to the label $ k $. To see this, it is easy to check that $ {\widetilde}{{\mathfrak{v}}}_k := {\mathfrak{v}}_{-k} $ satisfy the same equation . Note that the equation is linear and $ {\widetilde}{{\mathfrak{v}}}_k(0)={\mathfrak{v}}_k(0) $ for all $ k $. Lemma \[l.MaxPrin\] then gives us $ {\mathfrak{v}}_k(t) = {\widetilde}{{\mathfrak{v}}}_k(t) = {\mathfrak{v}}_{-k}(t) $ for all $ k $ and $ t {\geqslant}0 $.

We now consider the observable $$f_t(z)=e^{-\frac{t}{2\xi}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k(t)}{x_k(t)-z}.$$ A key observation is that the quadratic singularities in will disappear when combined with the Dyson Brownian motion, so that the evolution of $ f_t $ has no shocks similarly to a result from [@bourgade2018extreme]. Denoting $$s_t(z)=\dfrac{1}{2N}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{1}{x_k(t)-z}$$ the Stieltjes transform of the (symmetrized) empirical spectral measure, then the observable $ f_t $ satisfies the following dynamics.

\[l.dynamics\] For any $  {\text{\rm Im}\hspace{0.1cm}}z \neq 0 $, we have $$\label{e.dynamics}
\begin{aligned}
{\mathrm{d}}f_t &=\left(s_t(z)+\dfrac{z}{2\xi}\right)(\partial_z f_t){\mathrm{d}}t+\dfrac{1}{4N}(\partial_{zz}f_t){\mathrm{d}}t+\left[\dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2(x_k+z)}\right]{\mathrm{d}}t\\
& \quad + \left[\left( 1- \dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \left( \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{3z {\mathfrak{v}}_k}{2 x_k^2 (x_k -z)(x_k+z)} + \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z^3 {\mathfrak{v}}_k}{x_k^2 (x_k-z)^2 (x_k+z)^2} \right)\right] {\mathrm{d}}t\\
& \quad -\dfrac{e^{-\frac{t}{2\xi}}}{\sqrt{N}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2}{\mathrm{d}}B_k.
\end{aligned}$$

This can be proved by direct computation via the Itô’s formula. First, we have $${\mathrm{d}}f=-\dfrac{f}{2\xi}{\mathrm{d}}t+e^{-t/2}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathrm{d}}{\mathfrak{v}}_k}{x_k-z}+e^{-t/2}\sum_{-N {\leqslant}k {\leqslant}N}{\mathfrak{v}}_k {\mathrm{d}}\dfrac{1}{x_k-z} =: 