Fractional Brownian motion with Hurst index \({H = 0}\) and the Gaussian unitary ensemble. (English) Zbl 1393.60037
Summary: The goal of this paper is to establish a relation between characteristic polynomials of \(N\times N\) GUE random matrices \(\mathcal{H}\) as \(N\rightarrow\infty\), 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 \(D_{N}(z)=-\log|\det(\mathcal{H}-zI)|\) on mesoscopic scales as \(N\rightarrow\infty\). By employing a Fourier integral representation, we use this to prove a continuous analogue of a result by P. Diaconis and M. Shahshahani [J. Appl. Probab. 31A, 49–62 (1994; Zbl 0807.15015)]. 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.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60B20 | Random matrices (probabilistic aspects) |
| 60F17 | Functional limit theorems; invariance principles |
| 60F05 | Central limit and other weak theorems |
| 60G15 | Gaussian processes |
| 60G25 | Prediction theory (aspects of stochastic processes) |
Keywords:
random matrix theory; mesoscopic regime; logarithmically correlated; fractional Brownian motion; generalized processesCitations:
Zbl 0807.15015Software:
DLMFReferences:
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