Hermon Jonathan et Pymar Richard, The exclusion process mixes (almost) faster than independent particles, Annals of probability (Online) , 48(6), 2020, pp. 3077-3123
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.