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Random cascade models of multifractality : real-space renormalization and travelling-waves

Random multifractals occur in particular at critical points of disordered systems. For Anderson localization transitions, Mirlin and Evers [PRB 62,7920 (2000)] have proposed the following scenario (a) the Inverse Participation Ratios (I.P.R.) $Y_q(L)$ display the following fluctuations between the disordered samples of linear size $L$ : with respect to the typical value $Y^{typ}_q(L) = e^{\bar{\ln Y_q(L)}} \sim L^{- τ_{typ}(q)} $ that involve the typical multifractal spectrum $τ_{typ}(q)$, the rescaled variable $y=Y_q(L)/Y^{typ}_q(L) $ is distributed with a scale-invariant distribution presenting the power-law tail $1/y^{1+β_q}$, so that the disorder-averaged I.P.R. $\bar{Y_q(L)} \sim L^{- τ_{av}(q)} $ have multifractal exponents $τ_{av}(q) $ that differ from the typical ones $τ_{typ}(q)$ whenever $β_q<1$; (b) the tail exponents $β_q$ and the multifractal exponents are related by the relation $β_q τ_{typ}(q)=τ_{av}(q β_q)$. Here we show that this scenario can be understood by considering the real-space renormalization equations satisfied by the I.P.R. For the simplest multifractals described by random cascades, these renormalization equations are formally similar to the recursion relations for disordered models defined on Cayley trees and they admit travelling-wave solutions for the variable $(\ln Y_q)$ in the effective time $t_{eff}=\ln L$ : the exponent $τ_{typ}(q)$ represents the velocity, whereas the tail exponent $β_q$ represents the usual exponential decay of the travelling-wave tail. In addition, we obtain that the relation (b) above can be obtained as a self-consistency condition from the self-similarity of the multifractal spectrum at all scales. Our conclusion is thus that the Mirlin-Evers scenario should apply to other types of random critical points, and even to random multifractals occurring in other fields.