where the $\sup$ is taken over all measurable sets $A$.

The mixing time associated to the threshold ${\varepsilon}\in (0,1)$ is defined by $${T_{\rm mix}}^{N,k}({\varepsilon}):=\inf\{ t \ge 0 \ : \ d^{N,k}(t)\le {\varepsilon}\},$$ where $d^{N,k}(t)$ denotes the total-variation distance to equilibrium at time $t$ starting from the worst possible initial condition $$\label{tvdis}
 d^{N,k}(t):= \max_{\xi\in {\Omega}_{N,k}^0} \| P^{N,k}_t(\xi, \cdot)-\pi_{N,k}\|_{TV}.$$

We want to study the asymptotic behavior of the mixing time for this system when both the size of the system and the number of particles tend to infinity. A natural case to consider is when there is a non-trivial density of particles, that is $k/N\to \alpha \in (0,1)$, but we decide to also treat the boundary cases of vanishing density ($\alpha=0$) and full density ($\alpha=1$). By symmetry we can restrict to the case when $k=k_N\le N/2$: indeed, permuting the roles played by particles and empty sites boils down to reversing the direction of the asymmetry of the jump rates. Note that we will always impose $k\ge 1$ since when $k=0$ the process is trivial.

The asymptotic behavior of ${T_{\rm mix}}^{N,k}({\varepsilon})$ in the case of constant bias ($p>1/2$ is fixed when $N$ goes to infinity) has been obtained in a previous work.

We 