Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs
| Autor: | Benjamin Landon, Jiaoyang Huang, Horng-Tzer Yau |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
60B20
Statistics and Probability sparse random matrices Degree (graph theory) Gaussian 05C80 010102 general mathematics Spectrum (functional analysis) Zero (complex analysis) 15B52 01 natural sciences Sparse random graphs Combinatorics 010104 statistics & probability symbols.namesake Distribution (mathematics) symbols Adjacency matrix 05C50 0101 mathematics Statistics Probability and Uncertainty extreme eigenvalue distributions Random matrix Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Ann. Probab. 48, no. 2 (2020), 916-962 |
| ISSN: | 0091-1798 |
| DOI: | 10.1214/19-aop1378 |
| Popis: | 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 Erdős–Rényi graph $G(N,p)$. Tracy–Widom fluctuations of the extreme eigenvalues for $p\gg N^{-2/3}$ was proved in (Probab. Theory Related Fields 171 (2018) 543–616; Comm. Math. Phys. 314 (2012) 587–640). 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 Erdős–Rényi graphs are less rigid than those of random $d$-regular graphs (Bauerschmidt et al. (2019)) of the same average degree. |
| Databáze: | OpenAIRE |
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