Universal Correlations in the random matrices and 1D particles with long range interactions in a confinement potential

TL;DR

This paper investigates universal eigenvalue correlations in large random matrices and applies these findings to the ground state and phase transition phenomena of 1D particles with long-range interactions in a confinement potential.

Contribution

It demonstrates the universality of eigenvalue correlations and derives the exact ground state and correlation functions for 1D particles with long-range interactions.

Findings

Eigenvalue correlations support universality as proposed by Brézin and Zee.

Exact ground state of 1D particles with long-range interactions obtained.

Identification of a phase transition with explicit density-density correlations.

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 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.