We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. We also study the space--time correlations of those eigenvalues.

Then, motivated by the long-time behaviour of the ODE $\dot{u}=Xu$, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of $X$, and prove the universality of their Gumbel fluctuations.