## Electronic Communications in Probability

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

#### Abstract

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

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

Dates
Accepted: 19 November 2018
First available in Project Euclid: 15 December 2018

https://projecteuclid.org/euclid.ecp/1544843115

Digital Object Identifier
doi:10.1214/18-ECP198

Mathematical Reviews number (MathSciNet)
MR3896830

Zentralblatt MATH identifier
1405.05162

#### Citation

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

#### References

• [1] G. W. Anderson, A. Guionnet, and O. Zeitouni, An Introduction to Random Matrices, Cambridge University Press, 2010.
• [2] M. Bauer and O. Golinelli, Random incidence matrices: moments of the spectral density, J. Statist. Phys. 103 (2001), no. 1-2, 301–337.
• [3] P. Billingsley, Probability and measure, second ed., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986.
• [4] C. Bordenave and M. Lelarge, Resolvent of large random graphs, Random Structures Algorithms 37 (2010), no. 3, 332–352.
• [5] C. Bordenave, P. Caputo, and D. Chafaï, Spectrum of large random reversible Markov chains: heavy-tailed weights on the complete graph, Ann. Probab. 39 (2011), no. 4, 1544–1590.
• [6] C. Bordenave, P. Caputo, and D. Chafaï, Spectrum of Non-Hermitian Heavy Tailed Random Matrices Comm. Math. Phys. 307 (2011), no. 2, 513–560.
• [7] C. Bordenave, A. Sen, and B. Virág, Mean quantum percolation, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3679–3707.
• [8] Y. Dekel, J. R. Lee, and N. Linial, Eigenvectors of random graphs: nodal domains, Random Structures Algorithms 39 (2011), no. 1, 39–58.
• [9] R. Durrett, Probability: Theory and Examples, Cambridge University Press, 2010.
• [10] N. Enriquez and L. Ménard, Spectra of large diluted but bushy random graphs, Random Structures Algorithms 49 (2016), no. 1, 160–184.
• [11] L. Erdős, A. Knowles, H.-T. Yau, and J. Yin, Spectral statistics of Erdős-Rényi graphs I: Local semicircle law, Ann. Probab. 41 (2013), no. 3B, 2279–2375.
• [12] L. Erdős, B. Schlein, and H.-T. Yau, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann. Probab. 37 (2009), no. 3, 815–852.
• [13] P. Jung, Lévy-Khintchine random matrices and the Poisson weighted infinite skeleton tree, Trans. Amer. Math. Soc. 370 (2018), no. 1, 641–668.
• [14] P. Massart, Concentration inequalities and model selection, Lecture Notes in Mathematics, vol. 1896, Springer, Berlin, 2007, Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003, With a foreword by Jean Picard.
• [15] M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972.
• [16] B. Simon, Trace Ideals and Their Applications, Cambridge University Press, 1979.
• [17] T. Tao, Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, 2012.
• [18] T. Tao and V. Vu, Random matrices: universality of local eigenvalue statistics, Acta Math. 206 (2011), no. 1, 127–204.
• [19] L. V. Tran, V. H. Vu, and K. Wang, Sparse random graphs: eigenvalues and eigenvectors, Random Structures Algorithms 42 (2013), no. 1, 110–134.
• [20] E. P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2) 62 (1955), 548–564.
• [21] I. Zakharevich, A generalization of Wigner’s law, Comm. Math. Phys. 268 (2006), no. 2, 403–414.