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

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journal contribution
posted on 2023-06-12, 08:54 authored by Y V Fyodorov, B A Khoruzhenko, Nicholas SimmNicholas Simm
The goal of this paper is to establish a relation between characteristic polynomials of N×N GUE random matrices H as N?8, 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?8. 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.

History

Publication status

  • Published

File Version

  • Published version

Journal

Annals of Probability

ISSN

0091-1798

Publisher

Institute of Mathematical Statistics

Issue

4

Volume

44

Page range

2980-3031

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Probability and Statistics Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-05-17

First Open Access (FOA) Date

2018-05-24

First Compliant Deposit (FCD) Date

2018-05-17

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