The circular law for random matrices
Authors:
- Friedrich GötzeFGFriedrich Götze
- Alexander TikhomirovAlexander Tikhomirov
Top Cited Papers
Open Access
- 1 July 2010
- journal article
- Published by Institute of Mathematical Statistics in The Annals of Probability
- Vol. 38 (4) , 1444-1491
Abstract
We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent 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 two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.Keywords
All Related Versions
This publication has 20 references indexed in Scilit:
- Inverse Littlewood–Offord theorems and the condition number of random discrete matricesAnnals of Mathematics, 2009
- Invertibility of random matrices: norm of the inverseAnnals of Mathematics, 2008
- The Littlewood–Offord problem and invertibility of random matricesAdvances in Mathematics, 2008
- Smallest singular value of random matrices and geometry of random polytopesAdvances in Mathematics, 2005
- Concentration of permanent estimators for certain large matricesThe Annals of Applied Probability, 2004
- A limit theorem at the edge of a non-Hermitian random matrix ensembleJournal of Physics A: General Physics, 2003
- Circular lawThe Annals of Probability, 1997
- The Edgeworth Expansion for $U$-Statistics of Degree TwoThe Annals of Statistics, 1986
- SPECTRA OF RANDOM SELF ADJOINT OPERATORSRussian Mathematical Surveys, 1973
- Statistical Ensembles of Complex, Quaternion, and Real MatricesJournal of Mathematical Physics, 1965