Delocalization transition in unconventional random matrices
Critical Power-Law Banded Random Matrices (CRMT) are a powerful toy model which allows one to study universal features of the Anderson transition. Wave functions of CRMT are multifractal and the fractal dimensions are controlled by the matrix band- width. The \sigma-model can be successfully applied to explore CRMT with a large band-width (a weak multifractality regime). Recently, an alternative method of the virial expansion (VE) has been suggested to describe the opposite case of RMT with the small band-width. VE is an expansion in the number of localized states weakly interacting in the energy space due to the matrix off-diagonal elements. VE answers the question whether such weak interaction can lead to the criticality (in the strong multifractality regime) and to the delocalization. We briefly review VE, and use it to study spectral properties and statistics of wave-functions of CRMT. Analytical results will be compared with numerical ones.