  This paper studies the mixing behavior of the Asymmetric Simple Exclusion
Process (ASEP) on a segment of length $N$. Our main result is that for particle
densities in $(0,1),$ the total-variation cutoff window of ASEP is $N^{1/3}$
and the cutoff profile is $1-F_{\mathrm{GUE}},$ where $F_{\mathrm{GUE}}$ is the
Tracy-Widom distribution function. This also gives a new proof of the cutoff
itself, shown earlier by Labb\'{e} and Lacoin. Our proof combines coupling
arguments, the result of Tracy-Widom about fluctuations of ASEP started from
the step initial condition, and exact algebraic identities coming from
interpreting the multi-species ASEP as a random walk on a Hecke algebra.
