the time at which macroscopic equilibrium is attained.

The aim of this paper is to understand better the role of the asymmetry in mixing and how one interpolates between the symmetric and asymmetric regimes. This leads us to consider a model with an asymmetry that vanishes with the scale of observation, usually referred to as Weakly Asymmetric Exclusion Process (WASEP). While hydrodynamic limit [@Demasi89; @Gartner88; @KipLan] and fluctuations scaling limits [@DG91; @BG97; @LabbeKPZ] for WASEP are now well understood, much less is known about how a weak asymmetry affects the mixing time of the system.

A first step in this direction was made in [@LevPer16]. Therein the order of magnitude for the mixing time was identified for all possible intensities of vanishing bias, but with different constant for the upper and the lower bounds. Three regimes where distinguished (in the case where there is a density of particles):

-   When $b_N\le 1/N$, the mixing time remains of the same order as that of the symetric case $N^2\log N$.

-   When $1/N\le b_N\le (\log N)/N$, the mixing time is of order $(b_N)^{-2}\log N$.

-   When $(\log N)/N\le b_N\le 1$, the mixing time is of order $(b_N)^{-1} N$.

The transition occurring around 