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arXiv · 1906.03554

Extreme Eigenvalue Distributions of Jacobi Ensembles: New Exact Representations, Asymptotics and Finite Size Corrections

Laureano Moreno-Pozas·David Morales-Jimenez·Matthew R. McKay
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Abstract

Let $\mathbf{W}_1$ and $\mathbf{W}_2$ be independent $n\times n$ 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 $(\mathbf{W}_1+\mathbf{W}_2)^{-1}\mathbf{W}_1$, which are analogous to those of F matrices ${\bf W}_1 {\bf W}_2^{-1}$ and those of the Jacobi unitary ensemble (JUE). Defining $α_1=m_1-n$ and $α_2=m_2-n$, we derive new exact distribution formulas in terms of $(α_1+α_2)$-dimensional matrix determinants, with elements 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 $n \to \infty$ 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, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlevé differential equations, or hypergeometric functions of matrix arguments.

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Laureano Moreno-Pozas, David Morales-Jimenez, Matthew R. McKay. 2019-06-09. Extreme Eigenvalue Distributions of Jacobi Ensembles: New Exact Representations, Asymptotics and Finite Size Corrections. https://doi.org/10.1016/j.nuclphysb.2019.114724

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