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

Abstract

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 $Y_q^{typ}(L)=e^{\overline{\ln Y_q(L)}}\sim L^{-{\tau }_{typ}(q)}$ that involve the typical multifractal spectrum ${\tau }_{typ}(q)$, the rescaled variable $y=Y_q(L)/Y_q^{typ}(L)$ is distributed with a scale-invariant distribution presenting the power-law tail $1/y^{1+{\beta }_q}$, so that the disorder-averaged I.P.R. $\overline{Y_q(L)}\sim L^{-{\tau }_{av}(q)}$ have multifractal exponents ${\tau }_{av}(q)$ that differ from the typical ones ${\tau }_{typ}(q)$ whenever ${\beta }_q<1$; (b) the tail exponents ${\beta }_q$ and the multifractal exponents are related by the relation ${\beta }_q{\tau }_{typ}(q)={\tau }_{av}(q{\beta }_q)$. 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 ${\tau }_{typ}(q)$ represents the velocity, whereas the tail exponent ${\beta }_q$ 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.

v2=final version, 12 pages

References in corpus (13)

• Anderson Transitions
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Multifractal analysis with the probability density function at the three-dimensional Anderson transition
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model I: Symmetry relation under typical averaging
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model II: Symmetry relation under ensemble averaging
• Conformal Random Geometry
• Monte Carlo Study of Phase Transitions in the Bond-Diluted 3D 4-State Potts Model
• Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
• Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions
• Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices
• On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces
• Directed polymer in a random medium of dimension 1+3 : multifractal properties at the localization/delocalization transition
• On the multifractal statistics of the local order parameter at random critical points : application to wetting transitions with disorder

Cited by in corpus (2)

• Multifractal wave functions of simple quantum maps
• A critical Dyson hierarchical model for the Anderson localization transition