The exclusion process mixes (almost) faster than independent particles

Oliveira conjectured that the order of the mixing time of the exclusion process with k-particles on an arbitrary n-vertex graph is at most that of the mixing-time of k independent particles. We verify this up to a constant factor for d-regular graphs when each edge rings at rate 1/d in various cases: (1) when d=.(logn/kn) , (2) when gap:= the spectral-gap of a single walk is O(1/log4n) and k.n.(1) , (3) when k.na for some constant 0<a<1 . In these cases, our analysis yields a probabilistic proof of a weaker version of Aldous. famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound of O(lognloglogn/gap) , which is within a loglogn factor from Oliveira.s conjecture when k.n.(1) . As applications, we get new mixing bounds: (a) O(lognloglogn) for expanders, (b) order dlog(dk) for the hypercube {0,1}d , (c) order (Diameter)2logk for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus.