Resistance distributions of the random resistor network near the percolation threshold

We study the generalized resistive susceptibility, ()x[exp[-1/22R(xx)]]av where [] av denotes an average over all configurations of clusters with weight appropriate to bond percolation, R(x,x) is the resistance between nodes x and x when occupied bonds are assigned unit resistance and vacant bonds infinite resistance. For bond concentration p near the percolation threshold at pc, we give a simple calculation in 6- dimensions of () from which we obtain the distribution of resistances between two randomly chosen terminals. From () we also obtain the qth-order resistive susceptibility (q)x[(x,x) R(x,x)q]av, where (x,x) is an indicator function which is unity when sites x and x are connected and is zero otherwise. In the latter case, (x,x)R(x,x)q is interpreted to be zero. Our universal amplitude ratios, qlimppc(q) ((0))q-1((1))q, reproduce previous results and agree beautifully with our new low-concentration series results. We give a simple numerical approximation for the (q)s in all dimensions. The relation of the scaling function for () with that for the susceptibility of the diluted xy model for p near pc is discussed.