Electronic Journal of Probability

Traffic distributions of random band matrices

Benson Au

Full-text: Open access

Abstract

We study random band matrices within the framework of traffic probability. As a starting point, we revisit the familiar case of permutation invariant Wigner matrices and compare the situation to the general case in the absence of this invariance. Here, we find a departure from the usual free probabilistic universality of the joint distribution of independent Wigner matrices. We further prove general Markov-type concentration inequalities for the joint traffic distribution. We then extend our analysis to random band matrices and investigate the extent to which the joint traffic distribution of independent copies of these matrices deviates from the Wigner case.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 77, 48 pp.

Dates
Received: 3 December 2017
Accepted: 25 July 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717736

Digital Object Identifier
doi:10.1214/18-EJP205

Mathematical Reviews number (MathSciNet)
MR3858905

Zentralblatt MATH identifier
06964771

Subjects
Primary: 15B52: Random matrices 46L53: Noncommutative probability and statistics 46L54: Free probability and free operator algebras 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
free probability random band matrix traffic probability Wigner matrix

Rights
Creative Commons Attribution 4.0 International License.

Citation

Au, Benson. Traffic distributions of random band matrices. Electron. J. Probab. 23 (2018), paper no. 77, 48 pp. doi:10.1214/18-EJP205. https://projecteuclid.org/euclid.ejp/1536717736


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References

  • [1] 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.
  • [2] Benson Au, Guillaume Cébron, Antoine Dahlqvist, Franck Gabriel, and Camille Male, Large permutation invariant random matrices are asymptotically free over the diagonal, Preprint. arXiv:1805.07045v1
  • [3] Z. D. Bai and Y. Q. Yin, Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix, Ann. Probab. 16 (1988), no. 4, 1729–1741.
  • [4] Zhidong Bai and Jack W. Silverstein, Spectral analysis of large dimensional random matrices, second ed., Springer Series in Statistics, Springer, New York, 2010.
  • [5] Jinho Baik, Percy Deift, and Toufic Suidan, Combinatorics and random matrix theory, Graduate Studies in Mathematics, vol. 172, American Mathematical Society, Providence, RI, 2016.
  • [6] Anis Ben Ghorbal and Michael Schürmann, Non-commutative notions of stochastic independence, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 3, 531–561.
  • [7] L. V. Bogachev, S. A. Molchanov, and L. A. Pastur, On the density of states of random band matrices, Mat. Zametki 50 (1991), no. 6, 31–42, 157.
  • [8] Paul Bourgade, Random band matrices, To appear in Proceedings of the International Congress of Mathematicians. arXiv:1807.03031v1
  • [9] Guillaume Cébron, Antoine Dahlqvist, and Camille Male, Universal constructions for spaces of traffics, Preprint. arXiv:1601.00168v1
  • [10] Aurélien Deya and Ivan Nourdin, Convergence of Wigner integrals to the tetilla law, ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), 101–127.
  • [11] Ken Dykema, On certain free product factors via an extended matrix model, J. Funct. Anal. 112 (1993), no. 1, 31–60.
  • [12] László Erdős and Horng-Tzer Yau, A dynamical approach to random matrix theory, Courant Lecture Notes in Mathematics, vol. 28, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2017.
  • [13] Fumio Hiai and Dénes Petz, The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs, vol. 77, American Mathematical Society, Providence, RI, 2000.
  • [14] Camille Male, Traffic distributions and independence: permutation invariant random matrices and the three notions of independence, To appear in Mem. Amer. Math. Soc. arXiv:1111.4662v8
  • [15] Camille Male, The limiting distributions of large heavy Wigner and arbitrary random matrices, J. Funct. Anal. 272 (2017), no. 1, 1–46.
  • [16] Camille Male and Sandrine Péché, Uniform regular weighted graphs with large degree: Wigner’s law, asymptotic freeness and graphons limit, Preprint. arXiv:1410.8126v1
  • [17] James A. Mingo and Roland Speicher, Sharp bounds for sums associated to graphs of matrices, J. Funct. Anal. 262 (2012), no. 5, 2272–2288.
  • [18] James A. Mingo and Roland Speicher, Free probability and random matrices, Fields Institute Monographs, vol. 35, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017.
  • [19] Alexandru Nica and Roland Speicher, Commutators of free random variables, Duke Math. J. 92 (1998), no. 3, 553–592.
  • [20] 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.
  • [21] Dimitri Shlyakhtenko, Random Gaussian band matrices and freeness with amalgamation, Internat. Math. Res. Notices (1996), no. 20, 1013–1025.
  • [22] Roland Speicher, On universal products, Free probability theory (Waterloo, ON, 1995), Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 257–266.
  • [23] Roland Speicher, Free probability theory: and its avatars in representation theory, random matrices, and operator algebras; also featuring: non-commutative distributions, Jahresber. Dtsch. Math.-Ver. 119 (2017), no. 1, 3–30.
  • [24] Terence Tao, Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, 2012.
  • [25] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992, A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups.
  • [26] Dan Voiculescu, Symmetries of some reduced free product $C^\ast $-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588.
  • [27] Dan Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), no. 1, 201–220.
  • [28] Eugene P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2) 62 (1955), 548–564.
  • [29] Eugene P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. (2) 67 (1958), 325–327.