Linear Cover Time is Exponentially Unlikely

TL;DR

This paper proves that the probability of a simple random walk covering a bounded degree graph in linear time is exponentially small, highlighting the rarity of such rapid coverage.

Contribution

It establishes an exponential bound on the probability of linear cover time for bounded degree graphs, extending understanding of random walk cover times.

Findings

Probability of linear cover time is exponentially small.

Bound holds uniformly over all graphs with given degree and size.

Conjecture that the bound is independent of maximum degree for simple graphs.

Abstract

We show that the probability that a simple random walk covers a finite, bounded degree graph in linear time is exponentially small. More precisely, for every D and C, there exists a=a(D,C)>0 such that for any graph G, with n vertices and maximal degree D, the probability that a simple random walk, started anywhere in G, will visit every vertex of G in its first Cn steps is at most exp(-an). We conjecture that the same holds for a=a(C)>0 that does not depend on D, provided that the graph G is simple.