Invertibility of Sparse non-Hermitian matrices

@article{Basak2015InvertibilityOS,
  title={Invertibility of Sparse non-Hermitian matrices},
  author={Anirban Basak and Mark Rudelson},
  journal={arXiv: Probability},
  year={2015}
}
We consider a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered random variables, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, and prove a quantitative estimate on the smallest singular value for $p_n = \Omega(\frac{\log n}{n})$, under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse… 
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