  In this article, addressing large $n$ systems, we report that in numerous
systems hosting long and short range interactions, multiple correlation lengths
may appear. The largest correlation lengths often monotonically increase with
temperature and diverge in the high temperature limit. Notwithstanding, the
magnitude of the correlations themselves decreases with increasing temperature.
We examine correlation function in the presence of competing interactions of
long and short ranges. The behavior of the correlation and modulation lengths
as a function of temperature provides us with selection rules on the possible
underlying microscopic interactions. As a concrete example of these notions, we
consider the correlations in a system of screened Coulomb interactions
coexisting with attractive short range interactions.
