The Littlewood-Offord problem and invertibility of random matrices

@article{Rudelson2007TheLP,
  title={The Littlewood-Offord problem and invertibility of random matrices},
  author={M. Rudelson and R. Vershynin},
  journal={Advances in Mathematics},
  year={2007},
  volume={218},
  pages={600-633}
}
Abstract We prove two basic conjectures on the distribution of the smallest singular value of random n × n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n − 1 / 2 , which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood–Offord problem: for i.i.d. random variables X k and real numbers a k… Expand
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