# Limit law for the cover time of a random walk on a binary tree

## TL;DR

This paper establishes a limit law for the normalized cover time of a modified binary tree by a simple random walk, revealing its asymptotic distribution as the tree depth grows.

## Contribution

It proves the convergence in distribution of the normalized cover time for a binary tree with an extra edge, identifying the explicit limit distribution.

## Key findings

- Normalized cover time converges in distribution as tree depth increases
- Explicit constant $m_n$ is identified for normalization
- Limit distribution of the normalized cover time is characterized

## Abstract

Let $T_n$ denote the binary tree of depth $n$ augmented by an extra edge connected to its root. Let $C_n$ denote the cover time of $T_n$ by simple random walk. We prove that $\sqrt{ \mathcal{C}_{n} 2^{-(n+1) } } - m_n$ converges in distribution as $n\to \infty$, where $m_n$ is an explicit constant, and identify the limit.