Spiked models and BBP transitions

Take normalized GUE $H_N$ as defined earlier, a deterministic unit vector $v_N$, and a fixed positive parameter theta. Form $A_N=H_N+\theta v_Nv_N^*$. Unitary invariance makes the direction arbitrary; an independent random unit vector gives the same conclusions. For fixed theta, the top eigenvalue converges in probability to two at or below threshold one, and to theta plus its reciprocal above threshold. The corresponding squared overlap with the planted direction has the limits in the display.

Squared overlap is unchanged by the arbitrary complex phase of an eigenvector, so it is the natural alignment observable. Below threshold it vanishes in the limit even though finite samples generally have a small nonzero overlap. Above threshold it remains less than one at every fixed finite spike strength, reflecting residual noise. A rank-one perturbation changes the empirical cumulative spectral distribution by at most one over dimension; its limiting empirical law therefore stays semicircular while one extreme eigenvalue can separate.

Away from eigenvalues of the unperturbed matrix, the matrix determinant lemma factors the perturbed characteristic determinant as shown. An outlier must make the scalar second factor vanish. The isotropic resolvent quadratic form approaches the semicircle transform outside the limiting support. Because this module uses denominator x minus z, that transform is negative on the real axis above two. The sign in the secular equation is therefore a plus before theta times the transform.

The transform approaches minus one as the argument decreases to the upper edge from outside. A root strictly above that edge exists exactly when theta exceeds one. Substituting the semicircle quadratic into the root equation then gives the outlier location. This calculation motivates the formula, but making it a proof requires control of the extreme noise spectrum and convergence of the directional resolvent. Differentiating the resolvent in the eigenvector normalization gives the overlap formula; merely solving a scalar location equation does not by itself establish alignment.

In a sample-covariance model, the population matrix might instead be $\Sigma=I+\theta vv^*$, with feature-over-sample ratio tending to gamma. Here theta is the excess above the background population eigenvalue one. For Gaussian observations, a positive excess must exceed the square root of gamma to separate at the upper edge. Thus the threshold for the full population eigenvalue is one plus that square root. This is the covariance setting associated with the original Baik–Ben Arous–Péché transition, not the additive model’s threshold one.

The critical parameter region also changes fluctuation laws, so an edge limit cannot be carried unchanged across a spike transition. The location and overlap limits stated here do not specify those finer critical laws or finite-dimensional error bars. For applications, first identify whether noise is added to a Hermitian matrix or enters through observations multiplied by a population covariance. Then state the dimension ratio, normalization, and whether the parameter denotes a full population eigenvalue or an excess. Those choices determine what a reported signal threshold actually means.