Local law of addition of random matrices on optimal scale

Understand

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 a Haar orthogonal matrix.