Towards rigorous analysis of the Levitov-Mirlin-Evers recursion

TL;DR

This paper provides a rigorous asymptotic analysis of the Levitov-Mirlin-Evers recursion related to eigenvector multifractality in critical powerlaw random band matrices, highlighting phase transition phenomena and open problems.

Contribution

It adapts methods from multiplicative cascades and branching random walks to analyze the LME recursion, revealing phase transition behavior.

Findings

LME recursion exhibits a phase transition likely to be a freezing transition.

Methods from cascade and branching walk theories can be adapted to analyze LME recursion.

Open problems identified for further rigorous analysis.

Abstract

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