\dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \left( \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{3z {\mathfrak{v}}_k}{2 x_k^2 (x_k -z)(x_k+z)} + \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z^3 {\mathfrak{v}}_k}{x_k^2 (x_k-z)^2 (x_k+z)^2} \right).\end{aligned}$$ Note that now all the singularities are removed.

The desired result then follows by combining the previous results and the term $ I_1 $.

Recall that in Section \[s.LocalLaw\] we have shown that the Stieltjes transform of the empirical measure for singular values satisfies the local law , so that the leading term of the stochastic differential equation satisfied by $ f_t $ is close to $$\dfrac{z}{2\xi} + z m_{{\mathsf{MP}}}(z^2) = \dfrac{z}{2\xi} + \dfrac{1-\xi-z^2 + \sqrt{(z^2 - \lambda_-)(z^2 - \lambda_+)}}{2\xi z} = \dfrac{(1-\xi) + \sqrt{(z^2 - \lambda_-)(z^2 - \lambda_+)}}{2\xi z}.$$ Thus, the dynamics of $ f_t $ can be approximated by the following advection equation $$\label{e.AdvectionPDE}
\partial_t r = \dfrac{(1-\xi) + \sqrt{(z^2 - \lambda_-)(z^2 - \lambda_+)}}{2\xi z} \partial_z r.$$

Geometric properties of the characteristics {#s.geometry}
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In order to estimate the evolution of the observable, we analyze its dynamics by studying the characteristics of the approximate advection PDE , similarly to [@huang2018rigidity; @bourgade2018extreme]. To do this, we first need some bounds on the shape of the characteristics $ (z_t)_{t {\geqslant}0} $, and some estimates for the initial value. As 