  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.
