# A Scaling Limit for the Cover Time of the Binary Tree

## TL;DR

This paper proves that the normalized cover time of a binary tree by a continuous-time random walk converges in distribution to a Gumbel distribution shifted by a random variable, linking it to Gaussian free fields.

## Contribution

It establishes the weak limit of the cover time for binary trees and identifies the limiting distribution as a Gumbel shifted by a Gaussian free field-related random variable.

## Key findings

- Normalized cover time converges to a Gumbel distribution.
- The limit involves a random shift related to Gaussian free fields.
- The approach uses comparison with the extremal landscape of the Gaussian free field.

## Abstract

We consider a continuous time random walk on the rooted binary tree of depth $n$ with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by $2^{n+1} n$ and then centered by $(\log 2) n - \log n$, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated discrete Gaussian free fields on the infinite version of the tree. The existence of the limit and its overall form were conjectured in the literature. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field on the tree.