has a non-trivial scaling limit (the inviscid Burgers’ equation with zero-flux boundary conditions), which fixates at time $\frac{(\sqrt{\alpha}+\sqrt{1-\alpha})^2}{p-q}$. The first order asymptotic for the mixing time is thus determined by the time the density profile needs to reach equilibrium.

When $p=1/2$, the right-time scale to observe a macroscopic motion for the particles is $N^2$, and it is worth mentioning that the scaling limit obtained for the particle density (the heat equation) does not fixate in finite time. To reach equilibrium, however, we must wait for a longer time, of order $N^2\log N$, which is the time needed for local fluctuations in the particle density to come to equilibrium.

We are interested in studying the process when the drift tends to zero: this requires to understand the transition between these two patterns of relaxation to equilibrium. Hence we consider $p$ to be a function of $N$ which is such that the bias towards the right $b_N:=p_N-q_N=2p_N-1$ vanishes $$\label{vanishing}
\lim_{N\to \infty} b_N=0.$$ In this regime, the model is sometimes called WASEP for Weakly Asymetric Simple Exclusion Process. Its convergence to equilibrium has already been studied in[@LabbeKPZ; @LevPer16]. In [@LevPer16] the authors identify the order of magnitude of the mixing time as a 