Linear cover time is exponentially unlikely

Proving a 2009 conjecture of Itai Benjamini, we show:  For any C, there is a greater than 0 such that for any simple random walk on an n-vertex graph G, the probability that the first Cn steps of the walk hit every vertex of G is at most exp[-an].  A first ingredient of the proof is a similar statement for Markov chains in which all transition probabilities are less than a suitable function of C.  Joint with Jeff Kahn.