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Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble
Version 2 2023-06-12, 08:54
Version 1 2023-06-09, 13:20
journal contribution
posted on 2023-06-12, 08:54 authored by Y V Fyodorov, B A Khoruzhenko, Nicholas SimmNicholas SimmThe 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.
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Publication status
- Published
File Version
- Published version
Journal
Annals of ProbabilityISSN
0091-1798Publisher
Institute of Mathematical StatisticsExternal DOI
Issue
4Volume
44Page range
2980-3031Department affiliated with
- Mathematics Publications
Research groups affiliated with
- Probability and Statistics Research Group Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2018-05-17First Open Access (FOA) Date
2018-05-24First Compliant Deposit (FCD) Date
2018-05-17Usage metrics
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