Distribution of Eigenvalues for the Ensemble of Real Symmetric Toeplitz Matrices

Abstract

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variable from a fixed probability distributionpof 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 pThis 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 eigenvalues 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.