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

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

Published 2013-12-01, updated 2015-06-19Version 2

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. First, 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 $D_{N}(z):=-\log|\det(zI-\mathcal{H})|$ on mesoscopic scales as $N \to \infty$. By employing a Fourier integral representation, we show how this implies a continuous analogue of a result by Diaconis and Shahshahani \cite{DS94}. 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.