Nick Simm

Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble

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.

Moments of the transmission eigenvalues, proper delay times, and random matrix theory. I

We develop a method to compute the moments of the eigenvalue densities of matrices in the Gaussian, Laguerre, and Jacobi ensembles for all the symmetry classes β ∈ {1, 2, 4} and finite matrix dimension n. The moments of the Jacobi ensembles have a physical interpretation as the moments of the transmission eigenvalues of an electron through a quantum dot with chaotic dynamics. For the Laguerre ensemble we also evaluate the finite n negative moments. Physically, they correspond to the moments of the proper delay times, which are the eigenvalues of the Wigner-Smith matrix. Our formulae are well suited to an asymptotic analysis as n → ∞.

On the distribution of the maximum value of the characteristic polynomial of GUE random matrices

Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random N×N matrices H from the Gaussian Unitary Ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of DN(x):=−log|det(xI−H)| as N→∞ and x∈(−1,1). We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher-Hartwig asymptotics due to Krasovsky \cite{K07} with the heuristic {\it freezing transition} scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the Circular Unitary Ensemble, the present GUE case poses new challenges. In particular we show how the conjectured {\it self-duality} in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of DN(x).

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

For an

On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials

{N \times N}

Large-N expansion for the time-delay matrix of ballistic chaotic cavities

N×N Haar distributed random unitary matrix UN, we consider the random field defined by counting the number of eigenvalues of UN in a mesoscopic arc centered at the point u on the unit circle. We prove that after regularizing at a small scale

On the real spectrum of a product of Gaussian matrices

{\epsilon_{N} > 0}