Random matrices appear in various theoretical and practical fields, from quantum physics to applied finance. The spectrum of random matrices plays an important role in such areas, for instance by quantifying random energy levels or estimating covariance matrices. In the regime of a large number of entries, such a spectrum very often shows a universal limit behaviour, which depends on the ensemble at hand.
Another interesting problem is that of fluctuations of such a spectrum. This is an issue for instance in disordered systems, where one is interested in the spectrum conditioned on having a certain amount of positive eigenvalues (stable directions). Studying numerically this issue is made difficult because the conditioning event is generally very rare, so a naive rejection method drawing matrices is inefficient. In this talk, I will present briefly a numerical and theoretical study of such conditioned Coulomb gases in view of [Chafai, Ferré, Stoltz, 2019].