Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs

@article{Huang2020TransitionFT,
  title={Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–R{\'e}nyi graphs},
  author={Jiaoyang Huang and Benjamin Landon and H. T. Yau},
  journal={The Annals of Probability},
  year={2020}
}
We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd\H{o}s-R\'enyi graph $G(N,p)$. Tracy-Widom fluctuations of the extreme eigenvalues for $p\gg N^{-2/3}$ was proved in [17,46]. We prove that there is a crossover in the behavior of the extreme eigenvalues at $p\sim N^{-2/3}$. In the case that $N^{-7/9}\ll p\ll N^{-2/3}$, we prove that the extreme eigenvalues have asymptotically… 

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