Extreme eigenvalue distributions of sparse Erdős–Rényi graphs

Jiaoyang Huang
Harvard University

I will discuss the extreme eigenvalue distributions of sparse Erdos–Rényi graphs $G(N,p)$. We prove that there is a crossover in the behavior of the extreme eigenvalues at $p\sim N^{-2/3}$. When $p \gg N^{-2/3}$, the extreme eigenvalues have asymptotically Tracy-Widom fluctuations. When $N^{-7/9} \ll p \ll N^{-2/3}$, the extreme eigenvalues have asymptotically Gaussian fluctuations. 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. Our proof is based on constructing a higher order self-consistent equation for the Stieltjes transform of the empirical eigenvalue distributions. This is based on joint work with Benjamin Landon and Horng-Tzer Yau.

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