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arXiv · 1312.0212

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 $N\times N$ 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. 2016-09-02. Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble. https://doi.org/10.1214/15-aop1039

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