Levy Diffusion and Classes of Universal Parametric Correlations

A general formulation of translationally invariant, parametrically correlated random matrix ensembles, is used to classify universality in correlation functions. Surprisingly, the range of possible physical systems is bounded, and can be labeled by a parameter $\alpha\in (0,2]$, in a manner analogous to L\'evy diffusion. Universality is obtained after scaling by the (anomalous) diffusion constant $D_\alpha$ (the usual scaling is divergent for $\alpha<2$). For each $\alpha$, correlation functions are universal, and distinct. The previous results in the literature correspond to the limiting case of superdiffusion, $\alpha=2$.