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

Fyodorov, Y V, Khoruzhenko, B A and Simm, N J (2016) Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble. Annals of Probability, 44 (4). pp. 2980-3031. ISSN 0091-1798

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Abstract

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 DN(z)=−log|det(H−zI)| on mesoscopic scales as 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) 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
Depositing User: Nicholas Simm
Date Deposited: 17 May 2018 09:07
Last Modified: 02 Jul 2019 15:36
URI: http://sro.sussex.ac.uk/id/eprint/75905

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