A sharp estimate for cover times on binary trees

TL;DR

This paper precisely estimates the cover time for a binary tree by a random walk, revealing a second order correction term and its difference from the Gaussian free field maximum.

Contribution

It provides the second order correction for the cover time of binary trees, highlighting differences from the Gaussian free field maximum.

Findings

Second order correction for cover time computed

Correction differs from Gaussian free field maximum

High probability asymptotic behavior established

Abstract

We compute the second order correction for the cover time of the binary tree of depth $n$ by (continuous-time) random walk, and show that with probability approaching 1 as $n$ increases, $\sqrt{\tau_{\mathrm{cov}}}=\sqrt{|E|}[\sqrt{2\log 2}\cdot n - {\log n}/{\sqrt{2\log 2}} + O((\log\logn)^8]$, thus showing that the second order correction differs from the corresponding one for the maximum of the Gaussian free field on the tree.