Transition from TracyWidom to Gaussian fluctuations of extremal eigenvalues of sparse ErdősRényi graphs
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 ErdősRényi graph $G(N,p)$. TracyWidom 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 TracyWidom distribution. These results show that the eigenvalues at the edge of the spectrum of sparse ErdősRényi graphs are less rigid than those of random $d$regular graphs [4] of the same average degree.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1712.03936
 Bibcode:
 2017arXiv171203936H
 Keywords:

 Mathematics  Probability;
 05C80;
 05C50;
 60B20;
 15B52
 EPrint:
 1 figure