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- W2775375998 abstract "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 ErdH{o}s-R'enyi graph $G(N,p)$. Tracy-Widom fluctuations of the extreme eigenvalues for $pgg N^{-2/3}$ was proved in [17,46]. We prove that there is a crossover in the behavior of the extreme eigenvalues at $psim N^{-2/3}$. In the case that $N^{-7/9}ll pll N^{-2/3}$, 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 ErdH{o}s-R'enyi graphs are less rigid than those of random $d$-regular graphs [4] of the same average degree." @default.
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- W2775375998 date "2017-12-11" @default.
- W2775375998 modified "2023-10-18" @default.
- W2775375998 title "Transition from Tracy-Widom to Gaussian fluctuations of extremal eigenvalues of sparse ErdH{o}s-R'enyi graphs" @default.
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- W2775375998 doi "https://doi.org/10.48550/arxiv.1712.03936" @default.
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