2013

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

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

Understand

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.

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