Contact processes on random regular graphs

We show that the contact process on a random d-regular graph initiated by a single infected vertex obeys the "cutoff phenomenon" in its supercritical phase. In particular, we prove that, when the infection rate is larger than the lower critical value of the contact process on the infinite d-regular tree, there are positive constants C, p depending on the infection rate such that for any . > 0, when the number n of vertices is large then (a) at times t < (C . .)logn the fraction of infected vertices is vanishingly small, but (b) at time (C + .)logn the fraction of infected vertices is within . of p, with probability p.