FRACTIONAL BROWNIAN MOTION WITH HURST INDEX H=0 AND THE GAUSSIAN UNITARY ENSEMBLE

@article{Fyodorov2013FRACTIONALBM,
  title={FRACTIONAL BROWNIAN MOTION WITH HURST INDEX H=0 AND THE GAUSSIAN UNITARY ENSEMBLE},
  author={Yan V. Fyodorov and Boris A. Khoruzhenko and Nicholas J. Simm},
  journal={Annals of Probability},
  year={2013},
  volume={44},
  pages={2980-3031}
}
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. 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(H−zI)| on mesoscopic scales as N→∞. By employing a Fourier integral… 

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