## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

3-1-1993

#### Publication Source

Physical Review B

#### Volume

47

#### Issue

10

#### Start Page

5756

#### Last Page

5769

#### DOI

10.1103/PhysRevB.47.5756

#### Abstract

The distribution of currents, *i _{b}*, in the bonds

*b*of a randomly diluted resistor network at the percolation threshold is investigated through a study of the moments of the distribution

*P*

^{^}(i

^{2}) and the moments of the distribution

*P*(

*y*), where

*y*=-ln

*i*

_{b}^{2}. For

*q>q*the

_{c}*q*th moment of

*P*

^{^}(i

^{2}), M

_{q}(i.e., the average of i

^{2q}), scales as a power law of the system size

*L*, with a multifractal (noise) exponent ψ̃(

*q*)-ψ̃(0). Numerical data indicate that q

_{c}is negative, but becomes small for large

*L*. Assuming that all derivatives ψ̃(

*q*) exist at

*q*=0

^{+}, we show that for positive integer

*k*the

*k*th moment, μ

_{k}, of

*P*(

*y*) is given by

μ_{k}=(α_{0} ln*L*)^{k}{1+[*k*C_{1}+1/2*k*(*k*-1)*D*_{1}] (ln*L*)^{−1}+*O*[(ln*L*)^{−2}]},

where α_{0} and *D _{1}* (but not C

_{1}) are

*universal*constants obtained from ψ̃(

*q*). A second independent argument, requiring an assumed analyticity property of the asymptotic multifractal function,

*f*(α), leads to the same equation for

*all*

*k*. This latter argument allows us to include finite-size corrections to

*f*(α), which are of order (ln

*L*)

^{−1}. These corrections must be taken into account in interpreting numerical studies of

*P*(

*y*). We note that data for

*P*(-ln

*i*) seem to show power-law behavior as a function of

^{2}*i*

^{2}for small

*i*. Values of the exponents are directly related to the values of

*q*, and the numerical data in two dimensions indicate it to be small (but probably nonzero). We suggest, in view of the nature of the finite-size corrections in the expression for μ

_{c}_{k}, that the asymptotic regime may not have been reached in the numerical work. For

*d*=6 we find that M

_{q}(

*L*)~(ln

*L*)

^{θ(q)}, where θ(

*q*)→∞ for

*q*→

*q*

_{c}=-1/2.

#### Recommended Citation

Aharony, A.,
Blumenfeld, R.,
&
Harris, A.
(1993).
Distribution of the Logarithms of Currents in Percolating Resistor Networks. I. Theory.
*Physical Review B,*
*47*
(10),
5756-5769.
http://dx.doi.org/10.1103/PhysRevB.47.5756

**Date Posted:** 12 August 2015

This document has been peer reviewed.