Universal correlations in random matrices and one-dimensional particles with long-range interactions in a confinement potential

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Abstract
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 Brézin 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.

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