On the asymptotic distribution of the largest eigenvalue of unbalanced complex Jacobi ensembles
Abstract
Let A ∼ ℂ𝒲p(I,m) and B ∼ ℂ𝒲p(I,n) be two independent complex Wishart random matrices. In this work, we investigate the asymptotic distribution of the largest eigenvalue θmax of the random matrix J = (A+B)−1B. The high-dimensional regime is potentially unbalanced in the sense that, as p → ∞, m may grow faster than p, whereas n grows proportionally to p. The asymptotic behavior of θmax is characterized by means of a Riemann-Hilbert (RH) problem for Jacobi polynomials with modified Jacobi weights. We establish the type 2 Tracy--Widom asymptotic distribution of θmax. This provides an alternative, RH-based proof of the corresponding result of Johnstone [Ann. Statist. 36 (2008), 2638-2716] in the balanced regime, and establishes a conjecture of Jiang [Probab. Theory Relat. Fields 144 (2009), 221-246] in the unbalanced regime. In addition, we obtain an explicit correction of order p−2/3 to the Tracy--Widom approximation, with a remainder of order 𝒪(p−1), and show exponential decay, as a function of p, of the upper tail of the distribution of θmax.
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BibTeX
@misc{didier2026asymptotic,
title = {{On the asymptotic distribution of the largest eigenvalue of unbalanced complex Jacobi ensembles}},
author = {Gustavo Didier and Aikaterini Gkogkou and John Lopez Santander and Kenneth D. T-R McLaughlin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00577},
eprint = {2611.00577},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00577},
}
RIS
TY - UNPB
TI - On the asymptotic distribution of the largest eigenvalue of unbalanced complex Jacobi ensembles
AU - Gustavo Didier
AU - Aikaterini Gkogkou
AU - John Lopez Santander
AU - Kenneth D. T-R McLaughlin
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00577
UR - https://arxiv.org/abs/2611.00577
AB - Let A ∼ ℂ𝒲_p(I,m) and B ∼ ℂ𝒲_p(I,n) be two independent complex Wishart random matrices. In this work, we investigate the asymptotic distribution of the largest eigenvalue θ_max of the random matrix J = (A+B)^−1B. The high-dimensional regime is potentially unbalanced in the sense that, as p → ∞, m may grow faster than p, whereas n grows proportionally to p. The asymptotic behavior of θ_max is characterized by means of a Riemann-Hilbert (RH) problem for Jacobi polynomials with modified Jacobi weights. We establish the type 2 Tracy--Widom asymptotic distribution of θ_max. This provides an alternative, RH-based proof of the corresponding result of Johnstone [Ann. Statist. 36 (2008), 2638-2716] in the balanced regime, and establishes a conjecture of Jiang [Probab. Theory Relat. Fields 144 (2009), 221-246] in the unbalanced regime. In addition, we obtain an explicit correction of order p^−2/3 to the Tracy--Widom approximation, with a remainder of order 𝒪(p^−1), and show exponential decay, as a function of p, of the upper tail of the distribution of θ_max.
ER -