The extremal landscape for the C$\beta$E ensemble

We consider the extremes of the logarithm of the characteristic polynomial of matrices from the C$\beta$E 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.