The extremal landscape for the C${\hat{I}}^2$E ensemble

Abstract

We consider the extremes of the logarithm of the characteristic polynomial of matrices from the C${\hat{I}}^2$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.

Appendix B is an adaptation of estimates from arXiv:1607.00243 and uses their source file as template. V3 incorporates numerous comments of the referee and makes numerous corrections throughout the paper. To appear in Forum of Mathematics, Sigma