Random matrix representations of critical statistics

This article examines two random matrix ensembles that are useful for describing critical spectral statistics in systems with multifractal eigenfunction statistics: the Gaussian non-invariant ensemble and the invariant random matrix ensemble. It first provides an overview of non-invariant Gaussian random matrix theory (RMT) with multifractal eigenvectors and invariant random matrix theory (RMT) with log-square confinement before discussing self-unfolding and not self-unfolding in invariant RMT. It then considers a non-trivial unfolding and how it changes the form of the spectral correlations, along with the appearance of a ghost correlation dip in RMT and Hawking radiation. It also describes the correspondence between invariant and non-invariant ensembles and concludes by introducing a simple field theory in 1+1 dimensions which reproduces level statistics of both of the two random matrix models and the classical Wigner-Dyson spectral statistics in the framework of the unified formalism of Luttinger liquid.