RT Journal Article
T1 Extreme eigenvalue distributions of Jacobi ensembles: new exact representations, asymptotics and finite size corrections
A1 Moreno-Pozas, Laureano
A1 Morales-Jiménez, David
A1 Mckay, Matthew R.
K1 Ingeniería eléctrica
K1 Electrónica
K1 Variables aleatorias
K1 Autovalores
AB Let W1 and W2 be independent complex central Wishart matrices with m1 and m2 degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices (W1+W2)^-1 W1, which are analogous to those of F matrices W1 W2^-1 and those of the Jacobi unitary ensemble (JUE). Defining alpha1=m1-n and alpha2=m2-n with m1,m2>n,  we derive new exact distribution formulas in terms of (alpha1+alpha2)-dimensional matrix determinants, with entries involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-n analysis with alpha1 and alpha2 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 -> infinity in terms of alpha1- and alpha2-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.
PB Elsevier
YR 2019
FD 2019-10
LK https://hdl.handle.net/10630/34151
UL https://hdl.handle.net/10630/34151
LA eng
NO Laureano Moreno-Pozas, David Morales-Jimenez, Matthew R. McKay, Extreme eigenvalue distributions of Jacobi ensembles: New exact representations, asymptotics and finite size corrections, Nuclear Physics B, Volume 947, 2019.
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 20 ene 2026