On the distribution of the 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× 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 (xIH) as N\to ∞ 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 FisherHartwig asymptotics due to Krasovsky [34] with the heuristic 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 selfduality 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).
 Publication:

Nonlinearity
 Pub Date:
 September 2016
 DOI:
 10.1088/09517715/29/9/2837
 arXiv:
 arXiv:1503.07110
 Bibcode:
 2016Nonli..29.2837F
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Probability
 EPrint:
 18 pages, 5 figures. Typos corrected and some additional discussion added