Resistance Distributions of the Random Resistor Network Near the Percolation Threshold

We study the generalized resistive susceptibility, χ(λ)≡Σx’[exp[-1/2λ2R(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-ε dimensions of χ(λ) from which we obtain the distribution of resistances between two randomly chosen terminals. From χ(λ) we also obtain the qth-order resistive susceptibility χ(q)≡Σx’[ν(x,x’) R(x,x’)q]av, where ν(x,x’) is an indicator function which is unity when sites x and x ’ are connected and is zero otherwise. In the latter case, ν(x,x ’)R(x,x ’)q is interpreted to be zero. Our universal amplitude ratios, ρq≡limp→pcχ(q) (χ(0))q−1(χ(1))q, reproduce previous results and agree beautifully with our new low-concentration series results. We give a simple numerical approximation for the χ(q)’s in all dimensions. The relation of the scaling function for χ(λ) with that for the susceptibility of the diluted xy model for p near pc is discussed.