The logarithm of the characteristic polynomial of random matrices shares many characteristics, and some differences, with Gaussian logarithmically correlated fields. This statement, interpreted broadly, is valid for a large class of random matrices. I will review known results and eventually focus on one case, that of $\mathrm{C}\beta\mathrm{E}$ matrices, which leads to a partial resolution of the Keating, Hiary and Fyodorov conjecture.

Based on joint works with Paquette, with Bourgade and Lopato, and with Lambert and Leblé.