Renormalization-group treatment of the random resistor network in 6-ε dimensions

Abstract

We consider a hypercubic lattice in which neighboring points are connected by resistances which assume independently the random values σ_>^-1 and σ_<^-1 with respective probabilities p and 1-p. For σ_<=0 the lattice is viewed as consisting of irreducible nodes connected by chains of path length L. This geometrical length is distinct from the characteristic length L_r which sets a scale of resistance in the random network or L_m which sets a scale of effective exchange in a dilute magnet. Near the percolation concentration p_c one sets L~|p-p_c|^-ζ, L_r~|p-p_c|-^ζr and L_m~|p-p_c|^-ζm. Stephen and Grest (SG) have already shown that ζ_m=1+o(ε²) for spatial dimensionality d=6-ε. Here we show in a way similar to SG that ζ_r=1+o(ε²). Thus it is possible that ζ_m=ζ_r=1 for a continuous range of d below 6. However, increasing evidence suggests that this equality does not hold for d<4, and in particular a calculation in 1+ε dimensions analogous to that of SG for ζ_m does not seem possible.