Distribution of the logarithms of currents in percolating resistor networks. I. Theory

The distribution of currents, ib, in the bonds b of a randomly diluted resistor network at the percolation threshold is investigated through a study of the moments of the distribution P^(i2) and the moments of the distribution P(y), where y=-lnib2. For q>qc the qth moment of P^(i2), Mq (i.e., the average of i2q), scales as a power law of the system size L, with a multifractal (noise) exponent ψ(q)-ψ(0). Numerical data indicate that qc is negative, but becomes small for large L. Assuming that all derivatives ψ(q) exist at q=0+, we show that for positive integer k the kth moment, μk, of P(y) is given by μk=(α0 lnL)k{1+[kC1+1/2k(k-1)D1] (lnL)-1+O[(lnL)-2]}, where α0 and D1 (but not C1) are universal constants obtained from ψ(q). A second independent argument, requiring an assumed analyticity property of the asymptotic multifractal function, f(α), leads to the same equation for all k. This latter argument allows us to include finite-size corrections to f(α), which are of order (lnL)-1. These corrections must be taken into account in interpreting numerical studies of P(y). We note that data for P(-lni2) seem to show power-law behavior as a function of i2 for small i. Values of the exponents are directly related to the values of qc, and the numerical data in two dimensions indicate it to be small (but probably nonzero). We suggest, in view of the nature of the finite-size corrections in the expression for μk, that the asymptotic regime may not have been reached in the numerical work. For d=6 we find that Mq(L)∼(lnL)θ(q), where θ(q)→ for q→qc=-1/2.