The MEGA Autumn School will bring together for one week young mathematicians interested in topics related to random matrices and random graphs.

The 2025 edition will take place from Monday, October 13 to Friday, October 17, 2025, in Les Plantiers, a village nestled in the heart of the Cévennes, at the CNRS CAES – La Maison Clément – Faveyrolles, 3012 Les Plantiers. The number of available spots is limited, but all applications are welcome.

Two mini-courses will be delivered, including lectures by David Garcia-Zelada and Jonathan Husson (see the Planning section). Participants will also be encouraged to contribute by presenting their own research work.

The MEGA axis of the Mathematics and Physics thematic network will cover accommodation and meals for around fifteen participants, as well as the shuttle from Nîmes train station. The rest of the travel expenses must be covered by your home laboratory.

For any questions regarding the organization, feel free to contact the organizers:

• Aniss FARES (Aniss.fares[at]polytechnique.edu)

• Mohammed-Younes GUEDDARI (mohammed-younes.gueddari[at]univ-eiffel.fr)

• Zikun OUYANG (ouyang[at]ceremade.dauphine.fr)

David Garcia-Zelada

Outliers in Random Matrices and Polynomials

We will discuss the existence or non-existence of outliers in two-dimensional systems formed by the eigenvalues of certain random matrices or by the zeros of certain random polynomials. We will focus in particular on the works [arXiv:1811.12225] and [arXiv:2012.05602], whose proofs rely on relatively elementary arguments but, in my opinion, are particularly interesting.

Jonathan Husson

Since its inception by the statistician John Wishart in the 1920s and the subsequent work of Eugene Wigner in the 1950, random matrix theory has expanded to many other fields both in mathematics (with applications for instance to number number theory, integrable systems,...) and outside (in physics, telecommunication, machine learning or biology,..). One of the main question tackled by this theory is the behavior of the spectrum of large random matrices when the dimension goes to infinity. In particular, for Wigner matrices, one has laws of large numbers for both the whole spectrum and for the extremal eigenvalues. But beyond, one can also ask about the probability of large deviations of those spectral quantities (that is how does the probability that they are close to some atypical value decay). This question has known numerous developments in the last ~15 years, in particular for the challenging cases of non-Gaussian matrices. In this mini-course, we will review some of those results and in particular how to use spherical integrals to prove large deviation principles for the largest eigenvalues of random matrices.