## The Annals of Probability

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

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2980-3031.

Dates
Revised: May 2015
First available in Project Euclid: 2 August 2016

https://projecteuclid.org/euclid.aop/1470139158

Digital Object Identifier
doi:10.1214/15-AOP1039

Mathematical Reviews number (MathSciNet)
MR3531684

Zentralblatt MATH identifier
06631788

#### Citation

Fyodorov, Y. V.; Khoruzhenko, B. A.; Simm, N. J. Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble. Ann. Probab. 44 (2016), no. 4, 2980--3031. doi:10.1214/15-AOP1039. https://projecteuclid.org/euclid.aop/1470139158

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