In this review of the QHE, Steve Girvin makes the following statement (bottom of pg. 6, beginning of Sec. 1.1.1):

As one learns in the study of scaling in the localization transition, resistivity (which is what theorists calculate) and resistance (which is what experimental- ists measure) for classical systems (in the shape of a [d-dimensional] hypercube) of size L are related by [12,13]

$$ R = \rho L^{(2-d)} $$

where $R$ is the *resistance* of the hypercube and $\rho$ its *resistivity*. For $d=2$ it is clear that the resistance and the resistivity of the sample are the same, and as Girvin goes on to say, this fact is crucial for the universality of the quantization of the resistivity plateaus in the QHE.

My question is: **Is there a simple physical explanation of the above relationship?** Girvin cites two papers. Both are RMP: "Disordered Electronic Systems" by Lee and Ramakrishnan and "Continuous Quantum Phase Transitions" by Sondhi et al. I am hoping that someone can spare me the effort of going through these two papers by providing an answer.

This post imported from StackExchange Physics at 2014-04-01 16:33 (UCT), posted by SE-user user346