Absence of Outliers via the Characteristic Polynomial

• David Garcia-Zelada (LPSM, Sorbonne Université)

For sequences of random polynomials with independent coefficients, some zeros typically remain outside the limiting support, and their behavior may depend on the law of the coefficients. After recalling this phenomenon, we will see that it does not occur when the random polynomials arise as characteristic polynomials of families of non-Hermitian random matrices. Moreover, a form of universality emerges, closely connected to the universality of eigenvalue fluctuations. These results are based on joint work with Charles Bordenave and Djalil Chafaï, as well as joint work with Quentin François.

β-ensembles at high temperature

• Charlie Dworaczek Guera (KTH, Stockholm)

The high temperature regime of β-ensembles refers to the usual probabilistic model describing the joint law of the spectrum of certain random matrices (when the parameter β=1,2,4), but with the parameter β scaled as β~1/N. In this regime, the macroscopic behaviour is no longer described asymptotically by a compactly supported measure but by a full-support measure. Indeed, as the parameter β becomes small, the entropy plays a significant role and allows a macroscopic number of particles to escape to infinity. I will discuss how the CLT for linear statistics, the asymptotics of the partition function and the large deviations at the edge can be obtained based on joint works with Ronan Memin (ENS). I will also explain how it is connected to many classical integrable systems and how it can be seen as a toy model to study the Generalized Gibbs Ensemble of the Calogero fluid.