time needed for local fluctuations to reach their equilibrium values are of the same order and we believe that there is an interaction between the two phenomena. We provide a more detailed conjecture in Section \[conjectos\]

We have not included here results concerning the biased card shuffling considered in [@Benjamini; @LabLac16]. Let us mention that while our analysis should also yield optimal bound for the mixing time of this process when holds (i.e. ${T_{\rm mix}}^{k,N}({\varepsilon})\sim 2 N /(p_N-q_N)$), it seems much more difficult to prove the equivalent of Theorem \[Th:smallbias\]. The main reason is that the coupling presented in Section \[Appendix:Coupling\] cannot be extended to a coupling on the permutation process. Building on and adapting the techniques presented in [@Lac16 Section 5] it should a priori be possible to obtain a result concerning the mixing time starting from an extremal condition (the identity or its symmetric), but this is out of the scope of the present paper.

Conjecture in the regime $b_N\asymp \log k_N/N$ {#conjectos}
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Let us here formulate, and heuristically support a conjecture concerning the mixing-time in the crossover regime where $$\label{limbeta}
\lim_{N\to \infty} \frac{b_N N}{\log k_N}={\beta}.$$ for some ${\beta}\in(0,\infty)$. For the ease of exposition, while it should 