Circular law for the sum of random permutation matrices

@article{Basak2017CircularLF,
  title={Circular law for the sum of random permutation matrices},
  author={Anirban Basak and Nicholas A. Cook and Ofer Zeitouni},
  journal={arXiv: Probability},
  year={2017}
}
Let $P_n^1,\dots, P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum_{\ell=1}^d P_n^\ell$. We show that if $\log^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d}$ converges weakly to the circular law in probability as $n \to \infty$. 
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