Local Law of Addition of Random Matrices on Optimal Scale

Abstract

Abstract:The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.

Local stability of the free additive convolution

“…Finally, we remark that the local stability result is also a key ingredient in [3], where we were able to prove a local law down to the smallest possible scale η ≫ N −1 , but with a weaker error bound than in Theorem 2.8; see Remark 2.4 for details. Acknowledgment.…”

“…Comparing with (2.28), we see that we can choose η in (2.31) almost as small as N −1 at the price of losing a factor √ N . The stability and perturbation analysis in [3] rely on the optimal results in Theorem 2.5 and Theorem 2.7 as well as in Sections 3-5 of the present paper.…”

“…In [3], we derive, with a different approach, the estimate (with the notation of (2.28)) 31) for N ≥ N 0 , with some N 0 sufficiently large, and with η m = N −1+γ . In fact, we obtain estimates for individual matrix elements of the resolvent G H as well.…”

“…In Section 3, we consider the stability of the system (2.5) when at least one of the measures µ 1 and µ 2 is supported at more than two points and we give the proof of Theorem 2.5. In Section 4, we consider perturbations of the system (2.5) and derive results that will be used in the proof of Theorem 2.8 and also in [3]. In Section 5, we prove Theorem 2.7.…”

Random Band Matrices

“…(iv) The convolution model D 1 + U * D 2 U , where D 1 , D 2 are diagonal and U is uniform on O(N ), appears in free probability theory. Its empirical spectral measure in understood up to the optimal scale [10], and GOE bulk statistics were proved in [34].…”