The paper treats theWigner distribution of scalar-valued stochastic processes defined on R. We show that if the process is Gaussian and weakly harmonizable then a stochastic Wigner distribution is well defined. The special case of stationary processes is studied, in which case the Wigner distribution is weakly stationary in the time variable and the variance is equal to the deterministic Wigner...

Random matrix theory (RMT) has wide applications in various areas of science. It can be traced back to sample covariance matrices studied by J. Wishart in data analysis in 1920s1930s. In 1951, E. Wigner associated the energy levels of heavy-nuclei atoms with Hermitian matrices with i.i.d entries. To simplify the complex Hamiltonians of heavy-nuclei atoms, Wigner introduced the ensemble of real ...

We study Wigner function value statistics of classically chaotic quantum maps on compact 2D phase space. We show that the Wigner function statistics of a random state is a Gaussian, with the mean value becoming negligible compared to the width in the semiclassical limit. Using numerical example of quantized sawtooth map we demonstrate that the relaxation of time-dependent Wigner function statis...

The on-axis image irradiance obtained from a radially symmetric optical system under broadband illumination is analysed in the phase-space domain of the Wigner distribution function associated with a modified one-dimensional pupil aperture. To this end, a coordinate transformation is employed in such a way that the varying on-axis irradiance can be derived from different slices of a two-dimensi...

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and re...