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We consider the simple exclusion process with $k$ particles on a segment oflength $N$ performing random walks with transition $p>1/2$ to the right and$q=1-p$ to the left. We focus on the case where the asymmetry in the jump rates$b=p-q>0$ vanishes in the limit when $N$ and $k$ tend to infinity, and obtainsharp asymptotics for the mixing times of this sequence of Markov chains in thetwo cases where the asymmetry is either much larger or much smaller than $(\logk)/N$. We show that in the former case ($b \gg (\log k)/N$), the mixing timecorresponds to the time needed to reach macroscopic equilibrium, like for thestrongly asymmetric (i.e.\ constant $b$) case studied in [LL18], while thelatter case ($b \ll (\log k)/N$) macroscopic equilibrium is not sufficient formixing and one must wait till local fluctuations equilibrate, similarly to whathappens in the symmetric case worked out in [Lac16b]. In both cases,convergence to equilibrium is abrupt: we have a cutoff phenomenon for thetotal-variation distance. We present a conjecture for the remaining regime whenthe asymmetry is of order $(\log k) / N$.

Mixing time and cutoff for the weakly asymmetric simple exclusion process