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arXiv · 1411.4091

Equilibrium problems for Raney densities

Peter J. Forrester·Dang-Zheng Liu·Paul Zinn-Justin
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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.

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Peter J. Forrester, Dang-Zheng Liu, Paul Zinn-Justin. 2014-11-15. Equilibrium problems for Raney densities. https://arxiv.org/abs/1411.4091

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