  We study the correlations between eigenvalues of the large random matrices by
a renormalization group approach. The results strongly support the universality
of the correlations proposed by Br\'ezin and Zee. Then we apply the results to
the ground state of the 1D particles with long range interactions in a
confinement potential. We obtain the exact ground state. We also show the
existence of a transition similar to a phase separation. Before and after the
transition, we obtain the density-density correlation explicitly. The
correlation shows nontrivial universal behavior.
