On High Moments and the Spectral Norm of Large Dilute Wigner Random Matrices

O. Khorunzhiy

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2014) no. 1, pp. 64-125

Résumé

We consider a dilute version of the Wigner ensemble of $n\times n$ random real symmetric matrices $H^{(n,\rho )}$, where $\rho$ denotes an average number of non-zero elements per row. We study the asymptotic properties of the spectral norm $\Vert H^{(n,\rho_n)}\Vert$ in the limit of infinite $n$ with $\rho_n = n^{2/3(1+\varepsilon)}$, $\varepsilon>0$. Our main result is that the probability $\mathbf{P}\left\{ \Vert H^{(n,\rho_n)} \Vert > 1+x n^{-2/3}\right\}$, $x>0$ is bounded for any $\varepsilon \in (\varepsilon_0, 1/2]$, $\varepsilon_0>0$ by an expression that does not depend on the particular values of the first several moments $V_{2l}, 2\le l\le 6$ and $V_{12+2\mathbf{P}hi_0}$, $\phi_0=\phi(\varepsilon_0)$ of the matrix elements of $H^{(n,\rho)}$ provided they exist and the probability distribution of the matrix elements is symmetric. The proof is based on the study of the upper bound of the averaged moments of random matrices with truncated random variables $ \mathbf{E}\{ \mathrm{Tr} (\hat H^{(n,\rho_n)})^{2s_n}\}$, $s_n = \lfloor \chi n^{2/3}\rfloor$ with $\chi>0$, in the limit $n\to\infty$. We also consider the lower bound of $\mathbf{E}\{ \mathrm{Tr} ( H^{(n,\rho_n)})^{2s_n}\}$ and show that in the complementary asymptotic regime, when $\rho_n = n^\epsilon$ with $ \epsilon\in(0, 2/3]$ and $n\to\infty$, the fourth moment $V_4$ enters the estimates from below and the scaling variable $n^{-2/3}$ at the border of the limiting spectrum is to be replaced by a variable related with $\rho_n^{-1}$.

Bibliographie

[1] G. W. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, 118, Cambridge University Press, Cambridge, 2010 | Zbl

[2] Z. D. Bai, Y. Q. Yin, “Necessary and Sufficient Conditions for the Almost Sure Convergence of the Largest Eigenvalue of Wigner Matrices”, Ann. Probab., 16 (1988), 1729–1741 | DOI | MR | Zbl

[3] O. N. Feldheim, S. Sodin, “A Universality Result for the Smallest Eigenvalues of Certain Sample Covariance Matrices”, Geom. Funct. Anal., 20 (2010), 88–123 | DOI | MR | Zbl

[4] Z. Füredi, J. Komlós, “The Eigenvalues of Random Symmetric Matrices”, Combinatorica, 1 (1981), 233–241 | DOI | MR | Zbl

[5] S. Geman, “A Limit Theorem for the Norm of Random Matrices”, Ann. Probab., 8 (1980), 252–261 | DOI | MR | Zbl

[6] V. L. Girko, Spectral Properties of Random Matrices, Nauka, M., 1988 (Russian)

[7] A. Khorunzhy, “Sparse Random Matrices: Spectral Edge and Statistics of Rooted Trees”, Adv. Appl. Probab., 33 (2001), 124–140 | DOI | MR | Zbl

[8] O. Khorunzhy, “High Moments of Large Wigner Random Matrices and Asymptotic Properties of the Spectral Norm”, Random Oper. Stoch. Eq., 20 (2012), 25–68

[9] O. Khorunzhiy, On Correlation Function of High Moments of Wigner Random Matrices, arXiv: 1011.3965

[10] O. Khorunzhiy, B. Khoruzhenko, L. Pastur, M. Shcherbina, “The large-n Limit in Statistical Mechanics and the Spectral Theory of Disordered Systems”, Phase Transitions and Critical Phenomena, 15, Academic Press, London, 1992, 74–239

[11] O. Khorunzhy, W. Kirsch, P. Müller, “Lifshitz Tails for Spectra of Erdős–Rényi Random Graphs”, Ann. Appl. Probab., 16 (2006), 295–309 | DOI | MR

[12] O. Khorunzhy, J.-F. Marckert, “Uniform Bounds for Exponential Moment of Maximum of a Dyck Path”, Electr. Commun. Probab., 14 (2009), 327–333

[13] O. Khorunzhy, M. Shcherbina, V. Vengerovsky, “Eigenvalue Distribution of Large Weighted Random Graphs”, J. Math. Phys., 45 (2004), 1648–1672 | DOI | MR | Zbl

[14] O. Khorunzhiy, V. Vengerovsky, Even Walks and Estimates of High Moments of Large Wigner Random Matrices, arXiv: 0806.0157

[15] V. A. Marchenko, L. A. Pastur, “Distribution of Eigenvalues for Some Sets of Random Matrices”, Math. USSR Sb., 1:4 (1967), 457–483 | DOI

[16] M. L. Mehta, Random Matrices, Elsevier/Academic Press, Amsterdam, 2004 | Zbl

[17] A. D. Mirlin, Ya. V. Fyodorov, “Universality of Level Correlation Function of Sparse Random Matrices”, J. Phys. A, 24 (1991), 2273–2286 | DOI | MR | Zbl

[18] L. Pastur, “On the Spectrum of Random Matrices”, Theor. Math. Phys., 10 (1972), 102–111 | DOI | MR

[19] C. Porter (ed.), Statistical Theories of Spectra: Fluctuations, Acad. Press, New-York, 1965

[20] G. J. Rodgers, A. J. Bray, “Density of States of a Sparse Random Matrix”, Phys. Rev. B, 37 (1988), 3557–3562 | DOI | MR

[21] A. Ruzmaikina, “Universality of the Edge Distribution of the Eigenvalues of Wigner Random Matrices with Polynomially Decaying Distributions of Entries”, Commun. Math. Phys., 261 (2006), 277–296 | DOI | MR | Zbl

[22] Ya. Sinai, A. Soshnikov, “Central Limit Theorem for Traces of Large Symmetric Matrices with Independent Matrix Elements”, Bol. Soc. Brazil. Mat., 29 (1998), 1–24 | DOI | MR | Zbl

[23] Ya. Sinai, A. Soshnikov, “A Refinement of Wigner's Semicircle Law in a Neighborhood of the Spectrum Edge for Random Symmetric Matrices”, Func. Anal. Appl., 32 (1998), 114–131 | DOI | MR

[24] S. Sodin, “The Tracy-Widom Law for Some Sparse Random Matrices”, J. Stat. Phys., 136 (2009), 834–841 | DOI | MR | Zbl

[25] A. Soshnikov, “Universality at the Edge of the Spectrum in Wigner Random Matrices”, Commun. Math. Phys., 207 (1999), 697–733 | DOI | MR | Zbl

[26] C. A. Tracy, H. Widom, “Level Spacing Distribution and the Airy Kernel”, Commun. Math. Phys., 159 (1994), 151–174 | DOI | MR | Zbl

[27] E. Wigner, “Characteristic Vectors of Bordered Matrices with Infinite Dimensions”, Ann. Math., 62 (1955), 548–564 | DOI | MR | Zbl