the following important estimate for the probability of these events.

\[l.rigidity\] There exists a fixed $ C_0 $ large enough such that the following holds. For any $ D>0 $, there exists $ N_0(D)>0 $ such that for any $ N > N_0 $ we have $${\mathbb{P}}({\mathscr{A}}) > 1-N^{-D}.$$

As mentioned in the last section, the rigidity estimates are proved in [@pillai2014universality] for fixed $ t $ and $ \nu=0,1 $. The extension to all $ t $ and $ \nu $ is based on the arguments in [@bourgade2018extreme; @erdHos2015gap]: (1) discretize in $ t $ and $ \nu $; (2) use Weyl’s inequality to bound the increments in small time intervals; (3) use the maximum principle to bound the increment in small $ \nu $-intervals.

Conditioned on the rigidity phenomenon, we have the following estimate for the initial conditions. The proof is the same as [@bourgade2018extreme Lemma 2.4].

\[l.InitialValue\] In the set $ {\mathscr{A}}$, for any $ z=E+{\text{\rm i}}\eta \in \mathscr{R} $, we have $ {\text{\rm Im}\hspace{0.1cm}}f_0(z) \lesssim \varphi^{1/2} $ if $ \eta > \max(E-\sqrt{\lambda_+},-E+\sqrt{\lambda_-}) $, and $ {\text{\rm Im}\hspace{0.1cm}}f_0(z) \lesssim \varphi^{1/2} \frac{\eta}{\kappa(z)} $ otherwise. The same bound also holds for $ |{\text{\rm Im}\hspace{0.1cm}}f_0| $.

Quantitative relaxation at the 