John Sinsheimer

TL;DR: In this article, it was shown that for the case where the first row is a palindrome, the limiting spectral measure converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian.

TL;DR: In this paper, it was shown that for real symmetric palindromic Toeplitz matrices, where the first row is a palindrome, the limiting spectral measure converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian.

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 Gaussian. The deviations from Gauss-ian 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), and the resulting spectral measures converge (weakly and almost surely) to the Gaussian.