Spectral density of generalized Wishart matrices and free multiplicative convolution.

We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, W=XX(†), where X stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, MP(⊠s), which for an integer s yield Fuss-Catalan distributions corresponding to a product of s-independent square random matrices, X=X(1)⋯X(s). New formulas for the level densities are derived for s=3 and s=1/3.

Operator solutions for fractional Fokker-Planck equations.

We obtain exact results for fractional equations of Fokker-Planck type using the evolution operator method. We employ exact forms of one-sided Lévy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators.

Lévy stable two-sided distributions: exact and explicit densities for asymmetric case.

We investigate functions g(α,β;x) which are heavy-tailed Lévy stable probability distributions of index 0<α≤2 and appropriate asymmetry parameter β. They are of central importance in physics of amorphous and disordered systems, econophysics, geology, hydrology, internet traffic, dynamics of human relations, etc. We present an ensemble of exact and explicit solutions of g(α,β;x) for all admissible rational values of α and β.

Product of Ginibre matrices: Fuss-Catalan and Raney distributions.

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P(s)(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P(s)(x) in terms of a combination of s hypergeometric functions of the type (s)F(s-1). The explicit formula derived here is exact for an arbitrary positive integer s, and for s=1 it reduces to the Marchenko-Pastur distribution.

Exact and explicit probability densities for one-sided Lévy stable distributions.

We study functions gα(x) which are one-sided, heavy-tailed Lévy stable probability distributions of index α, 0<α<1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expressions for gα(x), 0 ≤ x<∞, for all α=l/k<1, with k and l positive integers. We reproduce all the known results given by k ≤ 4 and present many new exact solutions for k > 4, all expressed in terms of known functions.