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

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variable 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 inter-preted 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 mea-sure’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. KEY WORDS: Random matrix theory; Toeplitz matrices; distribution of eigenvalues; diophantine obstructions; central limit theorem