Renormalization-Group Treatment of the Random Resistor Network in 6−ε Dimensions
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 Lr which sets a scale of resistance in the random network or Lm which sets a scale of effective exchange in a dilute magnet. Near the percolation concentration pc one sets L~|p−pc|−ζ, Lr~|p−pc|−ζr and Lm~|p−pc|−ζm. Stephen and Grest (SG) have already shown that ζm=1+o(ε2) for spatial dimensionality d=6−ε. Here we show in a way similar to SG that ζr=1+o(ε2). Thus it is possible that ζm=ζr=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 ζm does not seem possible.