zbMATH — the first resource for mathematics

Universality of Wigner random matrices: a survey of recent results. (English. Russian original) Zbl 1230.82032
Russ. Math. Surv. 66, No. 3, 507-626 (2011); translation from Usp. Mat. Nauk. 66, No. 3, 67-198 (2011).
It is not easy to summarize a survey paper (which looks like a monograph with many author’s results) dealing with the study of the universality of spectral statistics for large random matrices. The term of universality, loosely speaking, refers to the property that the statistics of eigenvalues of square matrices, with a large number of entries, are determined by the symmetry type of the ensembles but are independent of the details of the distributions. Following the introduction which defines Wigner matrices and introduces the reader to the local statistics of matrix eigenvalues, some topics of importance are successively addresses, namely the local semicircle law and delocalisation, the universality for Gaussian convolutions, the comparison theorems for Green functions and the universality of Wigner matrices. The Dyson Brownian motion plays a basic role in the paper, and, as one of the main results, it is shown that the density of the eigenvalues converges to the Wigner semi-circle law.

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
Full Text: DOI arXiv