Fyodorov_2016_Nonlinearity_29_2837.pdf (1.45 MB)
On the distribution of maximum value of the characteristic polynomial of GUE random matrices
journal contribution
posted on 2023-06-09, 13:43 authored by Y V Fyodorov, Nicholas SimmNicholas SimmMotivated 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
NonlinearityISSN
0951-7715Publisher
Institute of PhysicsExternal DOI
Volume
29Page range
2837-2855Department affiliated with
- Mathematics Publications
Research groups affiliated with
- Probability and Statistics Research Group Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2018-06-13First Open Access (FOA) Date
2018-06-13First Compliant Deposit (FCD) Date
2018-06-13Usage metrics
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