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

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posted on 2023-06-09, 13:43 authored by Y V Fyodorov, Nicholas SimmNicholas Simm
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 DN(x):=-log|det(xI-H)| as N?8 and x?(-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 DN(x).

History

Publication status

  • Published

File Version

  • Published version

Journal

Nonlinearity

ISSN

0951-7715

Publisher

Institute of Physics

Volume

29

Page range

2837-2855

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Probability and Statistics Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-06-13

First Open Access (FOA) Date

2018-06-13

First Compliant Deposit (FCD) Date

2018-06-13

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