Barrier estimates for a critical Galton--Watson process and the cover time of the binary tree

TL;DR

This paper develops precise barrier estimates for a critical Galton-Watson process with geometric offspring, which are then applied to analyze cover times of binary trees and two-dimensional manifolds, improving existing bounds.

Contribution

The paper introduces sharp barrier estimates for critical Galton-Watson processes with geometric offspring, enabling improved analysis of cover times in graphs and manifolds.

Findings

Proves tightness of the normalized cover time for binary trees.

Provides improved bounds on cover times compared to previous results.

Establishes barrier estimates useful for analyzing Brownian cover times on manifolds.

Abstract

For the critical Galton--Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As an application of the barrier estimates, we prove that if $C_L$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_L/2^{L+1}} -\sqrt{2\log 2} L+\log L/\sqrt{2\log 2}$ is tight. The latter improves results of Aldous (1991), Bramson and Zeitouni (2009) and Ding and Zeitouni (2012). In a subsequent article we use these barrier estimates to prove tightness of the Brownian cover time for the two-dimensional sphere.