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Fractional Brownian motion with Hurst index $H=0$ and the Gaussian unitary ensemble
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Fyodorov, Y. V., Khoruzhenko, B. A. and Simm, Nick (2016) Fractional Brownian motion with Hurst index $H=0$ and the Gaussian unitary ensemble. The Annals of Probability, 44 (4). pp. 2980-3031. doi:10.1214/15-AOP1039 ISSN 0091-1798.
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Official URL: http://dx.doi.org/10.1214/15-AOP1039
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 D_N(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, D_N(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: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Polynomials , Random variables, Brownian motion processes | ||||||||
Journal or Publication Title: | The Annals of Probability | ||||||||
Publisher: | Institute of Mathematical Statistics | ||||||||
ISSN: | 0091-1798 | ||||||||
Official Date: | 2 August 2016 | ||||||||
Dates: |
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Volume: | 44 | ||||||||
Number: | 4 | ||||||||
Page Range: | pp. 2980-3031 | ||||||||
DOI: | 10.1214/15-AOP1039 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 8 November 2016 | ||||||||
Date of first compliant Open Access: | 8 November 2016 | ||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||
Grant number: | EP/J002763/1 |
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