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Central limit theorem for the range of critical branching random walk

Tianyi Bai, Yueyun Hu

TL;DR

This work analyzes second-order fluctuations for the range of a critical BRW on $\mathbb{Z}^d$ indexed by the Kesten tree. It proves linear varinace growth for $d>8$ and a Gaussian CLT for the centered, rescaled range when $d>16$, using a stationary-depth-first exploration framework, a CLT for stationary sequences, a truncation scheme that exploits local independence, and a recursive moment method to control higher moments. The approach handles dependent increments inherent to BRWs and yields precise dimension thresholds, advancing the understanding of BRW range fluctuations in high dimensions. The techniques may extend to related range functionals, such as capacity, in similar dependent-tree indexed systems.

Abstract

In this paper, we study second order fluctuations for the size of the range of a critical branching random walk (BRW) in $\mathbb Z^d$. We consider the BRW with geometric offspring indexed by the Kesten tree, and show that the size of its range has linear variance when $d>8$, and satisfies a central limit theorem (CLT) with Gaussian limiting distribution when $d>16$. The proof combines the stationarity of the model under depth-first exploration, the general CLT of Dedecker and Merlevède [7], a truncation scheme exploiting the local independence of the tree, and a recursive method for controlling moments.

Central limit theorem for the range of critical branching random walk

TL;DR

This work analyzes second-order fluctuations for the range of a critical BRW on indexed by the Kesten tree. It proves linear varinace growth for and a Gaussian CLT for the centered, rescaled range when , using a stationary-depth-first exploration framework, a CLT for stationary sequences, a truncation scheme that exploits local independence, and a recursive moment method to control higher moments. The approach handles dependent increments inherent to BRWs and yields precise dimension thresholds, advancing the understanding of BRW range fluctuations in high dimensions. The techniques may extend to related range functionals, such as capacity, in similar dependent-tree indexed systems.

Abstract

In this paper, we study second order fluctuations for the size of the range of a critical branching random walk (BRW) in . We consider the BRW with geometric offspring indexed by the Kesten tree, and show that the size of its range has linear variance when , and satisfies a central limit theorem (CLT) with Gaussian limiting distribution when . The proof combines the stationarity of the model under depth-first exploration, the general CLT of Dedecker and Merlevède [7], a truncation scheme exploiting the local independence of the tree, and a recursive method for controlling moments.

Paper Structure

This paper contains 10 sections, 23 theorems, 195 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The contour process
  4. The contour process
  5. Estimates on $\mathcal{T}^\infty$.
  6. Linear growth of variance: proof of \ref{['kappaexistspositive']}
  7. Existence of $\kappa$
  8. Non-triviality of $\kappa$
  9. Proofs of \ref{['cond-a']} and \ref{['cond-b']}
  10. Proof of \ref{['cond-c']}: Uniform integrability by fourth moment estimate

Key Result

Theorem 1.1

Assume eq:assmption. When $d>8$, there is a constant $\kappa=\kappa_d>0$ such that When $d>16$, for every continuous real function $\varphi$ satisfying $\sup_{x\in\mathbb R}|\varphi(x)/(1+x^2)|<\infty$, we have where ${\tt g}_\kappa\sim {\mathcal{N}}(0, \kappa)$ denotes a centered Gaussian random variable with variance $\kappa$. In particular, $\frac{ Y_n- {\mathbb E}[Y_n]}{\sqrt{n}}$ converges

Figures (3)

  • Figure 1: The infinite tree $\mathcal{T}^\infty$ and its contour process. In the picture, $u_{-2}=u_0=\varnothing$, $u_{-3}=u_{-5}=\varnothing_1$.
  • Figure 2: As long as $m\ge k_1+k_2$, $\xi_j^{k_2}\mathbbm{1}_{\{{\tt d}(u_0,u_j)=m\}}$ is always independent of the first $k_1$ subtrees in the past of $u_0$.
  • Figure 3: In the left-side tree, we can enumerate $(\Delta C(2i-1,2i))_i$ as $((1,-1);(1,-1);(1,1);(-1,1);(1,-1);(-1,-1);(-1,1);(1,-1),\dots)$. There are four occurrences of $(1,-1)$, so we have four possible bad points, marked in blue. Suppose that the last two are bad, after deleting them we obtain the right-side tree with $\widehat{C}=((1,-1);(1,-1);(1,1);(-1,1);(-1,-1);(-1,1)\dots)$. In notation of Lemma \ref{['lem:YN']}, we have $X_4=X_6=1$, because we delete a bad point after the $4$-th and $6$-th parenthesis in $\widehat{C}$, or equivalently, after $\widehat{u}_8$ and $\widehat{u}_{12}$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 35 more