Linear Cover Time Is Exponentially Unlikely

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 α=α(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 C n steps is at most e−αn . We conjecture that the same holds for α that does not depend on D , provided that the graph G is simple.