Linear cover time is exponentially unlikely

TL;DR

This paper proves that the probability of a simple graph's random walk covering all vertices in linear time is exponentially small, confirming a 2009 conjecture and advancing understanding of cover times.

Contribution

It establishes an exponential bound on the probability of linear cover time for simple graphs, confirming a longstanding conjecture and extending to Markov chains with small transition probabilities.

Findings

Probability of linear cover time is exponentially small

Validates a 2009 conjecture by Itai Benjamini

Provides bounds for Markov chains with small transition probabilities

Abstract

Proving a 2009 conjecture of Itai Benjamini, we show: For any C there is an $\varepsilon>0$ such that for any simple graph $G$ on $V$ of size $n$, and $X_0,\ldots$ an ordinary random walk on $G$, $P(\{X_0,\dots, X_{Cn}\}= V) < e^{-\varepsilon n}.$ A first ingredient in the proof of this is a similar statement for Markov chains in which all transition probabilities are sufficiently small relative to $C$.