownian motion and strengthens prior results by Aïdékon–Hu–Shi. The authors connect the level-set deviations to the additive martingale limits and the global minimum of linearly transformed BBMs via a spine decomposition, revealing that the optimal path concentrates near a tangent line and that the event is governed by a minimum hitting mechanism. They establish a sharp exponential rate e^{-I(x,a)t} with explicit constants, and they further characterize the BBM conditioned on a large level set, uncovering entropy-repulsion phenomena: the overlap and the maximum display Gaussian fluctuations around a space-time curve, with the maximum displaying a ballistic speed smaller than the unconditioned maximum. Additionally, a Pareto-type limiting structure emerges for the rescaled level-set size, highlighting the heavy-tailed nature of the conditioned BBM. Overall, the paper advances the understanding of BBM under rare conditioning and links level-set large deviations to martingale limits and minimum-assembly dynamics.

Abstract

We study the precise large deviation probabilities for the sizes of intermediate level sets in branching Brownian motion (BBM). Our conclusions improve a result of Aïdekon, Hu and Shi in [J. Math. Sci. \textbf{238}(2019)]. Additionally, we analyze the typical behaviors of BBM conditioned on large level sets. Our approach relies on the connections between intermediate level sets, additive martingale limits of BBM, and the global minimum of linearly transformed BBMs.

Branching brownian motion conditioned on large level sets

TL;DR

This work derives precise large-deviation asymptotics for intermediate level sets in branching Brownian motion and strengthens prior results by Aïdékon–Hu–Shi. The authors connect the level-set deviations to the additive martingale limits and the global minimum of linearly transformed BBMs via a spine decomposition, revealing that the optimal path concentrates near a tangent line and that the event is governed by a minimum hitting mechanism. They establish a sharp exponential rate e^{-I(x,a)t} with explicit constants, and they further characterize the BBM conditioned on a large level set, uncovering entropy-repulsion phenomena: the overlap and the maximum display Gaussian fluctuations around a space-time curve, with the maximum displaying a ballistic speed smaller than the unconditioned maximum. Additionally, a Pareto-type limiting structure emerges for the rescaled level-set size, highlighting the heavy-tailed nature of the conditioned BBM. Overall, the paper advances the understanding of BBM under rare conditioning and links level-set large deviations to martingale limits and minimum-assembly dynamics.

Abstract

We study the precise large deviation probabilities for the sizes of intermediate level sets in branching Brownian motion (BBM). Our conclusions improve a result of Aïdekon, Hu and Shi in [J. Math. Sci. \textbf{238}(2019)]. Additionally, we analyze the typical behaviors of BBM conditioned on large level sets. Our approach relies on the connections between intermediate level sets, additive martingale limits of BBM, and the global minimum of linearly transformed BBMs.
Paper Structure (21 sections, 19 theorems, 208 equations, 3 figures)

This paper contains 21 sections, 19 theorems, 208 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work and further questions
  4. Proof ideas
  5. Preliminaries
  6. Spine Decomposition
  7. Global minimum and martingale limits
  8. High points of BBM
  9. Useful inequalities
  10. The Optimal Strategy
  11. The road to Proposition \ref{['prop-up-to-constant']}
  12. Proof of Lemmas in Section \ref{['sec-Up-to-cst']}
  13. Sharp Estimate
  14. Proof of Theorem \ref{['thm-LDP']}: substituting level sets with martingale limits in large deviations
  15. Change of measure
  16. ...and 6 more sections

Key Result

Theorem 1.1

Fix $x > 0$, $(1-\frac{x^2}{2})_{+} <a<1$, and $y> 0$. We have Above, the constant $eq-cst-a-x(\cdot,\cdot)$ is defined by

Figures (3)

  • Figure 1: The functions $\mathsf{F}_{t}$ and $\mathsf{L}_{t}$
  • Figure 2: The sets $\mathcal{B}^{>\ell}(\hat{u})$ and $\mathcal{B}^{\leq \ell}(\hat{u})$
  • Figure 3: Left: Event $G^{4}_{\ell}(w)$. Right: Event $G^{1}_{L}(w) \backslash G^{4}_{\ell}(w)$.

Theorems & Definitions (32)

  • Theorem 1.1: Precise Large Deviation Estimates
  • Theorem 1.2: Entropy Repulsion: Overlap
  • Theorem 1.3: Entropy Repulsion: Maximum
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2: CDM24
  • Lemma 2.3: CDM24
  • Lemma 2.4
  • Lemma 2.5: Overlap distribution
  • ...and 22 more