that $$\pi_{N,k}(Q_1(\xi)\ge 2m)\le \left\lfloor \frac{N}{m}\right\rfloor \exp\left(-m{\lambda}^{-N} \frac{k}{N}\right).$$ This remains true for $Q_2(\xi)$ upon replacing $k$ by $N-k$, and this concludes the proof of if one choses $m=\frac{{\lambda}^{N}  N u}{2k}$ .

Eigenfunctions and contractions {#sec:eigen}
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The exact expression of the principal eigenfunction / eigenvalue has been derived in previous works [@LevPer16; @LabLac16]. It turns out that it can be obtained by applying a discrete Hopf-Cole transform to the generator of our Markov chain. Let us recall some identities in that direction as they will be needed later on; the details can be found in [@LabLac16 Section 3.3]. We set $$\label{Eq:rho} \varrho := \big(\sqrt p - \sqrt q\big)^2 \sim \frac{b_N^2}{2}\;,$$ and we let $a_{N,k}$ be the unique solution of $$\begin{cases}
    (\sqrt{pq}\, {\Delta}-\varrho)a(x)=0\;,\quad x\in {\llbracket}1,N-1  {\rrbracket}\;,\\
     a(0)=1\;,\quad a(N)= {\lambda}^{\frac{2k-N}{2}}  \;,
 \end{cases}$$ where ${\Delta}$ denotes the discrete Laplace operator $$\label{laplace}
 {\Delta}(f)(x)=f(x+1)+f(x-1)-2f(x), \quad  x\in {\llbracket}1,N-1  {\rrbracket}.$$ If $(h^\zeta_t,t\ge 0)$ denotes the height function process starting from some arbitrary initial condition $\zeta\in\Omega_{N,k}$, then the map $$V(t,x) := {\mathbb{E}}[\lambda^{\frac12 h^{\zeta}_t(x)} - a_{N,k}(x)]\;,\quad t\ge 0\;,\quad x\in {\llbracket}0,N{\rrbracket}\;,$$ solves $$\label{Eq:V}
\begin{cases}
    \partial_t V(t,x)= (\sqrt{pq}\,{\Delta}- \varrho) V(t,x)\;,\quad x\in{\llbracket}1,N-1  {\rrbracket}\;.\\
    V(t,0) = V(t,N) = 0\;.
\end{cases}$$ This allows to identify $N-1$ eigenvalues and eigenfunctions of the generator ${\mathcal{L}}_{N,k}$ of the 