Random matrix representations of critical statistics

We consider two random matrix ensembles which are relevant for describing critical spectral statistics in systems with multifractal eigenfunction statistics. One of them is the Gaussian non-invariant ensemble which eigenfunction statistics is multifractal, while the other is the invariant random matrix ensemble with a shallow, log-square confinement potential. We demonstrate a close correspondence between the spectral as well as eigenfuncton statistics of these random matrix ensembles and those of the random tight-binding Hamiltonian in the point of the Anderson localization transition in three dimensions. Finally we present a simple field theory in 1+1 dimensions which reproduces level statistics of both of these random matrix models and the classical Wigner-Dyson spectral statistics in the framework of the unified formalism of Luttinger liquid. We show that the (equal-time) density correlations in both random matrix models correspond to the finite-temperature density correlations of the Luttinger liquid. We show that spectral correlations in the invariant ensemble with log-square confinement correspond to a Luttinger liquid in the 1+1 curved space-time with the event horizon, similar to the phonon density correlations in the sonic analogy of Hawking radiation in black holes.