Eigenvalue Spacing Distribution for the Ensemble of Real Symmetric Toeplitz Matrices

Details Eigenvalue Spacing Distribution for the Ensemble of Real Symmetric Toeplitz Matrices Eigenvalue Spacing Distribution for the Ensemble of Real Symmetric Toeplitz Matrices

2003-12-10

arxiv.org/abs/math/0312215v1

Journal of Theoretical Probability, Vol. 18 (2005), no. 3, 537 - 566

Mathematics

Probability

24 pages, 3 figures

Scientific paper

10.1007/s10959-005-3518-5

Consider the ensemble of Real Symmetric Toeplitz Matrices, each entry iidrv from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. The limiting spectral measure (the density of normalized eigenvalues) converges weakly to a new universal distribution with unbounded support, independent of p. This distribution's moments are almost those of the Gaussian's; the deficit may be interpreted in terms of Diophantine obstructions. With a little more work, we obtain almost sure convergence. An investigation of spacings between adjacent normalized eigenvalues looks Poissonian, and not GOE.

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