Random cascade models of multifractality: real-space renormalization and travellingwaves

Abstract

Random multifractals occur in particular at critical points of disordered systems.For Anderson localization transitions, Mirlin and Evers (2000 Phys. Rev. B 627920) have proposed the following scenario: (a) the inverse participation ratios (IPR)Yq(L) display the following fluctuations between the disordered samples of linear sizeL: with respect to the typical value that involve the typical multifractal spectrumτtyp(q), the rescaledvariable y = Yq(L)/Yqtyp(L) is distributed with a scale-invariant distribution presenting the power-law tail1/y1 + βq, so the disorder-averaged IPR have multifractal exponents τav(q) that differ from the typical onesτtyp(q) wheneverβq < 1; (b) the tailexponents βq and the multifractal exponents are related by the relationβqτtyp(q) = τav(qβq). Here we show that this scenario can be understood by considering thereal-space renormalization equations satisfied by the IPR. For the simplestmultifractals described in terms of random cascades, these renormalizationequations are formally similar to the recursion relations for disordered modelsdefined on Cayley trees and they admit travelling-wave solutions for the variable(lnYq) in the effectivetime teff = lnL: theexponent τtyp(q) represents the velocity, whereas the tail exponentβq represents the usual exponential decay of the travelling-wave tail. In addition, we obtainthat the relation in (b) above can be obtained as a self-consistency condition from theself-similarity of the multifractal spectrum at all scales. Our conclusion is thus that theMirlin–Evers scenario should apply to random critical points of other types, and even torandom multifractals occurring in other fields.