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.

Published at http://dx.doi.org/10.1214/15-AOP1039 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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