Towards rigorous analysis of the Levitov-Mirlin-Evers 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 [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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• Anderson Transitions
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Convergence of the centered maximum of log-correlated Gaussian fields
• High values of disorder-generated multifractals and logarithmically correlated processes
• Multi-Scale Jacobi Method for Anderson Localization
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Cited by in corpus (6)

• Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach
• Resonant energy scales and local observables in the many-body localised phase
• Many Body Localization Transition in the strong disorder limit : entanglement entropy from the statistics of rare extensive resonances
• Statistical properties of the Green function in finite size for Anderson Localization models with multifractal eigenvectors
• Multifractality in the generalized Aubry-Andre quasiperiodic localization model with power-law hoppings or power-law Fourier coefficients
• Renormalization Group Analysis of the Hierarchical Anderson Model