right that vanishes when $N$ tends to infinity. This setup is often referred to as the Weakly Asymmetric Simple Exclusion Process (WASEP): it interpolates between the symmetric case (SSEP) and the one with a positive constant bias (ASEP).

While convergence to equilibrium for a particle system can be considered on a macroscopic scale via the evolution of the particle density or hydrodynamic profile (see e.g. [@KipLan] and references therein), an alternative and complementary viewpoint (when the system is of finite size) consists in measuring the so-called ${\varepsilon}$-Total Variation Mixing Time [@LevPerWil]. It is defined as the first time at which the total variation distance to the stationary state, starting from the “worst" initial condition, falls below a given threshold $\epsilon$. Compared to the hydrodynamic profile, this provides a much more microscopic information on the particle system.

The problem of mixing time of the simple exclusion process on the segment has been extensively studied both in the symmetric [@Wil04; @Lac16; @Lac162] and the asymmetric setup [@Benjamini; @LabLac16] and it has been proved in [@Lac16] and [@LabLac16] respectively that in both cases, the worst case total variation distance drops abruptly from its maximal value $1$ to $0$, so that the mixing time 