Dang-Zheng Liu

Raney Distributions and Random Matrix Theory

Recent works have shown that the family of probability distributions with moments given by the Fuss–Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition has its moments given by a generalization of the Fuss–Catalan numbers. Such computations begin with an algebraic equation satisfied by the Stieltjes transform, which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the Raney distribution. For the Fuss–Catalan distribution, an equilibrium problem characterizing the density is identified. The Stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from

On explicit probability densities associated with Fuss-Catalan numbers

q

Asymptotics for Products of Characteristic Polynomials in Classical β -Ensembles

q inverse standard Gaussian matrices, and

Scaling Limits of Correlations of Characteristic Polynomials for the Gaussian β-Ensemble with External Source

s

Selberg integrals, super-hypergeometric functions and applications to β-ensembles of random matrices

s standard Gaussian matrices, is shown to satisfy a variant of the algebraic equation relating to the Raney distribution. Supported on