### Abstract

Let W_{1} and W_{2} be independent n×n complex central Wishart matrices with m_{1} and m_{2} degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices (W_{1}+W_{2})^{−1}W_{1}, which are analogous to those of F matrices W_{1}W_{2} ^{−1} and those of the Jacobi unitary ensemble (JUE). Defining α_{1}=m_{1}−n and α_{2}=m_{2}−n with m_{1}, m_{2}≥n, we derive new exact distribution formulas in terms of (α_{1}+α_{2})-dimensional matrix determinants, with entries involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-n analysis with α_{1} and α_{2} fixed (i.e., under the so-called “hard-edge” scaling limit). The analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as n→∞ in terms of α_{1}- and α_{2}-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-n corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations and properties of Legendre polynomials, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlevé differential equations, or hypergeometric functions of matrix arguments.

Original language | English |
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Article number | 114724 |

Number of pages | 41 |

Journal | Nuclear Physics B |

Volume | 947 |

Early online date | 14 Aug 2019 |

DOIs | |

Publication status | Published - 01 Oct 2019 |

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

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## Cite this

*Nuclear Physics B*,

*947*, [114724]. https://doi.org/10.1016/j.nuclphysb.2019.114724