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Raney Distributions and Random Matrix Theory

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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 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.

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Acknowledgments

The work of P.J. Forrester was supported by the Australian Research Council, for the project ‘Characteristic polynomials in random matrix theory’. The work of D.-Z. Liu was supported by the National Natural Science Foundation of China under Grants 11301499 and 11171005, and by CUSF WK 0010000026. He would also like to express his sincere thanks to the Department of Mathematics and Statistics, The University of Melbourne for its hospitality during his stay. We thank Arno Kuijlaars and Thorsten Neuschel for alerting us to the work of Haagerup and Möller, and Lun Zhang for helpful comments on the first draft.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Peter J. Forrester

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Dang-Zheng Liu

  3. Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, 230026, People’s Republic of China

    Dang-Zheng Liu

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  1. Peter J. Forrester
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Forrester, P.J., Liu, DZ. Raney Distributions and Random Matrix Theory. J Stat Phys 158, 1051–1082 (2015). https://doi.org/10.1007/s10955-014-1150-4

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