Random Matrix Model and the Calogero-Sutherland Model: A Novel Current-Density Mapping

We investigate the relation between the invariant correlators of random matrix theory and correlators of the integrable one-dimensional systems. Starting from the relation between correlators for the coupling strengths $\lambda =\frac{1}{2}, 1, \mathrm{and} 2$, we explore the local current-density mapping applicable to arbitrary $\lambda$ including irrational values, which results from the novel structure of the Calogero-Sutherland model. We find an interesting and novel relationship between equal time current and density correlations for any coupling, which exist in addition to the usual Ward identities for this class of systems.