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Maximum displacement of critical centered branching random walks under minimal assumptions

Thomas Lehéricy

TL;DR

This work establishes sharp tail asymptotics for the maximum displacement in a critical centered branching random walk under minimal structural assumptions, showing $\mathbb{P}(\sup_{v} \Lambda_v > r) \sim \frac{6\eta^2}{\sigma^2 r^2}$ and extending the classical Lalley–Shao results to a vastly broader class of reproduction schemes. The authors develop a robust Markov-property-based martingale approach, culminating in a Feynman–Kac representation and a ratio-limit for the tail, which also yields a precise discrete-case scale $\mathbb{P}(\sup_v \Lambda_v = r) \sim \frac{12\eta^2}{\sigma^2 r^3}$ when the maximum lies on $\mathbb{Z}$. Beyond the tail, they prove convergence in distribution for the total progeny and related quantities under conditioning on a large maximum, and extend the framework to multitype BRWs, enabling applications to generic critical Boltzmann planar maps via mobiles. The work also delineates connections to the Brownian snake under stronger hypotheses and provides a blueprint for studying almost-critical subcritical BRWs to access Brownian-snake-type limits. Together, these results offer a minimal-assumption, versatile toolkit for extreme-value analysis in BRWs and related combinatorial models, with implications for random maps and continuum limits.

Abstract

We study the critical centered branching random walk with offspring and displacement distributions having finite variance, under minimal assumptions on its structure. We show that the probability that the position of the right-most particle is larger than $r$ decays like an explicit constant times $r^{-2}$; this generalizes an earlier result by Lalley and Shao. In addition, we obtain the convergence in distribution of the progeny of the branching random walk conditioned on the position of the right-most particle being large. Our results are applied to multitype branching random walks under minimal assumptions.

Maximum displacement of critical centered branching random walks under minimal assumptions

TL;DR

This work establishes sharp tail asymptotics for the maximum displacement in a critical centered branching random walk under minimal structural assumptions, showing and extending the classical Lalley–Shao results to a vastly broader class of reproduction schemes. The authors develop a robust Markov-property-based martingale approach, culminating in a Feynman–Kac representation and a ratio-limit for the tail, which also yields a precise discrete-case scale when the maximum lies on . Beyond the tail, they prove convergence in distribution for the total progeny and related quantities under conditioning on a large maximum, and extend the framework to multitype BRWs, enabling applications to generic critical Boltzmann planar maps via mobiles. The work also delineates connections to the Brownian snake under stronger hypotheses and provides a blueprint for studying almost-critical subcritical BRWs to access Brownian-snake-type limits. Together, these results offer a minimal-assumption, versatile toolkit for extreme-value analysis in BRWs and related combinatorial models, with implications for random maps and continuum limits.

Abstract

We study the critical centered branching random walk with offspring and displacement distributions having finite variance, under minimal assumptions on its structure. We show that the probability that the position of the right-most particle is larger than decays like an explicit constant times ; this generalizes an earlier result by Lalley and Shao. In addition, we obtain the convergence in distribution of the progeny of the branching random walk conditioned on the position of the right-most particle being large. Our results are applied to multitype branching random walks under minimal assumptions.
Paper Structure (45 sections, 31 theorems, 390 equations, 2 figures)

This paper contains 45 sections, 31 theorems, 390 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Planar trees.
  5. Bienaymé--Galton--Watson trees and processes.
  6. Point processes.
  7. Decorated Branching Random Walk.
  8. Statement of the theorems
  9. On the minimality of our hypotheses
  10. Multitype branching random walks
  11. Mobiles of generic critical pointed and rooted Boltzmann planar maps
  12. Connection with the Brownian snake
  13. Proof of the tail estimate
  14. The Markov property of the Branching Random Walk
  15. Continuity of the tail
  16. ...and 30 more sections

Key Result

Theorem 1

Under Assumptions assum, asymptotically as $r\to\infty$,

Figures (2)

  • Figure 1: Lower bound on the $\liminf$ (in red) and upper bound on the $\limsup$ (in blue) in Lemma \ref{['lem:continuity-w-broad']}, when $\eta = \sigma$. Note that the expressions in \ref{['eq:continuity-ratio-right']} for $y\geq 0$ and \ref{['eq:continuity-ratio-left']} for $y\leq 0$ define a function that is continuous and with continuous derivative at $y=0$. The actual limit from Theorem \ref{['th:main-tail']} is drawn in green for reference; it diverges at $-\sqrt{6} \approx -2.45$ while the blue one has finite value $3$ and infinite derivative at $-\frac{\sqrt{\pi}}{3\sqrt{6}} \approx -0.2412$.
  • Figure 2: Dependency structure of the results of Section \ref{['sec:2Volume-general']}, starting from Section \ref{['sec:2continuity-tail']}. Results marked with an asterisk are new to this section, the others are adaptations of the result in parenthesis. The bold arrows represent the main line of the proof.

Theorems & Definitions (56)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Proposition 6
  • Lemma 7
  • Proposition 8
  • proof
  • ...and 46 more