Limits of spiked random matrices I

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 β analogue, proving a conjecture of Baik et al. (Ann Probab 33:1643–1697, 2005). We also treat shifted mean Gaussian orthogonal and β 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 Schrödinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which β appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at β = 2,4, yielding in particular a new and simple proof of the Painlevé representations for these Tracy–Widom distributions.