The Circular Law for Random Matrices
Authors:
Preprint
Abstract
We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than four and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending results to a larger class of sparse matrices.Keywords
All Related Versions
This publication has 18 references indexed in Scilit:
- The Littlewood–Offord problem and invertibility of random matricesAdvances in Mathematics, 2008
- The Noise in the Circular Law and the Gaussian Free FieldInternational Mathematics Research Notices, 2007
- Smallest singular value of random matrices and geometry of random polytopesAdvances in Mathematics, 2005
- Topological Speed Limits to Network SynchronizationPhysical Review Letters, 2004
- Rate of convergence to the semi-circular lawProbability Theory and Related Fields, 2003
- A limit theorem at the edge of a non-Hermitian random matrix ensembleJournal of Physics A: General Physics, 2003
- The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular LawJournal of Multivariate Analysis, 1997
- On the limit of the largest eigenvalue of the large dimensional sample covariance matrixProbability Theory and Related Fields, 1988
- SPECTRA OF RANDOM SELF ADJOINT OPERATORSRussian Mathematical Surveys, 1973
- Statistical Ensembles of Complex, Quaternion, and Real MatricesJournal of Mathematical Physics, 1965