UNIVERSAL CORRELATIONS IN RANDOM MATRICES AND ONE-DIMENSIONAL PARTICLES WITH LONG-RANGE INTERACTIONS IN A CONFINEMENT POTENTIAL

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 Brezin and Zee. Then we apply the results to the ground state of the one-dimensional 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.