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Intersections of branching random walks on $\mathbb{Z}^8$

Zsuzsanna Baran

TL;DR

This work analyzes intersections of branching random walks on $\mathbb{Z}^8$ indexed by the infinite invariant tree (a spine with attached GW trees on both sides). It extends Lawler’s classic SRW results to BRWs by developing a magic formula, a depth-first queue representation of the Galton–Watson trees, and a concentration framework that yields sharp $\log n$-scaling for intersection counts and non-intersection probabilities in the critical dimension. The authors prove a precise $1/\log n$ decay for the non-intersection probability between a one-sided and a two-sided BRW, and establish a weak law of large numbers for the branching capacity of BRW ranges, namely $(\log n)/n \cdot \mathrm{BCap}(\mathcal{T}[0,n]) \to c_8$ in probability. These results illuminate the geometric structure of BRW ranges in high dimension and pave the way for further BRW intersection and capacity analyses, including potential extensions to other BRW configurations and questions of effective resistance.

Abstract

We consider random walks on $\Z^8$ indexed by the infinite invariant tree, which consists of an infinite spine and finite random trees attached to it on both sides. We establish the precise order of the non-intersection probability between one walk indexed by one side of the tree, and an independent one indexed by both sides of an independent tree. This is analogous to the result by Lawler from the '90s for two independent simple random walks on $\Z^4$. We also prove a weak law of large numbers for the branching capacity of the range of a branching random walk.

Intersections of branching random walks on $\mathbb{Z}^8$

TL;DR

This work analyzes intersections of branching random walks on indexed by the infinite invariant tree (a spine with attached GW trees on both sides). It extends Lawler’s classic SRW results to BRWs by developing a magic formula, a depth-first queue representation of the Galton–Watson trees, and a concentration framework that yields sharp -scaling for intersection counts and non-intersection probabilities in the critical dimension. The authors prove a precise decay for the non-intersection probability between a one-sided and a two-sided BRW, and establish a weak law of large numbers for the branching capacity of BRW ranges, namely in probability. These results illuminate the geometric structure of BRW ranges in high dimension and pave the way for further BRW intersection and capacity analyses, including potential extensions to other BRW configurations and questions of effective resistance.

Abstract

We consider random walks on indexed by the infinite invariant tree, which consists of an infinite spine and finite random trees attached to it on both sides. We establish the precise order of the non-intersection probability between one walk indexed by one side of the tree, and an independent one indexed by both sides of an independent tree. This is analogous to the result by Lawler from the '90s for two independent simple random walks on . We also prove a weak law of large numbers for the branching capacity of the range of a branching random walk.

Paper Structure

This paper contains 28 sections, 34 theorems, 178 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Further directions
  3. Overview
  4. Organisation
  5. Some notation and preliminary results
  6. Further notation regarding BRWs
  7. Green's functions
  8. Shift invariance of BRWs and the magic formula
  9. Depth-first queue process
  10. Other notation
  11. Proof of the concentration of $U_n$
  12. Estimates for one $\mathrm{GW}$-tree
  13. Expectation of $G_n^+$ and $G_n^-$
  14. Variance of $G_n^+$ and $G_n^-$
  15. Bounding $\mathbb{E}\left[\mathrm{Var}\left(U_n\;\middle\vert\;\mathcal{T}\right)\right]$
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\mu$ be a distribution on $\mathbb{Z}_{\ge0}$ with mean 1 and variance $\sigma^2\in(0,\infty)$. Let $\mathcal{T}$ and $\widetilde{\mathcal{T}}$ be two independent branching random walks on $\mathbb{Z}^8$ with offspring distribution $\mu$, as in def:RW_on_tree. Then for all sufficiently large $n where $c$ and $C$ are positive constants depending only on $\sigma^2$. The same result also holds w

Figures (4)

  • Figure 1: Illustration of $\mathcal{T}$ and the indexing of its vertices.
  • Figure 2:
  • Figure 3:
  • Figure 4:

Theorems & Definitions (63)

  • Definition 1.1: Infinite invariant tree
  • Definition 1.2: Branching random walk
  • Theorem 1
  • Theorem 2
  • Definition 1.3: Branching capacity
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 53 more