Liouville Theory as a Model for Prelocalized States in Disordered Conductors

It is established that the distribution of the zero energy eigenfunctions of ( $2+1$)-dimensional Dirac electrons in a random gauge potential is described by the Liouville model. This model has a line of critical points parametrized by the strength of disorder and the scaling dimensions of the inverse participation ratios coincide with the dimensions obtained in the conventional localization theory. From this fact we conclude that the renormalization group trajectory of the latter theory lies in the vicinity of the line of critical points of the Liouville model.