Raney Distributions and Random Matrix Theory

Abstract

Abstract. 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 q inverse standard Gaussian matrices, and s standard Gaussian matrices, is shown to satisfy a variant of the algebraic equation relating to the Raney distribution. Supported on (0, ∞), we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization. As an application, the leading asymptotic form of the density as the endpoints of the support are approached is computed, and is shown to have some universal features.

Comparison of eigeninference based on one- and two-point Green's functions

“…The case r = 1 is also singular at z = 0 as visible from (4). Since now we consider only the cases r < 1, when the eigenvalues are strictly positive, so S −1 does exist.…”

“…where the second line comes after explicit differentiation of the first formula and the corresponding Green's functions and their derivatives origin from (4). Similarly, one can define two-point Green's function for the inverse matrix S, generating double dual spectral moments α…”

“…Explicitly, Wishart sample covariance matrix reads S = 1 T XX † , where X ia is valued either in real or complex numbers space, i = 1, ...N spans the size of the sample and a = 1, ...T counts the number of measurements. Since that time, Wishart ensemble (sometimes called also Laguerre ensemble) found broad applications in several branches of physics, ranging from chaotic scattering [2], through conductance in mesoscopic physics [3], quantum information theory [4] to description of universal chiral properties of Quantum Chromodynamics [5]. With the advent of computer era, original ideas of Wishart met new challenges.…”

Fox H-kernel and θ-deformation of the Cauchy Two-Matrix Model and Bures Ensemble

“…The RHS in the case θ = 1 is recognised as the n-th Catalan number; for general θ ∈ Z + one has instead the n-th Fuss-Catalan number with parameter θ + 1. It was proposed in [15], and later verified in [17], that the Raney generalisation of the Fuss-Catalan numbers, as specified by the sequence R p,r (n) := r pn + r pn + r n , (n = 0, 1, 2, . .…”

“…Another is the global scaling limit, corresponding to a scaling of the eigenvalues for which in the N → ∞ limit the eigenvalue density is supported on some finite interval [0, L], L > 0. For the Muttalib-Borodin ensemble with Laguerre weight it is known that after the change of variables x l = y 1/θ l , the global density ρ(y) minimises the energy functional [9,15,17]…”