  Consider the ensemble of Real Symmetric Toeplitz Matrices, each entry iidrv
from a fixed probability distribution p of mean 0, variance 1, and finite
higher moments. 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; the deficit may be interpreted in terms of Diophantine
obstructions. With a little more work, we obtain almost sure convergence. An
investigation of spacings between adjacent normalized eigenvalues looks
Poissonian, and not GOE.
