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

Dasgupta, C. ; Harris, A. B. ; Lubensky, T. C. (1978) Renormalization-group treatment of the random resistor network in 6-ε dimensions Physical Review B: Condensed Matter and Materials Physics, 17 (3). pp. 1375-1382. ISSN 1098-0121

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Official URL: http://prb.aps.org/abstract/PRB/v17/i3/p1375_1

Related URL: http://dx.doi.org/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 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|, 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 ζmr=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.

Item Type:Article
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ID Code:83435
Deposited On:20 Feb 2012 12:09
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