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

Physics – Condensed Matter – Disordered Systems and Neural Networks

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

v2=final version, 12 pages

Scientific paper

10.1088/1742-5468/2010/06/P06014

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^{- \tau_{typ}(q)} $ that involve the typical multifractal spectrum $\tau_{typ}(q)$, the rescaled variable $y=Y_q(L)/Y^{typ}_q(L) $ is distributed with a scale-invariant distribution presenting the power-law tail $1/y^{1+\beta_q}$, so that the disorder-averaged I.P.R. $\bar{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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Random cascade models of multifractality : real-space renormalization and travelling-waves does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Random cascade models of multifractality : real-space renormalization and travelling-waves, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Random cascade models of multifractality : real-space renormalization and travelling-waves will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-221186

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.