Local spectral statistics of the addition of random matrices

Abstract

We consider the local statistics of $$H = V^* X V + U^* Y U$$ where V and U are independent Haar-distributed unitary matrices, and X and Y are deterministic real diagonal matrices. In the bulk, we prove that the gap statistics and correlation functions coincide with the GUE in the limit when the matrix size $$N \rightarrow \infty $$ under mild assumptions on X and certain rigidity assumptions on Y (the latter being an assumption on the convergence of the eigenvalues of Y to the quantiles of its limiting spectral measure which we assume to have a density). Our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to Dyson Brownian motion. Our method also applies to the case when V and U are drawn from the orthogonal group. Our proof relies on the local law for H proved in Bao et al. (Commun Math Phys 349(3):947–990, 2017; J Funct Anal 271(3):672–719, 2016; Adv Math 319:251–291, 2017) as well as the DBM convergence results of Landon and Yau (Commun Math Phys 355(3):949–1000, 2017) and Landon et al. (Adv Math 346:1137–1332, 2019).