Electronic Communications in Probability

Delocalization and limiting spectral distribution of Erdős-Rényi graphs with constant expected degree

Paul Jung and Jaehun Lee

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For fixed $\lambda >0$, it is known that Erdős-Rényi graphs $\{G(n,\lambda /n),n\in \mathbb{N} \}$, with edge-weights $1/\sqrt{\lambda } $, have a limiting spectral distribution, $\nu _{\lambda }$. As $\lambda \to \infty $, $\{\nu _{\lambda }\}$ converges to the semicircle distribution. For large $\lambda $, we find an orthonormal eigenvector basis of $G(n,\lambda /n)$ where most of the eigenvectors have small infinity norms as $n\to \infty $, providing a variant of an eigenvector delocalization result of Tran, Vu, and Wang (2013).

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 92, 13 pp.

Received: 21 August 2018
Accepted: 19 November 2018
First available in Project Euclid: 15 December 2018

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 15B52: Random matrices 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Erdős-Rényi random graph semicircle law delocalization

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Jung, Paul; Lee, Jaehun. Delocalization and limiting spectral distribution of Erdős-Rényi graphs with constant expected degree. Electron. Commun. Probab. 23 (2018), paper no. 92, 13 pp. doi:10.1214/18-ECP198. https://projecteuclid.org/euclid.ecp/1544843115

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