Multifractal structure of the incipient infinite percolating cluster

By analyzing the voltage distribution of a random resistor network, we show that the backbone of the percolating cluster can be partitioned into an infinity of subsets, each one characterized by a fixed value of x≡lnV/ln$V_{\max }$, where V is the voltage across each bond and $V_{\max }$ is its maximum value. Each subset is characterized by a distinct value of the fractal dimension φ(x), and as a consequence an infinite set of order parameters is required to describe the backbone structure. A new scaling approach and a real-space renormalization-group treatment are presented to treat the novel aspects of this problem. The mechanism for multifractality based on an underlying multiplicative process is illustrated on a hierarchical model.