Extremal eigenvalues and eigenvectors of deformed Wigner matrices

Abstract

We consider random matrices of the form H = W + λV , λ ∈ R + , where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d. entries that are independent of W . We assume subexponential decay of the distribution of the matrix entries of W and we choose λ ∼ 1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+ ∈ R + such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ > λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N → ∞ if λ > λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ > λ+, while they are completely delocalized for λ < λ+. Similar results hold for the lowest eigenvalues.

Bulk universality for deformed Wigner matrices

“…In the Appendix, we collect several technical results on the deformed semicircle law and its Stieltjes transform. Some of these results have previously appeared in [52] and [39,40].…”

Edge universality for deformed Wigner matrices

“…In this setting the deformed semicircle law is still supported on a single interval, but the square root behavior at the edge may fail. We refer to [24,25] for a detailed discussion.…”

“…Assuming that (v i ) are i.i.d. random variables with law given by a Jacobi measure as in (2.7) this has been studied in [25]. For example, when b > 1 then there exists 0 < λ + < ∞ such that if λ 0 < λ + then the Assumption 2.4 holds and the law of the rescaled largest eigenvalues converges to the Tracy-Widom distribution.…”

“…When λ 0 > λ + , the Assumption 2.4 is not satisfied, the deformed semicircle law ρ f c does not have a square root behavior at the upper edge and the law of the rescaled largest eigenvalue converges to a Weibull distribution. Correspondingly, the eigenvectors associated to the largest eigenvalues are completely delocalized for λ 0 < λ + (see [24]), while they are (partially) localized for λ 0 > λ + (see [25]).…”

“…To complete the proof of the desired theorem, we notice that it was proved in Lemma C.1 of [25] that there exists a random variable X ≡ X(N ), which converges to the Gaussian random variable with mean 0 and variance…”