Research output: Contribution to journal › Article

Y. V Fyodorov, B. A. Khoruzhenko, N. J. Simm

Original language | English |
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Pages (from-to) | 2980-3031 |

Journal | ANNALS OF PROBABILITY |

Volume | 44 |

Issue number | 4 |

Early online date | 2 Aug 2016 |

DOIs | |

Publication status | Published - 2016 |

**Fractional Brownian motion with_FYODORV_Publishedonline2August2016_GREEN AAM**AnnalsProb2016.pdf, 585 KB, application/pdf

13/07/2017

Accepted author manuscript

The goal of this paper is to establish a relation between characteristic polynomials of N×NN×N GUE random matrices HH as N→∞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 DN(z)=−log|det(H−zI)|DN(z)=−log|det(H−zI)| on mesoscopic scales as N→∞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, DN(x)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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