is not the one displayed in [@LabLac16] and for this reason we provide an explicit construction in Appendix \[Appendix:Coupling\].

Once a coupling is specified, by enlarging our probability space, one can also define the process $h^{\pi}_t$ which is started from an initial condition sampled from the equilibrium measure $\pi_{N,k}$, independently of $h_t^{\zeta}, \zeta\in {\Omega}_{N,k}$.

Let us end up this section introducing the (less canonical) notation $$\label{eq:strict}
 \zeta<\zeta' \quad \Leftrightarrow \quad \left( \zeta\le \zeta' \text{ and } \zeta\ne \zeta' \right).$$ We say that a function $f$ on $\Omega_{N,k}$ is increasing (strictly) if $f(\zeta)\le f(\zeta')$ ($f(\zeta)< f(\zeta')$) whenever $\zeta<\zeta'$. The minimal increment of an increasing function is defined by $$\label{eq:minincr}
\delta_{\min}(f)=\min_{\zeta,\zeta'\in {\Omega}_{N,k}, \zeta<\zeta'} f(\zeta')-f(\zeta).$$

The equilibrium measure in the large-bias case
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For $\xi\in {\Omega}_{N,k}^0$ we set $$\label{def1:lknrkn}
\begin{split}
\ell_{N}(\xi)&=\min\{ x\in {\llbracket}1, N{\rrbracket}\ : \  \xi(x)= 1 \},\\
r_{N}(\xi)&=\max\{ x\in {\llbracket}1, N{\rrbracket}\ : \  \xi(x)= 0 \}.
\end{split}$$

A useful observation on the invariant measure is the following. Given $\xi$, we define $\chi(\xi)$ as the sequence of particle spacings: $$\chi_i:= \xi_{i+1}-\xi_i\;,\; \text{ for } i\in {\llbracket}1,k-1{\rrbracket}, \quad  \chi_k=N+1-\xi_k.$$ From , under $\pi_{N,k}$ the probability of a given configuration is proportional to $$\label{eq:weights}
 {\lambda}^{-\chi_1} {\lambda}^{-2\chi_2} \ldots {\lambda}^{-k \chi_k},$$ In other terms, under the 