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We consider the biased card shuffling and the Asymmetric Simple ExclusionProcess (ASEP) on the segment. We obtain the asymptotic of their mixing times:our result show that these two continuous-time Markov chains display cutoff.Our analysis combines several ingredients including: a study of thehydrodynamic profile for ASEP, the use of monotonic eigenfunctions, stochasticcomparisons and concentration inequalities.

Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shuffling