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On the distribution of the maximum value of the characteristic polynomial of GUE random matrices
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Fyodorov, Y. V. and Simm, Nick (2016) On the distribution of the maximum value of the characteristic polynomial of GUE random matrices. Nonlinearity, 29 (9). pp. 2837-2855. doi:10.1088/0951-7715/29/9/2837 ISSN 0951-7715.
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Official URL: http://dx.doi.org/10.1088/0951-7715/29/9/2837
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 [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 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).
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Gaussian processes, Extreme value theory, Asymptotic distribution (Probability theory), Polynomials | ||||||||
Journal or Publication Title: | Nonlinearity | ||||||||
Publisher: | Institute of Physics Publishing Ltd. | ||||||||
ISSN: | 0951-7715 | ||||||||
Official Date: | 10 August 2016 | ||||||||
Dates: |
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Volume: | 29 | ||||||||
Number: | 9 | ||||||||
Page Range: | pp. 2837-2855 | ||||||||
DOI: | 10.1088/0951-7715/29/9/2837 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Date of first compliant deposit: | 3 October 2016 | ||||||||
Date of first compliant Open Access: | 4 October 2016 | ||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||
Grant number: | EP/J002763/1 |
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