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

The goal of this paper is to establish a relation between characteristic polynomials of N×N GUE random matrices H as N?8, 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 DN(z)=-log|det(H-zI)| on mesoscopic scales as N?8. 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, DN(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.