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Towards rigorous analysis of the Levitov-Mirlin-Evers recursion

Research output: Contribution to journalArticle

Yan V Fyodorov, Antti Kupiainen, Christian Webb

Original languageEnglish
Article number3871
JournalNONLINEARITY
Volume29
Issue number12
DOIs
Publication statusPublished - 10 Nov 2016

King's Authors

Abstract

This paper aims to develop a rigorous asymptotic analysis of an approximate renormalization group recursion for inverse participation ratios Pq 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

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