On the distribution of maximum value of the characteristic polynomial of GUE random matrices

TL;DR

This paper characterizes the asymptotic distribution of the maximum of a specific logarithmic function related to GUE matrices, combining rigorous asymptotics with heuristic freezing transition ideas, and confirms findings with numerical simulations.

Contribution

It provides an explicit asymptotic distribution for the maximum of the characteristic polynomial of GUE matrices, extending previous methods to a new challenging case.

Findings

Derived explicit asymptotic probability density for the maximum

Confirmed theoretical predictions with numerical simulations

Extended freezing transition analysis to GUE matrices

Abstract

Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random $N \times N$ matrices $H$ from the Gaussian Unitary Ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of $D_{N}(x):=-\log|\det(xI-H)|$ as $N \to \infty$ and $x\in (-1,1)$. We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher-Hartwig asymptotics due to Krasovsky \cite{K07} with the heuristic {\it freezing transition} scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the Circular Unitary Ensemble, the present GUE case poses new challenges. In particular we show how the conjectured {\it self-duality} in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of $D_{N}(x)$.