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Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

Author

Listed:
  • Steven Jackson

    (Williams College)

  • Steven J. Miller

    (Williams College)

  • Thuy Pham

    (Williams College)

Abstract

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work (Bryc et al., Ann. Probab. 34(1):1–38, 2006; Hammond and Miller, J. Theor. Probab. 18(3):537–566, 2005) showed that the spectral measures (the density of normalized eigenvalues) converge almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed (see Massey et al., J. Theor. Probab. 20(3):637–662, 2007) by making the first row palindromic. In this paper we study the case where there is more than one palindrome in the first row of real symmetric Toeplitz matrices. Using the method of moments and an analysis of the resulting Diophantine equations, we show that the spectral measures converge almost surely to a universal distribution. Assuming a conjecture on the resulting Diophantine sums (which is supported by numerics and some theoretical arguments), we prove that the limiting distribution has a fatter tail than any previously seen limiting spectral measure.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:2:d:10.1007_s10959-010-0311-x
    DOI: 10.1007/s10959-010-0311-x
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    References listed on IDEAS

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    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.
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