be in principle possible to extend the heuristic to the case of positive density (see Remark \[pluscomplique\] below) we restrict ourselves to the case $\lim_{N\to \infty }k_N/N=0$. The justification we provide for the conjecture might be better understood after a first reading of the entire paper.

\[jecture\] When $b_N$ and $k_N$ display the asymptotic behavior given by , we have for every ${\varepsilon}>0$ $$\lim_{N \to \infty} \frac{{T_{\rm mix}}^{N,k_N}({\varepsilon}) \log k_N}{N^2}= \begin{cases}       \frac{2}{{\beta}}+\frac{1}{{\beta}^2}, \quad &\text{ if } {\beta}\le 1/2,\\
       \left( \frac{\sqrt{2}+2\sqrt{{\beta}}}{2{\beta}} \right)^2, \quad &\text{ if } {\beta}\ge 1/2.
                                                              \end{cases}$$

To motivate this conjecture let us first describe the equilibrium measure and its dependence on ${\beta}$. As we are in the low-density regime, the equilibrium measure is quite close to the product measure one would obtain for the system without exclusion rules: the $k$ particles are approximately IID distributed with the distance from the right extremity being a geometric of parameter $q_N/p_N\approx e^{- \frac{2{\beta}\log k}{N}}$.

Hence the probability of having a particle at site $\lfloor zN \rfloor$ for $z\in [0,1]$ is roughly of order $k^{[1-2{\beta}(1-z)]+o(1)}/N$. Thus, while particles are concentrated near the right extremity at equilibrium, the equilibrium “logarithmic density” of particles exhibit a non trivial profile in the sense that for 