Transition from Tracy-Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erd\H{o}s-R\'enyi graphs

11 Dec 2017  ·  Huang Jiaoyang, Landon Benjamin, Yau Horng-Tzer ·

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 Gaussian fluctuations. Under a mean zero condition and when \$p=CN^{-2/3}\$, we find that the fluctuations of the extreme eigenvalues are given by a combination of the Gaussian and the Tracy-Widom distribution. These results show that the eigenvalues at the edge of the spectrum of sparse Erd\H{o}s-R\'enyi graphs are less rigid than those of random \$d\$-regular graphs [4] of the same average degree. read more

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