Fyodorov, Hiary and Keating have conjectured that the maximum of the characteristic polynomial of random matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the size of local maxima of L-function along the critical axis. I will explain the origins of this conjecture and some partial rigorous understanding, for unitary random matrices and the Riemann zeta function, relying on branching structures.

This talk is based on works in collaboration with Arguin, Belius, Radziwill and Soundararajan.