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Mutifractal Analysis of Broadly Distributed Observables at Criticality

The multifractal analysis of disorder induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent $β$ and the correlation length exponent $ν$ is investigated. We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent $ν$ are obtained by the typical order parameter exponent $α_0$ and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments.