We consider the extremes of the logarithm of the characteristic polynomial of matrices from the CβE ensemble. We prove convergence in distribution of the centered maxima (of the real and imaginary parts) towards the sum of a Gumbel variable and another independent variable, which we characterize as the total mass of a “derivative martingale”. We also provide a description of the landscape near extrema points.
(Joint work with Elliot Paquette)