  This paper aims to develop a rigorous asymptotic analysis of an approximate
renormalization group recursion for inverse participation ratios $P_q$ of
critical powerlaw random band matrices. The recursion goes back to the work by
Mirlin and Evers [37] and earlier works by Levitov [32, 33] and is aimed to
describe the ensuing multifractality of the eigenvectors of such matrices. We
point out both similarities and dissimilarities of LME recursion to those
appearing in the theory of multiplicative cascades and branching random walks
and show that the methods developed in those fields can be adapted to the
present case. In particular the LME recursion is shown to exhibit a phase
transition, which we expect is a freezing transition, where the role of
temperature is played by the exponent $q$. However, the LME recursion has
features that make its rigorous analysis considerably harder and we point out
several open problems for further study
