Towards rigorous analysis of the LevitovMirlinEvers recursion
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 (2000 Phys. Rev. B 62 7920) and earlier works by Levitov (1990 Phys. Rev. Lett. 64 547, 1999 Ann. Phys. 8 697706) and is aimed to describe the ensuing multifractality of the eigenvectors of such matrices. We point out both similarities and dissimilarities between the LME recursion and 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.
 Publication:

Nonlinearity
 Pub Date:
 December 2016
 DOI:
 10.1088/09517715/29/12/3871
 arXiv:
 arXiv:1509.01366
 Bibcode:
 2016Nonli..29.3871F
 Keywords:

 Mathematical Physics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Mathematics  Probability
 EPrint:
 doi:10.1088/09517715/29/12/3871