Mixing and generation in simple groups

Let G be a finite simple group. We show that a random walk on G with respect to the conjugacy class xG of a random element x G has mixing time 2. In particular it follows that (xG)2 covers almost all of G, which could be regarded as a probabilistic version of a longstanding conjecture of Thompson. We also show that if w is a non-trivial word, then almost every pair of values of w in G generates G.