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July 2016 Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble
Y. V. Fyodorov, B. A. Khoruzhenko, N. J. Simm
Ann. Probab. 44(4): 2980-3031 (July 2016). DOI: 10.1214/15-AOP1039


The goal of this paper is to establish a relation between characteristic polynomials of $N\times N$ GUE random matrices $\mathcal{H}$ as $N\to\infty$, 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 $D_{N}(z)=-\log|\det(\mathcal{H}-zI)|$ on mesoscopic scales as $N\to\infty$. 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, $D_{N}(x)$ 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.


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Y. V. Fyodorov. B. A. Khoruzhenko. N. J. Simm. "Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble." Ann. Probab. 44 (4) 2980 - 3031, July 2016.


Received: 1 December 2013; Revised: 1 May 2015; Published: July 2016
First available in Project Euclid: 2 August 2016

zbMATH: 06631788
MathSciNet: MR3531684
Digital Object Identifier: 10.1214/15-AOP1039

Primary: 60B20
Secondary: 15B52, 60F05, 60F17

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 4 • July 2016
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