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Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

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Abstract

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption \({p N \gg N^{2/3}}\), we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.

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Authors and Affiliations

  1. Institute of Mathematics, University of Munich, Theresienstrasse 39, 80333, Munich, Germany

    László Erdős

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Antti Knowles, Horng-Tzer Yau & Jun Yin

Authors
  1. László Erdős
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  2. Antti Knowles
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  3. Horng-Tzer Yau
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  4. Jun Yin
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Correspondence to László Erdős.

Additional information

Communicated by P. Forrester

Partially supported by SFB-TR 12 Grant of the German Research Council.

Partially supported by NSF grant DMS-0757425.

Partially supported by NSF grant DMS-0757425 and DMS-0804279.

Partially supported by NSF grant DMS-1001655.

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Erdős, L., Knowles, A., Yau, HT. et al. Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues. Commun. Math. Phys. 314, 587–640 (2012). https://doi.org/10.1007/s00220-012-1527-7

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  • DOI: https://doi.org/10.1007/s00220-012-1527-7

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