Covering a graph with independent walks

Let P be an irreducible and reversible transition matrix on a finite state space V with invariant distribution π. We let k chains start by choosing independent locations distributed according to π and then they evolve independently according to P . Let τcov(k) be the first time that every vertex of V has been visited at least once by at least one chain and let tcov(k) = E[τcov(k)] with tcov = tcov(1). We prove that tcov(k) . tcov/k. When k ≤ tcov/trel, where trel is the inverse of the spectral gap, we show that this bound is sharp. For k ≤ tcov/tmix with tmix the total variation mixing time of (P + I)/2 we prove that k ·maxx1,...,xk Ex1,...,xk [τcov(k)] ≍ tcov.