Victor Schön: Gaussian fluctuations of single eigenvalues in time-dependent GUE

A random matrix is a matrix whose elements are random variables. The field of applications of random matrices is vast with examples being nuclear physics, number theory and numerical analysis. If the joint distribution of the elements is specified in a certain way a matrix ensemble known as the Gaussian Unitary Ensemble is obtained, which is a random Hermitian matrix. Furthermore, by defining a Wiener process on the space of Hermitian matrices, a stochastic process on the spectrum is induced, known as Dyson Brownian motion (DBM).

With the aim of further characterising DBM, this thesis investigates the distribution of the fluctuations of single eigenvalues of the spectrum using statistical hypothesis tests such as Kolmogorov-Smirnov and Mardia’s test for multivariate normality. The results seem to indicate a Gaussian distribution of the fluctuations.