Limits of spiked random matrices I

TL;DR

This paper investigates the asymptotic behavior of the largest eigenvalues in high-dimensional spiked random matrices, proving new results in the real case and extending understanding of phase transitions and limit laws.

Contribution

It establishes the asymptotic distributions of top eigenvalues near phase transitions in real Wishart and beta ensembles, confirming a conjecture and providing new characterizations.

Findings

Asymptotic distributions near phase transition are characterized for real Wishart matrices.

New proofs of Painlevé representations for Tracy-Widom distributions at beta=2,4.

Extended results to beta ensembles with boundary condition dependence.

Abstract

Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank-one spiked real Wishart setting and its general beta analogue, proving a conjecture of Baik, Ben Arous and P\'ech\'e (2005). We also treat shifted mean Gaussian orthogonal and beta ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schr\"odinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which beta appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at beta=2,4, yielding in particular a new and simple proof of the Painlev\'e representations for these Tracy-Widom distributions.