We study mixing times for the totally asymmetric simple exclusion process
(TASEP) on a segment of size N with open boundaries. We focus on the maximal
current phase, and prove that the mixing time is of order N3/2, up to
logarithmic corrections. In the triple point, where the TASEP with open
boundaries approaches the Uniform distribution on the state space, we show that
the mixing time is precisely of order N3/2. This is conjectured to be the
correct order of the mixing time for a wide range of particle systems with
maximal current. Our arguments rely on a connection to last-passage
percolation, and recent results on moderate deviations of last-passage times.Comment: 42 pages, 10 figures, accepted versio