Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble

Y. V Fyodorov, B. A. Khoruzhenko, N. J. Simm

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Abstract

The goal of this paper is to establish a relation between characteristic polynomials of N×NN×N GUE random matrices HH as N→∞N→∞, and Gaussian processes with logarithmic correlations. We introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of DN(z)=−log|det(H−zI)|DN(z)=−log⁡|det(H−zI)| on mesoscopic scales as N→∞N→∞. By employing a Fourier integral representation, we use this to prove a continuous analogue of a result by Diaconis and Shahshahani [J. Appl. Probab. 31A (1994) 49–62]. On the macroscopic scale, DN(x)DN(x) gives rise to yet another type of Gaussian process with logarithmic correlations. We give an explicit construction of the latter in terms of a Chebyshev–Fourier random series.
Original languageEnglish
Pages (from-to)2980-3031
JournalANNALS OF PROBABILITY
Volume44
Issue number4
Early online date2 Aug 2016
DOIs
Publication statusPublished - Aug 2016

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