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Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$

Xinxin Chen, Shen Lin

TL;DR

This paper analyzes the occupation time of a critical branching random walk on $\mathbb{Z}^d$ started far from a compact set and conditioned to hit it. It establishes dimension-dependent asymptotics: $\|x\|^{4-d}$ for $d\le3$, $\log\|x\|$ at $d=4$, and $1$ for $d\ge5$, along with corresponding limit laws. The authors develop distinct techniques across dimensions: spinal decomposition for high $d$, a spatial reduced-tree framework at the critical dimension, and Brownian-snake/ISE connections in low dimensions, providing a discrete analogue to Le Gall and Merle’s results. They prove Yaglom-type limits, identify the most recent common ancestor structure, and quantify convergence via Wasserstein distances, highlighting deep links between branching capacity, hitting probabilities, and continuum scaling limits. The work advances understanding of occupation-time phenomena in CBRW and offers tools for studying related branching-interacting spatial processes.

Abstract

We study the critical branching random walk on $\mathbb{Z}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in $\mathbb{Z}^d$. We are interested in the occupation time in $K$ and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\|x\|^{4-d}$ in dimensions $d\leq 3$, of order $\log\|x\|$ in dimension $d=4$, and of order 1 in dimensions $d\geq 5$. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265).

Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$

TL;DR

This paper analyzes the occupation time of a critical branching random walk on started far from a compact set and conditioned to hit it. It establishes dimension-dependent asymptotics: for , at , and for , along with corresponding limit laws. The authors develop distinct techniques across dimensions: spinal decomposition for high , a spatial reduced-tree framework at the critical dimension, and Brownian-snake/ISE connections in low dimensions, providing a discrete analogue to Le Gall and Merle’s results. They prove Yaglom-type limits, identify the most recent common ancestor structure, and quantify convergence via Wasserstein distances, highlighting deep links between branching capacity, hitting probabilities, and continuum scaling limits. The work advances understanding of occupation-time phenomena in CBRW and offers tools for studying related branching-interacting spatial processes.

Abstract

We study the critical branching random walk on started from a distant point and conditioned to hit some compact set in . We are interested in the occupation time in and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order in dimensions , of order in dimension , and of order 1 in dimensions . The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265).
Paper Structure (18 sections, 17 theorems, 365 equations, 3 figures, 1 table)

This paper contains 18 sections, 17 theorems, 365 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Context
  3. Main results
  4. Further discussions
  5. Outline of the paper
  6. Preliminaries
  7. Change of measure and Spinal decomposition
  8. Results for random walk on $\mathbb{Z}^d$
  9. Occupation times when $d\ge5$: proof of Theorem \ref{['highd']}
  10. Occupation times in the critical dimension $d=4$
  11. Moments of $L_K$ and $Z_\mathbb{T}(K)$
  12. Yaglom-type limit: Theorem \ref{['4d']}
  13. Most recent common ancestor $\mathcal{A}_x(K)$
  14. $L_2$-Wasserstein distance on probability measures
  15. Proof of Theorem \ref{['4d']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

When $d=4$, we assume offspring and that $\mu$ is symmetric and has finite support. If $K\subset\mathbb{Z}^d$ is compact, then under $\mathbf{P}_x(\cdot\vert Z_\mathbb{T}(K)\ge1)$, the following joint convergence in law holds: where $Y$ has exponential distribution with parameter $1$, and the constant $\mathbf{c}_4=\frac{1}{8\pi^2\sqrt{\det \Gamma}}$.

Figures (3)

  • Figure 1: The most recent common ancestor $\mathcal{A}_x(K)$ and its spatial location $H_x$.
  • Figure 2: The left most path entering $K$.
  • Figure 3: Spine behavior in the space $\mathbb{Z}^4$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1
  • Theorem 1.4
  • Remark 2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 28 more