## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

3-1-1990

#### Publication Source

Physical Review B

#### Volume

41

#### Issue

7

#### Start Page

4610

#### Last Page

4618

#### DOI

10.1103/PhysRevB.41.4610

#### Abstract

We study the generalized resistive susceptibility, χ(λ)≡Σ_{x’}[exp[-1/2λ^{2}*R*(**xx**’)]]_{av} where [ ]_{av} denotes an average over all configurations of clusters with weight appropriate to bond percolation, *R*(**x**,**x**’) is the resistance between nodes **x** and **x**’ when occupied bonds are assigned unit resistance and vacant bonds infinite resistance. For bond concentration *p* near the percolation threshold at *p _{c}*, we give a simple calculation in 6-ε dimensions of χ(λ) from which we obtain the distribution of resistances between two randomly chosen terminals. From χ(λ) we also obtain the

*q*th-order resistive susceptibility χ

^{(q)}≡Σ

_{x’}[

*ν*(

**x**,

**x**’)

*R*(

**x**,

**x**’)

^{q}]

_{av}, where

*ν*(

**x**,

**x**’) is an indicator function which is unity when sites

**x**and

**x**’ are connected and is zero otherwise. In the latter case,

*ν*(

**x**,

**x**’)

*R*(

**x**,

**x**’)

^{q}is interpreted to be zero. Our universal amplitude ratios,

*ρ*

_{q}≡lim

_{p→pc}χ

^{(q)}(χ

^{(0)})

^{q−1}(χ

^{(1)})

^{q}, reproduce previous results and agree beautifully with our new low-concentration series results. We give a simple numerical approximation for the χ

^{(q)}’s in all dimensions. The relation of the scaling function for χ(λ) with that for the susceptibility of the diluted

*xy*model for

*p*near

*p*

_{c}is discussed.

#### Recommended Citation

Harris, A.,
Meir, Y.,
&
Aharony, A.
(1990).
Resistance Distributions of the Random Resistor Network Near the Percolation Threshold.
*Physical Review B,*
*41*
(7),
4610-4618.
http://dx.doi.org/10.1103/PhysRevB.41.4610

**Date Posted:** 12 August 2015

This document has been peer reviewed.

## Comments

At the time of publication, author A. Brooks Harris was affiliated with Tel Aviv University. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania.