and say that the particles perform random walks with jump rates $p$ to the right and $q=1-p$ to the left: These random walks are independent from one another except that any jump that would put a particle at a location already occupied by another particle is cancelled. Having in mind this particle representation, we let for $i\in {\llbracket}1, k{\rrbracket}$, $\xi_i$ denote the position of the $i$-th leftmost particle $$\xi_i:= \min\left\{ y\in {\llbracket}1,N {\rrbracket}\ : \ \sum_{x=1}^y \xi(x)=i \right\}.$$

We let $P^{N,k}_t$ denote the associated semi-group and $(\eta^\xi(t,\cdot))_{t\ge 0}$ denote the trajectory of the Markov chain starting from initial condition $\xi \in {\Omega}_{N,k}^0$. This Markov chain is irreducible, and admits a unique invariant (and reversible) probability measure $\pi_{N,k}$ given by $$\pi_{N,k}(\xi):= \frac{1}{Z_{N,k}}{\lambda}^{-A(\xi)}.$$ where $\lambda = p/q$, $Z_{N,k}:= \sum_{\xi\in {\Omega}_{N,k}^0} {\lambda}^{-A(\xi)}$, and $$\label{def:A}
A(\xi):= \sum_{i=1}^k (N-k+i-\xi_i)\ge 0$$ denotes the minimal number of moves that are necessary to go from $\xi$ to the configuration $\xi^{\min}$ where all the particles are on the right $\xi^{\min}(x):={\mathbf{1}}_{[N-k+1,N]}(x)$ (this terminology is justified by the fact that $\xi^{\min}$ is minimal for the order introduced in Section \[Sec:Prelim\]).

Recall that the total-variation distance between two probability measures defined on the same state-space ${\Omega}$ is defined by $$\|\alpha-{\beta}\|_{TV}=\sup_{A\subset {\Omega}} \alpha(A)-{\beta}(A),$$ 