The isotropic semicircle law and deformation of Wigner matrices (Q2856961)

A well-known problem in random matrix theory asks how the local spectral properties of Wigner matrices are affected when a finite-rank deterministic Hermitian matrix is added. To discuss but the rank-one case, it is known from work of \textit{S. Péché} [Probab. Theory Relat. Fields 134, No. 1, 127--173 (2006; Zbl 1088.15025)] that for a GUE matrix \(H\) with off-diagonal variance \(1/N\), a unit vector \(v\) and \(d > 1\), the spectrum of \(H + d v v^{\ast}\) consists of a bulk part that is asymptotically contained in \([-2, 2]\), and an outlier at \(d + d^{-1}\) with Gaussian fluctuations, whereas for \(d < 1\) there is no such outlier.NEWLINENEWLINEThe work under review studies the phase transition as \(d\) increases from \(1 - \epsilon\) to \(1 + \epsilon\) on the scale \(d = 1 + w N^{-1/3}\) for \(H\) a Wigner matrix. Specifically, \(k\)-rank deterministic perturbations \(A\) are considered, where the eigenvalues \(d_1, \dots, d_k\) of \(A\) may depend on \(N\), satisfying \(||d_i| - 1| \geq (\log N)^{C \log\log N} N^{-1/3}\), and the eigenvectors of \(A\) constitute an arbitrary orthogonal matrix. Under these assumptions, it is proven that the extremal bulk eigenvalues of the perturbed Wigner matrix are universal, and the distribution of the outliers of the perturbed matrix is identified.NEWLINENEWLINEThe proof rests on a generalization of the local semicircle law, i.e.\ of the tool used by Erdős, Schlein and Yau to prove the universality of Wigner matrices, as well as the more recent Green function comparison method due to Erdős, Yau and Yin.