Wigner matrices

This article reviews some of the important results in the study of the eigenvalues and the eigenvectors of Wigner random matrices, that is. random Hermitian (or real symmetric) matrices with iid entries. It first provides an overview of the Wigner matrices, introduced in the 1950s by Wigner as a very simple model of random matrices to approximate generic self-adjoint operators. It then considers the global properties of the spectrum of Wigner matrices, focusing on convergence to the semicircle law, fluctuations around the semicircle law, deviations and concentration properties, and the delocalization of the eigenvectors. It also describes local properties in the bulk and at the edge before concluding with a brief analysis of the known universality results showing how much the behaviour of the spectrum is insensitive to the distribution of the entries.

Spectra of overlapping Wishart matrices and the Gaussian free field

Consider a doubly-infinite array of i.i.d. centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in [A. Borodin, Clt for spectra of submatrices of Wigner random matrices, Mosc. Math. J. 14(1) (2014) 29–38; A. Borodin and V. Gorin, General beta Jacobi corners process and the Gaussian free field, preprint (2013), arXiv:1305.3627; T. Johnson and S. Pal, Cycles and eigenvalues of sequentially growing random regular graphs, Ann. Probab. 42(4) (2014) 1396–1437], can be described in terms of the height function, and leads to a connection with the Gaussian Free Field on the upper half-plane. Finally, we generalize our results from univariate polynomials to a special class of planar functions.

On the universality of the non-singularity of general Ginibre and Wigner random matrices

We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. Moreover, no moment assumptions are made about the distributions governing the entries. Our proofs are based on a concentration function inequality due to Kolmogorov, Rogozin and Kesten, which allows us to improve universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.