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On the maximal displacement of critical branching random walk in random environment

Wenxin Fu, Wenming Hong

TL;DR

The paper analyzes the maximal displacement of a critical branching random walk in random environment, showing that conditioned on non-extinction the appropriately scaled maximal position converges in distribution to a functional $A_\Lambda$ linked to Brownian meander. It develops a pathwise moderate deviation framework for time-inhomogeneous BRWs and proves both a conditional limit theorem for $M_n$ and a power-law tail for the rightmost point $M$, highlighting a substantially faster growth and heavier tail in the random environment than in the constant-environment setting. The approach hinges on reduced BRWs conditioned on survival, spine decompositions, and Skorohod coupling to Brownian meander limits. Together, these results illuminate the impact of environmental randomness on the extremal behavior of critical BRWs and quantify the interplay between survival, path deviations, and maximal displacement.

Abstract

In this article, we study the maximal displacement of critical branching random walk in random environment. Let $M_n$ be the maximal displacement of a particle in generation $n$, and $Z_n$ be the total population in generation $n$, $M$ be the rightmost point ever reached by the branching random walk. Under some reasonable conditions, we prove a conditional limit theorem, \begin{equation*} \mathcal{L}\left( \dfrac{M_n}{\sqrtσ n^{\frac{3}{4}}} |Z_n>0\right) \dcon \mathcal{L}\left(A_Λ\right), \end{equation*} where random variable $A_Λ$ is related to the standard Brownian meander. And there exist some positive constant $C_1$ and $C_2$, such that \begin{equation*} C_1\leqslant\liminf\limits_{x\rightarrow\infty}x^{\frac{2}{3}}¶(M>x) \leqslant \limsup\limits_{x\rightarrow\infty} x^{\frac{2}{3}}¶(M>x) \leqslant C_2. \end{equation*} Compared with the constant environment case (Lalley and Shao (2015)), it revaels that, the conditional limit speed for $M_n$ in random environment (i.e., $n^{\frac{3}{4}}$) is significantly greater than that of constant environment case (i.e., $n^{\frac{1}{2}}$), and so is the tail probability for the $M$ (i.e., $x^{-\frac{2}{3}}$ vs $x^{-2}$). Our method is based on the path large deviation for the reduced critical branching random walk in random environment.

On the maximal displacement of critical branching random walk in random environment

TL;DR

The paper analyzes the maximal displacement of a critical branching random walk in random environment, showing that conditioned on non-extinction the appropriately scaled maximal position converges in distribution to a functional linked to Brownian meander. It develops a pathwise moderate deviation framework for time-inhomogeneous BRWs and proves both a conditional limit theorem for and a power-law tail for the rightmost point , highlighting a substantially faster growth and heavier tail in the random environment than in the constant-environment setting. The approach hinges on reduced BRWs conditioned on survival, spine decompositions, and Skorohod coupling to Brownian meander limits. Together, these results illuminate the impact of environmental randomness on the extremal behavior of critical BRWs and quantify the interplay between survival, path deviations, and maximal displacement.

Abstract

In this article, we study the maximal displacement of critical branching random walk in random environment. Let be the maximal displacement of a particle in generation , and be the total population in generation , be the rightmost point ever reached by the branching random walk. Under some reasonable conditions, we prove a conditional limit theorem, \begin{equation*} \mathcal{L}\left( \dfrac{M_n}{\sqrtσ n^{\frac{3}{4}}} |Z_n>0\right) \dcon \mathcal{L}\left(A_Λ\right), \end{equation*} where random variable is related to the standard Brownian meander. And there exist some positive constant and , such that \begin{equation*} C_1\leqslant\liminf\limits_{x\rightarrow\infty}x^{\frac{2}{3}}¶(M>x) \leqslant \limsup\limits_{x\rightarrow\infty} x^{\frac{2}{3}}¶(M>x) \leqslant C_2. \end{equation*} Compared with the constant environment case (Lalley and Shao (2015)), it revaels that, the conditional limit speed for in random environment (i.e., ) is significantly greater than that of constant environment case (i.e., ), and so is the tail probability for the (i.e., vs ). Our method is based on the path large deviation for the reduced critical branching random walk in random environment.

Paper Structure

This paper contains 12 sections, 20 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Time-inhomogeneous Branching Random Walk
  3. Model and Assumptions
  4. The Sample Path Moderate Deviation Principle
  5. Local Upper Bound
  6. Local lower bound
  7. Improving the "weak" deviation result
  8. The maximal displacement
  9. Critical branching processes in random environment
  10. Proof of Theorems
  11. Proof of \ref{['Conditional limit theorem']}
  12. Proof of \ref{['Tail prob of M']}

Key Result

Theorem 1

Under the A1, condition on the survival events $\left\{Z_n>0\right\}$, $\dfrac{M_n}{\sqrt{\sigma}n^{\frac{3}{4}}}$ convergence in law to some non-degenerated random variable $A_\Lambda$: Here $A_\Lambda:=\sup\{g(1): \forall r\in[0,1],\int_{0}^{r} \frac{1}{2}g^{\prime}(s)^2 \mathrm{d} s\leqslant \Lambda_r,g\in C_0([0,1])\}$, where $\Lambda_t:=\inf\{W_s^+;s\in[t,1]\}$ is the minimal process of a st

Theorems & Definitions (34)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Definition 1
  • Theorem 3
  • Remark 4
  • ...and 24 more