Raney distributions and random matrix theory

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 $(0,\infty )$, we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterisation. 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.

30 pages

References in corpus (30)

• Biorthogonal ensembles
• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Level Spacing of Random Matrices in an External Source
• Asymptotic expansion of beta matrix models in the one-cut regime
• Statistical properties of random density matrices
• Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit
• Product of Ginibre matrices: Fuss-Catalan and Raney distributions
• Singular value correlation functions for products of Wishart random matrices
• Exact Solution of the Six-Vertex Model on a Random Lattice
• Random Bures mixed states and the distribution of their purity
• A note on biorthogonal ensembles
• Eigenvalue statistics for product complex Wishart matrices
• Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
• Free Bessel laws
• Random graph states, maximal flow and Fuss-Catalan distributions
• Energy correlations for a random matrix model of disordered bosons
• Differential equations for singular values of products of Ginibre random matrices
• Asymptotic corrections to the eigenvalue density of the GUE and LUE
• The law of large numbers for the free multiplicative convolution
• Biorthogonal ensembles with two-particle interactions
• Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry
• The six-vertex model on random lattices
• Orbit measures, random matrix theory and interlaced determinantal processes
• Coherent States from Combinatorial Sequences
• Matrix averages relating to the Ginibre ensembles
• Universal shocks in the Wishart random matrix ensemble - 1
• A note on the limiting mean distribution of singular values for products of two Wishart random matrices
• Asymptotic forms for hard and soft edge general $\beta$ conditional gap probabilities
• Ladder operators and differential equations for multiple orthogonal polynomials
• Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations

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• Topological recursion for irregular spectral curves
• Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition
• Spectral density of generalized Wishart matrices and free multiplicative convolution
• Asymptotic zero distribution of Jacobi-Piñeiro and multiple Laguerre polynomials
• Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap
• On Wright's generalized Bessel kernel
• Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
• Eigenvalue distribution of nonlinear models of random matrices
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• Complete diagrammatics of the single ring theorem
• Free self-decomposability and unimodality of the Fuss-Catalan distributions
• Signal from noise retrieval from one and two-point Green's function - comparison
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions