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

TL;DR

This paper studies the extreme values of the logarithm of characteristic polynomials in the CβE ensemble, showing their convergence to a Gumbel distribution plus a derivative martingale term, and describes the landscape near these extrema.

Contribution

It establishes the distributional convergence of the maxima and characterizes the local landscape around extrema for the CβE ensemble.

Findings

Maxima converge to a Gumbel plus derivative martingale

Characterization of the landscape near extrema

Provides a detailed probabilistic description of the extremal landscape

Abstract

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.