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We consider random matrices of the form |$H_N=A_N+U_N B_N U^*_N$|⁠ , where |$A_N$| and |$B_N$| are two |$N$| by |$N$| deterministic Hermitian matrices and |$U_N$| is a Haar distributed random unitary matrix. We establish a universal central limit theorem for the linear eigenvalue statistics of |$H_N$| on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics and consists of two main steps: (1) generating Ward identities using the left-translation invariance of the Haar measure, along with a local law for the resolvent of |$H_N$| and analytic subordination properties of the free additive convolution, allows us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations. [ABSTRACT FROM AUTHOR]

We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. [ABSTRACT FROM AUTHOR]

We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law. [ABSTRACT FROM AUTHOR]

We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix. [ABSTRACT FROM AUTHOR]

This paper is a companion to Paper I [W. De Roeck, J. Fröhlich, and K. Schnelli, "Quantum diffusion with drift and the Einstein relation. I," J. Math. Phys.55, 075206 (2014)]. The purpose of this paper is to describe and prove a certain number of technical results used in Paper I, but not proven there. Both papers concern long-time properties (diffusion, drift) of the motion of a driven quantum particle coupled to an array of thermal reservoirs. The main technical results derived in the present paper are: (1) an asymptotic perturbation theory applicable for small driving force, and (2) the construction of time-dependent correlation functions of particle observables. [ABSTRACT FROM AUTHOR]

We consider Hermitian random matrices of the form H = W + λV, where W is a Wigner matrix and V is a diagonal random matrix independent of W. We assume subexponential decay for the matrix entries of W and we choose λ ∼ 1 so that the eigenvalues of W and λV are of the same order in the bulk of the spectrum. In this paper, we prove for a large class of diagonal matrices V that the local deformed semicircle law holds for H, which is an analogous result to the local semicircle law for Wigner matrices. We also prove complete delocalization of eigenvectors and other results about the positions of eigenvalues. [ABSTRACT FROM AUTHOR]