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Spectral statistics of Erdős–Rényi graphs I: Local semicircle law
We consider the ensemble of adjacency matrices of Erdős–Renyi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p≡p(N). We rescale the matrixExpand
Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. We study theExpand
Local law for random Gram matrices
We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries withExpand
Phase Transition in the Density of States of Quantum Spin Glasses
We prove that the empirical density of states of quantum spin glasses on arbitrary graphs converges to a normal distribution as long as the maximal degree is negligible compared with the total numberExpand
Random Matrices with Slow Correlation Decay.
We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent awayExpand
Correlated Random Matrices: Band Rigidity and Edge Universality
We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of theExpand
Fluctuations of Functions of Wigner Matrices
We show that matrix elements of functions of N ×N Wigner matrices fluctuate on a scale of order N−1/2 and we identify the limiting fluctuation. Our result holds for any function f of the matrix thatExpand
Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of theExpand
Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues
We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix H and its minor Ĥ and find that the fluctuation is much smaller than the fluctuations of the individualExpand
Fluctuation around the circular law for random matrices with real entries
We extend our recent result [Cipolloni, Erdős, Schroder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices $X$ with independent, identically distributedExpand