Central limit theorem for the range of critical branching random walk

TL;DR

This paper establishes a central limit theorem for the size of the range of a critical branching random walk in high dimensions, revealing Gaussian fluctuations when dimension exceeds 16.

Contribution

It proves a CLT for the range of critical BRW in high dimensions, extending understanding of fluctuations in branching processes.

Findings

Range size has linear variance for d>8

Range satisfies a CLT with Gaussian limit for d>16

Method combines stationarity, CLT, truncation, and recursive moment control

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 Merlev\`ede [7], a truncation scheme exploiting the local independence of the tree, and a recursive method for controlling moments.