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

By Jiaoyang Huang, Benjamin R. Landon, Horng-Tzer Yau. Annals of Probability, 2017

55 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/48179ecfa5ffdb6c70c8fad16b19985317d961e5

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Citations in the corpus

Dyson Brownian motion for general $$\beta $$ and potential at the edge (2018, background): “Establishing local laws near the edge is generally more involved; the case of correlated matrices was shown in [1,4,18]; the deformed Wigner matrices were considered in [42]; the sparse random graphs were studied in [5,37].”
Extremal eigenvalues of critical Erdős–Rényi graphs (2021, result): “The behaviour of the extremal eigenvalues is more subtle, and has been investigated in several recent works [2, 3, 7, 6, 9, 12, 8, 13, 14, 19, 11].”