Limits of spiked random matrices I

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.Comment: 34 pages; minor corrections, references update

Spiking the random matrix hard edge

“…Rumanov [22] has already used (1.12) to find formulas for F at β = 2, 4 in terms of Painlevé III. That same paper provides new proofs of the β = 2, 4 spiked soft edge laws using the analog of the PDE (1.12) derived in [3]. It is further worth mentioning that more recently Rumanov [23] has found a Painlevé II formula at the β = 6 soft edge (the first known outside β = 1, 2, 4) once more using the spiked/PDE picture of [3].…”

“…In [3,4] the outcome of the Riccati map is that the one-spiked soft edge distributions are described in terms of the probability that the process dp 16) begun at w, has a given number of explosions to −∞ (after subsequent restarts at +∞).…”

“…3 This comes down to a quantitative version of the law of the iterated logarithm for the random sum nx≤k≤ny log χ β(n−k) χ β(n+a−k) . And then by the law of the iterated logarithm for Brownian motion, we also have that the limit kernel k satisfies the same bound, but with C n replaced by a fixed (i.e., independent of n) random constant C < ∞ Now we can choose a subsequence n ′ → ∞ over which C n ′ converges, and then realize this convergence as almost sure convergence on some probability space.…”

“…In that case the noise term reduces to 2 √ β b ′ x for x → b x a standard Brownian motion, and H is recognized as the Stochastic Airy Operator from [19], but with a simple Robin f ′ (0) = wf (0) rather than Dirichlet boundary condition at the origin. Hence, for r = 1 the single parameter c interpolates between the general beta Tracy-Widom law at w = +∞ (which forces f (0) = 0) and the Gaussian law in the supercritical regime, w = −∞ (which requires a little perturbation argument to make precise, see [3]). …”

“…We also show that, under a natural limit, all spiked hard edge laws derived here degenerate to the critically spiked soft edge laws (or deformed Tracy-Widom laws). The latter were first described at β = 2 by Baik, Ben Arous, and Peché [2], and from a unified β random operator point of view by Bloemendal and Virág [3,4]. …”