Averaging of multifractals

Averages of multifractal moments over random structures (e.g., percolation clusters) may be dominated, in some limits, by a rare subset of these structures. This may yield atypical results, contrary to the usual expectations from average quantities. These effects are demonstrated by discussions of the distribution of currents in dilute resistor networks, of the resistance of nonlinear resistor networks, and of self-avoiding walks on percolating clusters.