Multifractal localization

We discuss scaling with many length scale exponents (multifractality) mainly in the context of localized excitations in a random (or fractal) environment. Many distributions of physical properties on percolating clusters may be quite sharp when expressed in terms of the chemical (shortest path) distance, but become very broad when expressed in terms of the real-space Euclidean displacement. In many cases this reasoning provides a simple physical picture of multifractality. Specifically, we discuss the distributions of localized wave functions on percolating clusters, on length scales larger than the localization length. For deep impurity states, we find multifractality with new crossover features. For states inside the band, for fractons and for random walks, the situation is more complicated and we analyze several recent relevant numerical studies.