Abstract
Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variable from a fixed probability distributionpof mean 0,variance 1, and finite moments of all order. The limiting spectral measure (the density of normalized eigenvalues) converges weakly to a new universal distribution with unbounded support, independent of pThis distribution’s moments are almost those of the Gaussian’s, and the deficit may be interpreted in terms of obstructions to Diophantine equations; the unbounded support follows from a nice application of the Central Limit Theorem. With a little more work, we obtain almost sure convergence. An investigation of spacings between adjacent normalized eigenvalues looks Poissonian, and not GOE. A related ensemble (real symmetric palindromic Toeplitz matrices) appears to have no Diophantine obstructions, and the limiting spectral measure’s first nine moments can be shown to agree with those of the Gaussian; this will be considered in greater detail in a future paper.
Similar content being viewed by others
References
Z. Bai (1999) ArticleTitleMethodologies in spectral analysis of largedimensional random matrices A review. Stat. Sinica 9 611–677
Bose, A., Chatterjee, S., and Gangopadhyay, S. (2003).Limiting spectral distributions of large dimensional random matrices
A. Bose J. Mitra (2002) ArticleTitleLimiting spectral distribution of a special circulant Statist.Probab. Lett 60 IssueID1 111–120 Occurrence Handle10.1016/S0167-7152(02)00289-4
Bryc, W., Dembo, A., and Jiang, T.Spectral Measure of Large Randm Hankel, Markov and Toeplitz Matrices, preprint
Jakobson, D., Miller, S. D., Rivin, I., and Rudnick, Z. (1999). Spacings for Regular Graphs, Emerging applications of number theory (Minneapolis, MN, 1996), IMA Vol. Math. Appl., 109, Springer, New York, pp. 317–327
M. Loéve (1963) Probability Theory D. Van Nostrand Company, Inc. Princeton
B. McKay (1981) ArticleTitleThe expected eigenvalue distribution of a large regular graph Linear Algebra Appl 40 203–216 Occurrence Handle10.1016/0024-3795(81)90150-6
M. Mehta (1991) Random Matrices EditionNumber2nd ed Academic Press Boston
Miller, S. J., and Sinsheimer, J. of Eigenvalues for the Ensemble of Real Symmetric Palindromic Toeplitz Matrices, preprint.
E. Wigner (1957) ArticleTitleOn the distribution of the roots of certain symmetric matrices Ann. Math 67 325–327
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hammond, C., Miller, S.J. Distribution of Eigenvalues for the Ensemble of Real Symmetric Toeplitz Matrices. J Theor Probab 18, 537–566 (2005). https://doi.org/10.1007/s10959-005-3518-5
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10959-005-3518-5