IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v26y2013i4d10.1007_s10959-011-0391-2.html
   My bibliography  Save this article

The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

Author

Listed:
  • Murat Koloğlu

    (Williams College)

  • Gene S. Kopp

    (University of Chicago)

  • Steven J. Miller

    (Williams College)

Abstract

Given an ensemble of N×N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N→∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a “dial” we can “turn” from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f m show a visually stunning convergence to the semi-circle as m→∞, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f m is the product of a Gaussian and a certain even polynomial of degree 2m−2; the formula is the same as that for the m×m Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending not only on the frequency at which each element appears, but also on the way the elements are arranged.

Suggested Citation

  • Murat Koloğlu & Gene S. Kopp & Steven J. Miller, 2013. "The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices," Journal of Theoretical Probability, Springer, vol. 26(4), pages 1020-1060, December.
  • Handle: RePEc:spr:jotpro:v:26:y:2013:i:4:d:10.1007_s10959-011-0391-2
    DOI: 10.1007/s10959-011-0391-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-011-0391-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10959-011-0391-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Bose, Arup & Mitra, Joydip, 2002. "Limiting spectral distribution of a special circulant," Statistics & Probability Letters, Elsevier, vol. 60(1), pages 111-120, November.
    2. Adam Massey & Steven J. Miller & John Sinsheimer, 2007. "Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices," Journal of Theoretical Probability, Springer, vol. 20(3), pages 637-662, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Alexander Tikhomirov & Sabina Gulyaeva & Dmitry Timushev, 2024. "Limit Theorems for Spectra of Circulant Block Matrices with Large Random Blocks," Mathematics, MDPI, vol. 12(14), pages 1-16, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Steven Jackson & Steven J. Miller & Thuy Pham, 2012. "Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices," Journal of Theoretical Probability, Springer, vol. 25(2), pages 464-495, June.
    2. Arup Bose & Joydip Mitra & Arnab Sen, 2012. "Limiting Spectral Distribution of Random k-Circulants," Journal of Theoretical Probability, Springer, vol. 25(3), pages 771-797, September.
    3. Dang-Zheng Liu & Zheng-Dong Wang, 2011. "Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices," Journal of Theoretical Probability, Springer, vol. 24(4), pages 988-1001, December.
    4. Arup Bose & Rajat Subhra Hazra & Koushik Saha, 2011. "Spectral Norm of Circulant-Type Matrices," Journal of Theoretical Probability, Springer, vol. 24(2), pages 479-516, June.
    5. Maurya, Shambhu Nath, 2024. "Limiting spectral distribution of Toeplitz and Hankel matrices with dependent entries," Statistics & Probability Letters, Elsevier, vol. 209(C).
    6. S. Chatterjee & A. Bose, 2004. "A New Method for Bounding Rates of Convergence of Empirical Spectral Distributions," Journal of Theoretical Probability, Springer, vol. 17(4), pages 1003-1019, October.
    7. Bose, Arup & Guha, Suman & Hazra, Rajat Subhra & Saha, Koushik, 2011. "Circulant type matrices with heavy tailed entries," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1706-1716, November.
    8. Adam Massey & Steven J. Miller & John Sinsheimer, 2007. "Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices," Journal of Theoretical Probability, Springer, vol. 20(3), pages 637-662, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:26:y:2013:i:4:d:10.1007_s10959-011-0391-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.