Towards rigorous analysis of the Levitov–Mirlin–Evers recursion

Yan Fyodorov, Antti Kupiainen, Christian Webb

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)


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. 8 697–706) 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.
Original languageEnglish
Pages (from-to)3871-3896
Issue number29
Publication statusPublished - 2016
MoE publication typeA1 Journal article-refereed

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