[@bourgade2018extreme Lemma 2.2].

Furthermore, we have the following lemma regarding the growth of the characteristics, which will be useful for the error estimates in the local relaxation.

\[l.ChracteristcsInt\] For any $ z=E+{\text{\rm i}}\eta \in {\mathscr{S}}$, we have $$\dfrac{\varphi^4}{N^2} \int_0^t ds \int \dfrac{d \rho(x)}{|z_{t-s} -x|^4 \max(\kappa(x),s^2)} \lesssim \dfrac{\kappa(E)}{\max(\kappa(E),t^2)}.$$

The proof is essentially the same as in [@bourgade2018extreme Lemma A.2], except now we need to consider the Stieltjes transform $ m(z) $ instead of the one for the semicircle law. Recall the relation $ m(z)=z m_{{\mathsf{MP}}}(z^2) $, and note that for the Stieltjes transform of the Marchenko-Pastur law we have the following results from [@bloemendal2016principal Lemma 3.6]: $$|m_{{\mathsf{MP}}}(z)| \sim 1,
\ \ \ \ \
{\text{\rm Im}\hspace{0.1cm}}m_{{\mathsf{MP}}}(z) \sim \left\{
\begin{aligned}
& \sqrt{\kappa(E)+\eta} & \mbox{if } & E \in [\lambda_-,\lambda_+],\\
&\frac{\eta}{\sqrt{\kappa(E)+\eta}} & \mbox{if } & E \notin [\lambda_-,\lambda_+].
\end{aligned}
\right.$$ The rest of the proof follows from the calculation in [@bourgade2018extreme Lemma A.2].

We now consider the initial value $ f_0 $ on the curve $ {\mathscr{S}}$. For this purpose we define the set of good trajectories such that the rigidity holds: $${\mathscr{A}}:= \left\{ |x_k^{(\nu)}(t) - \gamma_k|<\varphi^{\frac{1}{2}}N^{-\frac{2}{3}}({\widehat}{k})^{-\frac{1}{3}} \ \mbox{for all}\ 0 {\leqslant}t {\leqslant}1, k \in \llbracket 1,N \rrbracket,0 {\leqslant}\nu {\leqslant}1 \right\}.$$ We have 