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 prin
ciple for quantum systems whose components are modelled by Wigner matrices.
We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptoti
cally given by the free convolution of the laws of $A$ and $B$ as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescus theorem. Our previous works [3,4] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.
Let $A$ and $B$ be two $N$ by $N$ deterministic Hermitian matrices and let $U$ be an $N$ by $N$ Haar distributed unitary matrix. It is well known that the spectral distribution of the sum $H=A+UBU^*$ converges weakly to the free additive convolution
of the spectral distributions of $A$ and $B$, as $N$ tends to infinity. We establish the optimal convergence rate ${frac{1}{N}}$ in the bulk of the spectrum.
We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $Wsim N$. All previous results concerning universality of no
n-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.
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 convergen
ce 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.
We consider $Ntimes N$ Hermitian random matrices $H$ consisting of blocks of size $Mgeq N^{6/7}$. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to blo
ck to form a block-band structure, with an essential band width $M$. We show that the entries of the Greens function $G(z)=(H-z)^{-1}$ satisfy the local semicircle law with spectral parameter $z=E+mathbf{i}eta$ down to the real axis for any $eta gg N^{-1}$, using a combination of the supersymmetry method inspired by cite{Sh2014} and the Greens function comparison strategy. Previous estimates were valid only for $etagg M^{-1}$. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.
We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy
parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.
We prove the edge universality of the beta ensembles for any $betage 1$, provided that the limiting spectrum is supported on a single interval, and the external potential is $mathscr{C}^4$ and regular. We also prove that the edge universality holds f
or generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class $mathscr{C}^4$.
