Capacity of the range of tree-indexed random walk

TL;DR

This paper investigates the asymptotic growth of the capacity of critical branching random walks on trees, revealing linear growth in high dimensions and a logarithmic correction at the critical dimension.

Contribution

It introduces a new measure for the infinite Galton-Watson process and estimates Green's functions to analyze capacity growth.

Findings

Capacity grows linearly for dimensions d ≥ 7

Capacity grows as n/ log n at dimension d=6

Provides asymptotic behavior of capacity in critical branching random walks

Abstract

By introducing a new measure for the infinite Galton-Watson process and providing estimates for (discrete) Green's functions on trees, we establish the asymptotic behavior of the capacity of critical branching random walks: in high dimensions $d\ge 7$, the capacity grows linearly; and in the critical dimension $d=6$, it grows asymptotically proportional to $\frac{n}{\log n}$.