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

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    Submitted version (468.0Kb)
    Volume
    29
    Pagination
    2837 - 2855
    Publisher URL
    http://dx.doi.org/10.1088/0951-7715/29/9/2837
    DOI
    10.1088/0951-7715/29/9/2837
    Journal
    Nonlinearity v.
    Metadata
    Show full item record
    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 $D_{N}(x)$.
    Authors
    Fyodorov, YV; Simm, NJ
    URI
    http://qmro.qmul.ac.uk/xmlui/handle/123456789/10673
    Collections
    • Applied Mathematics [140]
    Licence information
    http://arxiv.org/abs/1503.07110
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