Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices |
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Authors: | Adam Massey Steven J Miller John Sinsheimer |
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Institution: | (1) Department of Mathematics, Brown University, Providence, RI 02912, USA;(2) Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA |
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Abstract: | Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a
fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the
density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising
from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real
symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved
for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property
and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a
consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these
Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant
matrices.
A. Massey’s current address: Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. e-mail: amassey3102@math.ucla.edu. |
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Keywords: | Random matrix theory Toeplitz matrices Distribution of eigenvalues |
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