## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

2-1-1978

#### Publication Source

Physical Review B

#### Volume

17

#### Issue

3

#### Start Page

1375

#### Last Page

1382

#### DOI

10.1103/PhysRevB.17.1375

#### 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(ε^{2}) for spatial dimensionality d=6−ε. Here we show in a way similar to SG that ζ_{r}=1+o(ε^{2}). 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.

#### Recommended Citation

Dasgupta, C.,
Harris, A.,
&
Lubensky, T. C.
(1978).
Renormalization-Group Treatment of the Random Resistor Network in 6−ε Dimensions.
*Physical Review B,*
*17*
(3),
1375-1382.
http://dx.doi.org/10.1103/PhysRevB.17.1375

**Date Posted:** 12 August 2015

This document has been peer reviewed.