Anomalous scaling of moments in a random resistor networks

2014 We consider a random resistor network on a square lattice at the bond percolation threshold pc = 1/2. We calculate the current distribution on the incipient infinite cluster using a Fourier accelerated conjugate gradient method. We compute the n-th moment of this distribution in both the constant current and constant voltage ensembles for lattices up to 256 x 256 in size, and examine how these moments scale with lattice size. When analysed in the conventional way, our scaling exponents agree with published results. Previous authors have assumed that these exponents are theoretically determined in the limit of large n by the singly connected bonds. This makes an implicit assumption about the order of limits for which we find no theoretical justification. We reanalyse our data by subtracting the contribution of the singly connected bonds before calculating the moments. In the limit of infinite lattice size, this can make no difference, but for our finite lattices, the apparent scaling exponents are strongly affected. The dilemma that we pose can not be resolved numerically. We discuss briefly how it might be studied theoretically. J. Physique 48 (1987) 771-779 "