Séminaire des jeunes chercheurs

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Intervenant : PESCE Valentin (CMAP)
Titre :
A PDE approach to study large random matrices

Le : 22/01/2026 de : 17:00 à : 18:00
In this talk I will present the spectral dominance approach introduced by Bertucci, Debbah, Lasry and Lions (in short BDLL) to deal with PDEs that appear as limits of large random matrices problems. First, I will talk about the Wigner theorem (1955) which can be considered as a law of large numbers for the eigenvalues of random matrices. Then, I shall present a continuous in time model introduced by Dyson in 1962. This approach gives a new point of view on some problems of large random matrices as the Wigner theorem. It leads to the study of a PDE called the Dyson equation. I will explain how this PDE is usually treated in litterature leading to a complex Burgers equation. Finally, I shall present how BDLL use the theory of viscosity solutions to introduce a new notion of solutions for the Dyson equation. This approach gives a new point of view for the uniqueness of the Dyson equation and allows new robust proof for properties of the solution (regularization, long time behaviour…).
Salle : A707