FRACTIONAL BROWNIAN MOTION WITH HURST INDEX H=0 AND THE GAUSSIAN UNITARY ENSEMBLE

• 成果类型: Article
• 署名作者: Fyodorov, Y. V.; Khoruzhenko, B. A.; Simm, N. J.
• 署名单位: University of London; Queen Mary University London
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1039
• 发表日期: 2016
• 页码: 2980-3031
• 关键词: altshuler-shklovskii formulas; linear eigenvalue statistics; central-limit-theorem; Orthogonal polynomials; strong asymptotics; RANDOM MATRICES; fluctuations; UNIVERSALITY; models

摘要

The goal of this paper is to establish a relation between characteristic polynomials of N x N GUE random matrices H as N ->infinity, 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 vertical bar det(H - zI) on mesoscopic scales as N -> infinity. 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.