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 → ∞, 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 DN (z): = − log |det(zI−H) | on mesoscopic scales as N →∞. By employing a Fourier integral representation, we show how this implies a continuous analogue of a result by Diaconis and Shahshahani [18]. On the macroscopic scale, DN (x) gives rise to yet another type of Gaussian process with logarithmic correlations. We giv