Equilibrium problems for Raney densities

TL;DR

This paper establishes equilibrium problems for Raney densities, a class of probability densities related to combinatorial numbers and random matrix theory, using two analytical techniques for general parameters.

Contribution

It proves the validity of equilibrium problems for Raney densities for a broad range of parameters and identifies these problems using Wiener--Hopf and algebraic methods.

Findings

Validated equilibrium problems for Raney densities for general  > 0.

Identified equilibrium problems for specific parameter families.

Extended methods to cover additional parameter cases.

Abstract

The Raney numbers are a class of combinatorial numbers generalising the Fuss--Catalan numbers. They are indexed by a pair of positive real numbers $(p,r)$ with $p>1$ and $0 < r \le p$, and form the moments of a probability density function. For certain $(p,r)$ the latter has the interpretation as the density of squared singular values for certain random matrix ensembles, and in this context equilibrium problems characterising the Raney densities for $(p,r) = (\theta +1,1)$ and $(\theta/2+1,1/2)$ have recently been proposed. Using two different techniques --- one based on the Wiener--Hopf method for the solution of integral equations and the other on an analysis of the algebraic equation satisfied by the Green's function --- we establish the validity of the equilibrium problems for general $\theta > 0$ and similarly use both methods to identify the equilibrium problem for $(p,r) = (\theta/q+1,1/q)$, $\theta > 0$ and $q \in \mathbb Z^+$. The Wiener--Hopf method is used to extend the latter to parameters $(p,r) = (\theta/q + 1, m+ 1/q)$ for $m$ a non-negative integer, and also to identify the equilibrium problem for a family of densities with moments given by certain binomial coefficients.