Journal of Theoretical Probability

, Volume 26, Issue 4, pp 1020–1060 | Cite as

The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

  • Murat Koloğlu
  • Gene S. Kopp
  • Steven J. MillerEmail author


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.


Limiting spectral measure Circulant and Toeplitz matrices Random matrix theory Convergence Method of moments Orientable surfaces Euler characteristic 

Mathematics Subject Classification (2000)

15B52 60F05 11D45 60F15 60G57 62E20 



M. Koloğlu and G.S. Kopp were partially supported by Williams College and NSF Grants DMS0855257 and DMS0850577, and S.J. Miller was partly supported by NSF Grant DMS0970067. It is a pleasure to thank our colleagues from the Williams College 2010 SMALL REU program as well as the participants of the ICM Satellite Meeting in Probability & Stochastic Processes (Bangalore, 2010) for many helpful conversations, especially Arup Bose and Rajat Hazra. We would also like to thank Elizabeth Townsend Beazley for comments on Wentao Xiong’s senior thesis, which is the basis of Appendix B.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965). Online at Google Scholar
  2. 2.
    Baik, J., Borodin, A., Deift, P., Suidan, T.: A model for the bus system in Cuernevaca (Mexico). Math. Phys. (2005), 1–9. Online available at
  3. 3.
    Basak, A., Bose, A.: Limiting spectral distribution of some band matrices. Periodica Math. Hungarica 63(1), 113–150 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Basak, A., Bose, A.: Balanced random Toeplitz and Hankel matrices. Electron. Comm. Probab. 15, 134–148 Google Scholar
  5. 5.
    Banerjee, S., Bose, A.: Noncrossing partitions, Catalan words and the semicircle law. J. Theor. Probab. doi: 10.1007/s10959-011-0365-4
  6. 6.
    Beckwith, O., Miller, S.J., Shen, K.: Distribution of eigenvalues of weighted, structured matrix ensembles, preprint Google Scholar
  7. 7.
    Bose, A., Chatterjee, S., Gangopadhyay, S.: Limiting spectral distributions of large dimensional random matrices. J. Indian Stat. Assoc. 41, 221–259 (2003) MathSciNetGoogle Scholar
  8. 8.
    Bose, A., Hazra, R.S., Saha, K.: Patterned random matrices and notions of independence, Technical Report R3/2010 (2010), Stat-Math Unit, Kolkata. Available online at
  9. 9.
    Bose, A., Mitra, J.: Limiting spectral distribution of a special circulant. Stat. Probab. Lett. 60(1), 111–120 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bryc, W., Dembo, A., Jiang, T.: Spectral measure of large random Hankel, Markov, and Toeplitz matrices. Ann. Probab. 34(1), 1–38 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dyson, F.: Statistical theory of the energy levels of complex systems: I, II, III. J. Math. Phys. 3, 140–156 (1962), 157–165, 166–175 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dyson, F.: The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3, 1199–1215 (1962) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Erdős, L., Ramirez, J.A., Schlein, B., Yau, H.-T.: Bulk Universality for Wigner Matrices, preprint.
  14. 14.
    Erdős, L., Schlein, B., Yau, H.-T.: Wegner estimate and level repulsion for Wigner random matrices, preprint.
  15. 15.
    Firk, F.W.K., Miller, S.J.: Nuclei, primes and the random matrix connection. Symmetry 1, 64–105 (2009). doi: 10.3390/sym1010064 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Forrester, P.J.: Log-Gases and Random Matrices. London Mathematical Society Monographs, vol. 34. Princeton University Press, Princeton (2010) zbMATHGoogle Scholar
  17. 17.
    Gradshteyn, I., Ryzhik, I.: Tables of Integrals, Series, and Products. Academic Press, New York (1965) Google Scholar
  18. 18.
    Hammond, C., Miller, S.J.: Eigenvalue spacing distribution for the ensemble of real symmetric Toeplitz matrices. J. Theor. Probab. 18(3), 537–566 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457–485 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) zbMATHGoogle Scholar
  21. 21.
    Hayes, B.: The spectrum of Riemannium. Am. Sci. 91(4), 296–300 (2003) Google Scholar
  22. 22.
    Jackson, S., Miller, S.J., Pham, V.: Distribution of eigenvalues of highly palindromic Toeplitz matrices.
  23. 23.
    Jakobson, D., Miller, S.D., Rivin, I., Rudnick, Z.: Eigenvalue spacings for regular graphs. In: Emerging Applications of Number Theory, Minneapolis, 1996. The IMA Volumes in Mathematics and its Applications, vol. 109, pp. 317–327. Springer, New York (1999) CrossRefGoogle Scholar
  24. 24.
    Kargin, V.: Spectrum of random Toeplitz matrices with band structure. Electron. Commun. Probab. 14, 412–421 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kopp, G.S., Koloğlu, M., Miller, S.J., Strauch, F., Xiong, W.: The limiting spectral measure for ensembles of symmetric block circulant matrices. arXiv version.
  26. 26.
    Krbalek, M., Seba, P.: The statistical properties of the city transport in Cuernavaca (Mexico) and Random matrix ensembles. J. Phys. A, Math. Gen. 33, L229–L234 (2000) CrossRefzbMATHGoogle Scholar
  27. 27.
    Ledoux, M.: A recursion formula for the moments of the Gaussian orthogonal ensemble. Ann. IHP 45(3), 754–769 (2009) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Liu, D.-Z., Wang, Z.-D.: Limit distribution of eigenvalues for random Hankel and Toeplitz band matrices, to appear in J Theor Probab.
  29. 29.
    Massey, A., Miller, S.J., Sinsheimer, J.: Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices. J. Theor. Probab. 20(3), 637–662 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Meckes, M.: The spectra of random abelian G-circulant matrices, preprint.
  31. 31.
    Miller, S.J., Novikoff, T., Sabelli, A.: The distribution of the second largest eigenvalue in families of random regular graphs. Exp. Math. 17(2), 231–244 (2008) CrossRefzbMATHGoogle Scholar
  32. 32.
    McKay, B.: The expected eigenvalue distribution of a large regular graph. Linear Algebra Appl. 40, 203–216 (1981) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Montgomery, H.: The pair correlation of zeros of the zeta function. In: Analytic Number Theory. Proceedings of Symposia in Pure Mathematics, vol. 24, pp. 181–193. AMS, Providence (1973) CrossRefGoogle Scholar
  34. 34.
    Stein, E., Shakarchi, R.: Fourier Analysis: An Introduction. Princeton University Press, Princeton (2003) Google Scholar
  35. 35.
    Stein, E., Shakarchi, R.: Complex Analysis. Princeton University Press, Princeton (2003) zbMATHGoogle Scholar
  36. 36.
    Takacs, L.: A moment convergence theorem. Am. Math. Mon. 98(8), 742–746 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tao, T., Vu, V.: From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices. Bull. Am. Math. Soc. 46, 377–396 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Tao, T., Vu, V.: Random matrices: Universality of local eigenvalue statistics up to the edge, preprint.
  39. 39.
    Wigner, E.: On the statistical distribution of the widths and spacings of nuclear resonance levels. Proc. Camb. Philol. Soc. 47, 790–798 (1951) CrossRefzbMATHGoogle Scholar
  40. 40.
    Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 2(62), 548–564 (1955) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wigner, E.: Statistical properties of real symmetric matrices. In: Canadian Mathematical Congress Proceedings, pp. 174–184. University of Toronto Press, Toronto (1957) Google Scholar
  42. 42.
    Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. II. Ann. Math. Ser. 2 65, 203–207 (1957) MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wigner, E.: On the distribution of the roots of certain symmetric matrices. Ann. Math. Ser. 2 67, 325–327 (1958) MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Wishart, J.: The generalized product moment distribution in samples from a normal multivariate population. Biometrika A 20, 32–52 (1928) Google Scholar
  45. 45.
    Xiong, W.: The limiting spectral measure for the ensemble of generalized real symmetric period m-circulant matrices, senior thesis (advisor S.J. Miller), Williams College (2011) Google Scholar
  46. 46.
    Zvonkin, A.: Matrix integrals and map enumeration: An accessible introduction. Math. Comput. Model. 26(8–10), 281–304 (1997) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Murat Koloğlu
    • 1
  • Gene S. Kopp
    • 2
  • Steven J. Miller
    • 1
    Email author
  1. 1.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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