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This paper aims to develop a rigorous asymptotic analysis of an approximaterenormalization group recursion for inverse participation ratios $P_q$ ofcritical powerlaw random band matrices. The recursion goes back to the work byMirlin and Evers [37] and earlier works by Levitov [32, 33] and is aimed todescribe the ensuing multifractality of the eigenvectors of such matrices. Wepoint out both similarities and dissimilarities of LME recursion to thoseappearing in the theory of multiplicative cascades and branching random walksand show that the methods developed in those fields can be adapted to thepresent case. In particular the LME recursion is shown to exhibit a phasetransition, which we expect is a freezing transition, where the role oftemperature is played by the exponent $q$. However, the LME recursion hasfeatures that make its rigorous analysis considerably harder and we point outseveral open problems for further study

Towards rigorous analysis of the Levitov-Mirlin-Evers recursion