Resistance distributions of the random resistor network near the percolation threshold

We study the generalized resistive susceptibility, χ(λ)≡${\mathrm{\Sigma }}_{\mathbf{x}\mathcal{’}}$[exp[-1/2${\lambda }^2$R(xx ’)]${]}_{\mathit{av}}$ where [ ${]}_{\mathit{av}}$ 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 ${\mathit{p}}_{\mathit{c}}$, 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 ${\mathrm{\chi }}^{\left(\mathit{q}\right)}$≡${\mathrm{\Sigma }}_{\mathbf{x}\mathcal{’}}$[ν(x,x ’) R(x,x ’ ${)}^{\mathit{q}}$ ${]}_{\mathit{a}\mathit{v}}$, 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 ’ ${)}^{\mathit{q}}$ is interpreted to be zero. Our universal amplitude ratios, ${\mathrm{\rho }}_{\mathit{q}}$≡${\mathit{lim}}_{\mathit{p}\to \mathit{p}\mathit{c}}$ ${\mathrm{\chi }}^{\left(\mathit{q}\right)}$ (${\mathrm{\chi }}^{\mathrm{}\left(0\right)\mathrm{}}$ ${)}^{\mathit{q}\mathrm{-}1}$(${\mathrm{\chi }}^{\mathrm{}\left(1\right)\mathrm{}}$ ${)}^{\mathit{q}}$, reproduce previous results and agree beautifully with our new low-concentration series results. We give a simple numerical approximation for the ${\mathrm{\chi }}^{\left(\mathit{q}\right)}$’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 ${\mathit{p}}_{\mathit{c}}$ is discussed.