  We discuss the limiting spectral density of real symmetric random matrices.
Other than in standard random matrix theory the upper diagonal entries are not
assumed to be independent, but we will fill them with the entries of a
stochastic process. Under assumptions on this process, which are satisfied,
e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures
on finite state spaces, or by Gaussian Markov processes, we show that the
limiting spectral distribution depends on the way the matrix is filled with the
stochastic process. If the filling is in a certain way compatible with the
symmetry condition on the matrix, the limiting law of the empirical eigenvalue
distribution is the well known semi-circle law. For other fillings we show that
the semi-circle law cannot be the limiting spectral density.
