On the distribution of maximum value of the characteristic polynomial of GUE random 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 $D_N(x)$.

18 pages, 5 figures. Typos corrected and some additional discussion added

References in corpus (4)

• Toeplitz determinants with merging singularities
• Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble
• Freezing and decorated Poisson point processes
• Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes

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