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Lévy-Khintchine random matrices and the Poisson weighted infinite skeleton tree

Author: Paul Jung
Journal: Trans. Amer. Math. Soc. 370 (2018), 641-668
MSC (2010): Primary 15B52, 60B20, 60G51
Published electronically: July 7, 2017
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Abstract: We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdős-Rényi random graphs with $ p_n\sim \frac 1 n$. Our $ n\times n$ random matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on $ n$, and we require row sums to converge in distribution. It is then well-known that the limit distribution must be infinitely divisible.

We show that a limiting empirical spectral distribution (LSD) exists and, via local weak convergence of associated graphs, that the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges called ``cords to infinity''. One example covered by the results are matrices with i.i.d. entries having infinite second moments but normalized to be in the Gaussian domain of attraction. In this case, the limiting graph is $ \mathbb{N}$ rooted at $ 1$, so the LSD is the semicircle law.

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  • [AGZ10] Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge, 2010. MR 2760897
  • [Ald92] David Aldous, Asymptotics in the random assignment problem, Probab. Theory Related Fields 93 (1992), no. 4, 507-534. MR 1183889,
  • [Ald01] David J. Aldous, The $ \zeta (2)$ limit in the random assignment problem, Random Structures Algorithms 18 (2001), no. 4, 381-418. MR 1839499,
  • [ALS07] Luigi Accardi, Romuald Lenczewski, and Rafał Sałapata, Decompositions of the free product of graphs, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), no. 3, 303-334. MR 2354364,
  • [AS04] David Aldous and J. Michael Steele, The objective method: probabilistic combinatorial optimization and local weak convergence, Probability on discrete structures, Encyclopaedia Math. Sci., vol. 110, Springer, Berlin, 2004, pp. 1-72. MR 2023650,
  • [BAG08] Gérard Ben Arous and Alice Guionnet, The spectrum of heavy tailed random matrices, Comm. Math. Phys. 278 (2008), no. 3, 715-751. MR 2373441,
  • [BC12] Charles Bordenave and Djalil Chafaï, Around the circular law, Probab. Surv. 9 (2012), 1-89. MR 2908617,
  • [BCC11a] Charles Bordenave, Pietro Caputo, and Djalil Chafaï, Spectrum of large random reversible Markov chains: heavy-tailed weights on the complete graph, Ann. Probab. 39 (2011), no. 4, 1544-1590. MR 2857250,
  • [BCC11b] Charles Bordenave, Pietro Caputo, and Djalil Chafaï, Spectrum of non-Hermitian heavy tailed random matrices, Comm. Math. Phys. 307 (2011), no. 2, 513-560. MR 2837123,
  • [BG01] M. Bauer and O. Golinelli, Random incidence matrices: moments of the spectral density, J. Statist. Phys. 103 (2001), no. 1-2, 301-337. MR 1828732,
  • [BG05] Florent Benaych-Georges, Classical and free infinitely divisible distributions and random matrices, Ann. Probab. 33 (2005), no. 3, 1134-1170. MR 2135315,
  • [BGGM13] Florent Benaych-Georges, Alice Guionnet, and Camille Male, Central limit theorems for linear statistics of heavy tailed random matrices, Comm. Math. Phys. 329 (2014), no. 2, 641-686. MR 3210147,
  • [Bil86] Patrick Billingsley, Probability and measure, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 830424
  • [BJN$^+$07] Zdzisław Burda, Jerzy Jurkiewicz, Maciej A. Nowak, Gabor Papp, and Ismail Zahed, Free random Lévy and Wigner-Lévy matrices, Phys. Rev. E (3) 75 (2007), no. 5, 051126, 11. MR 2361817,
  • [BL10] Charles Bordenave and Marc Lelarge, Resolvent of large random graphs, Random Structures Algorithms 37 (2010), no. 3, 332-352. MR 2724665,
  • [BS01] Itai Benjamini and Oded Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), no. 23, 13. MR 1873300,
  • [BS10] Zhidong Bai and Jack W. Silverstein, Spectral analysis of large dimensional random matrices, 2nd ed., Springer Series in Statistics, Springer, New York, 2010. MR 2567175
  • [CB94] P. Cizeau and J. P. Bouchaud, Theory of Lévy matrices, Physical Review E 50 (1994), no. 3, 1810.
  • [DS07] R. Brent Dozier and Jack W. Silverstein, On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices, J. Multivariate Anal. 98 (2007), no. 4, 678-694. MR 2322123,
  • [EK10] Noureddine El Karoui, On information plus noise kernel random matrices, Ann. Statist. 38 (2010), no. 5, 3191-3216. MR 2722468,
  • [FZ97] Joshua Feinberg and A. Zee, Non-Hermitian random matrix theory: method of Hermitian reduction, Nuclear Phys. B 504 (1997), no. 3, 579-608. MR 1488584,
  • [GL09] Adityanand Guntuboyina and Hannes Leeb, Concentration of the spectral measure of large Wishart matrices with dependent entries, Electron. Commun. Probab. 14 (2009), 334-342. MR 2535081,
  • [HO07] Akihito Hora and Nobuaki Obata, Quantum probability and spectral analysis of graphs, Theoretical and Mathematical Physics, Springer, Berlin, 2007. MR 2316893
  • [Kal02] Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169
  • [Kle98] Abel Klein, Extended states in the Anderson model on the Bethe lattice, Adv. Math. 133 (1998), no. 1, 163-184. MR 1492789,
  • [KSV04] O. Khorunzhy, M. Shcherbina, and V. Vengerovsky, Eigenvalue distribution of large weighted random graphs, J. Math. Phys. 45 (2004), no. 4, 1648-1672. MR 2043849,
  • [Küh08] Reimer Kühn, Spectra of sparse random matrices, J. Phys. A 41 (2008), no. 29, 295002, 21. MR 2455271,
  • [Kyp06] Andreas E. Kyprianou, Introductory lectures on fluctuations of Lévy processes with applications, Universitext, Springer-Verlag, Berlin, 2006. MR 2250061
  • [Mal12] Camille Male, The limiting distributions of large heavy Wigner and arbitrary random matrices, J. Funct. Anal. 272 (2017), no. 1, 1-46. MR 3567500,
  • [MP67] V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues in certain sets of random matrices, Mat. Sb. (N.S.) 72 (114) (1967), 507-536 (Russian). MR 0208649
  • [NS06] Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879
  • [RB88] G. J. Rodgers and A. J. Bray, Density of states of a sparse random matrix, Phys. Rev. B (3) 37 (1988), no. 7, 3557-3562. MR 932406,
  • [Res07] Sidney I. Resnick, Heavy-tail phenomena: Probabilistic and statistical modeling, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2007. MR 2271424
  • [RS80] Michael Reed and Barry Simon, Methods of modern mathematical physics. I: Functional analysis, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. MR 751959
  • [Rya98] Øyvind Ryan, On the limit distributions of random matrices with independent or free entries, Comm. Math. Phys. 193 (1998), no. 3, 595-626. MR 1624843,
  • [Sos04] Alexander Soshnikov, Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails, Electron. Comm. Probab. 9 (2004), 82-91. MR 2081462,
  • [Ste02] J. Michael Steele, Minimal spanning trees for graphs with random edge lengths, Mathematics and computer science, II (Versailles, 2002) Trends Math., Birkhäuser, Basel, 2002, pp. 223-245. MR 1940139
  • [Tao12] Terence Tao, Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, 2012. MR 2906465
  • [Wig55] Eugene P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2) 62 (1955), 548-564. MR 0077805,
  • [WS80] Joachim Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York-Berlin, 1980. MR 566954
  • [Zak06] Inna Zakharevich, A generalization of Wigner's law, Comm. Math. Phys. 268 (2006), no. 2, 403-414. MR 2259200,

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Additional Information

Paul Jung
Affiliation: Department of Mathematical Sciences, KAIST, Daejeon 34141, Republic of Korea

Keywords: Empirical spectral distribution, Wigner matrices, L\'evy matrices, heavy-tailed random matrices, sparse random matrices, Erd\H{o}s-R\'enyi graph, local weak convergence.
Received by editor(s): August 4, 2014
Received by editor(s) in revised form: March 12, 2015, February 13, 2016, and April 18, 2016
Published electronically: July 7, 2017
Additional Notes: The author’s research was partially supported by NSA grant H98230-14-1-0144 while at the University of Alabama Birmingham.
Article copyright: © Copyright 2017 American Mathematical Society

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