observed in the symmetric case.

\[Th:smallbias\]

When holds, we have

$$\lim_{N\to \infty} \frac{\operatorname{\mathrm{gap}}_{N} {T_{\rm mix}}^{N,k_N}({\varepsilon})}{\log k_N}= \frac12.$$

Using Taylor expansion for $\operatorname{\mathrm{gap}}_N$ we have, whenever $b_N$ tends to zero $$\label{eq:taylorgap}
\operatorname{\mathrm{gap}}_{N}\stackrel{N\to \infty}{\sim} \frac{1}{2}\left( b^2_N+ \left(\frac{\pi}{N}\right)^2 \right).$$ Thus in particular we have $${T_{\rm mix}}^{N,k_N}({\varepsilon})\sim
\begin{cases}
  \frac{\log k_N}{b_N^2} & \text{if } 1/N \ll b_N\ll \log k_N/N, \\
  \frac{1}{\pi^2} N^2 \log k_N &\text{ if } 0< b_N \ll 1/N, \\
\frac{1}{\pi^2+\beta^2} N^2 \log k_N  &\text{ if } b_N\sim \beta/N.
\end{cases}$$

Note that our classification of regimes - does not cover all possible choices of $b_N$. Two cases have been excluded for very different reasons:

-   When $b_N=O(N^{-1})$ and $k_N$ is bounded, then we have a system of $k$ diffusive interacting random walks. This system does not exhibit cutoff and has a mixing time of order $N^2$ (The upper bound can actually be deduced from argument presented in Section \[preuv2\] and the lower bound is achieved e.g. by looking at the expectation and variance of the number of particles on right half of the segment, like what is done in [@Morris06 Section 6] ).

-   When $b_N$ is of order $\log k_N/N$ the time at which the density profile reaches its equilibrium and the 