The Annals of Probability

Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

László Erdős, Antti Knowles, Horng-Tzer Yau, and Jun Yin

Full-text: Open access

Abstract

We consider the ensemble of adjacency matrices of Erdős–Rényi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erdős–Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than $N^{-1}$ (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the $\ell^{\infty}$-norms of the $\ell^{2}$-normalized eigenvectors are at most of order $N^{-1/2}$ with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős–Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that $pN\gg N^{2/3}$.

Article information

Source
Ann. Probab., Volume 41, Number 3B (2013), 2279-2375.

Dates
First available in Project Euclid: 15 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1368623526

Digital Object Identifier
doi:10.1214/11-AOP734

Mathematical Reviews number (MathSciNet)
MR3098073

Zentralblatt MATH identifier
1272.05111

Subjects
Primary: 15B52: Random matrices 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Erdős–Rényi graphs local semicircle law density of states

Citation

Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. Spectral statistics of Erdős–Rényi graphs I: Local semicircle law. Ann. Probab. 41 (2013), no. 3B, 2279--2375. doi:10.1214/11-AOP734. https://projecteuclid.org/euclid.aop/1368623526


Export citation

References

  • [1] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
  • [2] Bai, Z. D., Miao, B. and Tsay, J. (2002). Convergence rates of the spectral distributions of large Wigner matrices. Int. Math. J. 1 65–90.
  • [3] Benaych-Georges, F. and Nadakuditi, R. R. (2011). The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Adv. Math. 227 494–521.
  • [4] Bleher, P. and Its, A. (1999). Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 185–266.
  • [5] Capitaine, M., Donati-Martin, C. and Féral, D. (2009). The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. Ann. Probab. 37 1–47.
  • [6] Deift, P. and Gioev, D. (2009). Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Mathematics 18. Amer. Math. Soc., Providence, RI.
  • [7] Deift, P., Kriecherbauer, T., McLaughlin, K. T. R., Venakides, S. and Zhou, X. (1999). Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 1335–1425.
  • [8] Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X. (1999). Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 1491–1552.
  • [9] Deift, P. A. (1999). Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics 3. Amer. Math Soc., Providence, RI.
  • [10] Dekel, Y., Lee, R. L. and Linial, N. (2011). Eigenvectors of random graphs: Nodal domains. Random Structures Algorithms 39 39–58.
  • [11] Dumitriu, I. and Pal, S. (2011). Sparse regular random graphs: Spectral density and eigenvectors. Preprint. Available at arXiv:0910.5306.
  • [12] Dyson, F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191–1198.
  • [13] Erdős, L., Knowles, A., Yau, H. T. and Yin, J. (2012). Spectral statistics of Erdős–Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues. Comm. Math. Phys. 314 587–640.
  • [14] Erdős, L., Péché, S., Ramírez, J. A., Schlein, B. and Yau, H.-T. (2010). Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63 895–925.
  • [15] Erdős, L., Ramírez, J. A., Schlein, B. and Yau, H.-T. (2010). Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15 526–603.
  • [16] Erdős, L., Schlein, B. and Yau, H.-T. (2009). Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 815–852.
  • [17] Erdős, L., Schlein, B. and Yau, H.-T. (2009). Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 641–655.
  • [18] Erdős, L., Schlein, B. and Yau, H.-T. (2010). Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. IMRN 3 436–479.
  • [19] Erdős, L., Schlein, B. and Yau, H.-T. (2011). Universality of random matrices and local relaxation flow. Invent. Math. 185 75–119.
  • [20] Erdős, L., Schlein, B., Yau, H. T. and Yin, J. (2012). The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. H. Poincaré Probab. Statist. 48 1–46.
  • [21] Erdős, L., Yau, H.-T. and Yin, J. (2011). Universality for generalized Wigner matrices with Bernoulli distribution. J. Comb. 2 15–81.
  • [22] Erdős, L., Yau, H. T. and Yin, J. (2012). Bulk universality for generalized Wigner matrices. Probab. Theory Related Fields 154 341–407.
  • [23] Erdős, L., Yau, H.-T. and Yin, J. (2012). Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229 1435–1515.
  • [24] Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 290–297.
  • [25] Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.
  • [26] Féral, D. and Péché, S. (2007). The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys. 272 185–228.
  • [27] Gaudin, M. (1961). Sur la loi limite de l’espacement des valeurs propres d’une matrice aléatoire. Nuclear Phys. 25 447–458.
  • [28] Guionnet, A. (2009). Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Math. 1957. Springer, Berlin.
  • [29] Guionnet, A. and Zeitouni, O. (2000). Concentration of the spectral measure for large matrices. Electron. Commun. Probab. 5 119–136 (electronic).
  • [30] Mehta, M. L. (1991). Random Matrices, 2nd ed. Academic Press, Boston, MA.
  • [31] Pastur, L. and Shcherbina, M. (2008). Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130 205–250.
  • [32] Pizzo, A., Renfrew, D. and Soshnikov, A. (2013). On finite rank deformations of Wigner matrices. Ann. Inst. Henri Poincaré Probab. Stat. 49 64–94.
  • [33] Sinaĭ, Y. G. and Soshnikov, A. B. (1998). A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funct. Anal. Appl. 32 114–131.
  • [34] Sodin, S. (2010). The spectral edge of some random band matrices. Ann. of Math. (2) 172 2223–2251.
  • [35] Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697–733.
  • [36] Tao, T. and Vu, V. (2010). Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 549–572.
  • [37] Tao, T. and Vu, V. (2011). Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 127–204.
  • [38] Tran, L., Vu, V. and Wang, K. (2010). Sparse random graphs: Eigenvalues and eigenvectors. Preprint. Available at arXiv:1011.6646.
  • [39] Vu, V. H. (2007). Spectral norm of random matrices. Combinatorica 27 721–736.
  • [40] Wigner, E. P. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 548–564.