  We study spectral densities for systems on lattices, which, at a phase
transition display, power-law spatial correlations. Constructing the spatial
correlation matrix we prove that its eigenvalue density shows a power law that
can be derived from the spatial correlations. In practice time series are short
in the sense that they are either not stationary over long time intervals or
not available over long time intervals. Also we usually do not have time series
for all variables available. We shall make numerical simulations on a
two-dimensional Ising model with the usual Metropolis algorithm as time
evolution. Using all spins on a grid with periodic boundary conditions we find
a power law, that is, for large grids, compatible with the analytic result. We
still find a power law even if we choose a fairly small subset of grid points
at random. The exponents of the power laws will be smaller under such
circumstances. For very short time series leading to singular correlation
matrices we use a recently developed technique to lift the degeneracy at zero
in the spectrum and find a significant signature of critical behavior even in
this case as compared to high temperature results which tend to those of random
matrix models.
