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

• Y.V. Fyodorov ^[1] ; B.A. Khoruzhenko ^[1] ; N.J. Simm ^[1]
  1. [1] Queen Mary University of London

     Queen Mary University of London

     Reino Unido

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 4, 2016, págs. 2980-3031
• Idioma: inglés
• DOI: 10.1214/15-AOP1039
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• Resumen

  • The goal of this paper is to establish a relation between characteristic polynomials of N×NN×N GUE random matrices HH as N→∞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 DN(z)=−log|det(H−zI)|DN(z)=−log⁡|det(H−zI)| on mesoscopic scales as N→∞N→∞. 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)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.