Distribution of Eigenvalues for the Ensemble of Real Symmetric Palindromic Toeplitz Matrices, preprint

Abstract. Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variables from a fixed probability distribution p of 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 p. This 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 eigenval-ues 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. 1