  Consider the ensembles of real symmetric Toeplitz matrices and real symmetric
Hankel matrices whose entries are i.i.d. random variables chosen from a fixed
probability distribution p of mean 0, variance 1, and finite higher moments.
Previous work on real symmetric Toeplitz matrices shows that the spectral
measures, or densities of normalized eigenvalues, converge almost surely to a
universal near-Gaussian distribution, while previous work on real symmetric
Hankel matrices shows that the spectral measures converge almost surely to a
universal non-unimodal distribution. Real symmetric Toeplitz matrices are
constant along the diagonals, while real symmetric Hankel matrices are constant
along the skew diagonals. We generalize the Toeplitz and Hankel matrices to
study matrices that are constant along some curve described by a real-valued
bivariate polynomial. Using the Method of Moments and an analysis of the
resulting Diophantine equations, we show that the spectral measures associated
with linear bivariate polynomials converge in probability and almost surely to
universal non-semicircular distributions. We prove that these limiting
distributions approach the semicircle in the limit of large values of the
polynomial coefficients. We then prove that the spectral measures associated
with the sum or difference of any two real-valued polynomials with different
degrees converge in probability and almost surely to a universal semicircular
distribution.
