From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices
| dc.creator | Tao, Terence | |
| dc.creator | Vu, Van | |
| dc.date | 2008-10-16 | |
| dc.date | 2009-01-01 | |
| dc.date.accessioned | 2026-07-07T12:23:21Z | |
| dc.date.available | 2026-07-07T12:23:21Z | |
| dc.description | The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\frac{1}{\sqrt{n}} M_n$ converges almost surely to the uniform distribution on the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the full circular law was recently established in \cite{TVcir2}. In this survey we describe some of the key ingredients used in the establishment of the circular law, in particular recent advances in understanding the Littlewood-Offord problem and its inverse. | |
| dc.description | 25 pages, 8 figures, to appear, Bull. Amer. Math. Soc. Various corrections and referee suggestions incorporated | |
| dc.identifier | https://arxiv.org/abs/0810.2994 | |
| dc.identifier | http://arxiv.org/abs/0810.2994 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/213955 | |
| dc.subject | Probability | |
| dc.subject | 15A52, 60G50 | |
| dc.title | From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices | |
| dc.type | text |