## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

5-1-1987

#### Publication Source

Physical Review B

#### Volume

35

#### Issue

13

#### Start Page

6964

#### Last Page

6986

#### DOI

10.1103/PhysRevB.35.6964

#### Abstract

A formulation based on that of Stephen for randomly diluted systems near the percolation threshold is analyzed in detail. By careful consideration of various limiting procedures, a treatment of *xy* spin models and resistor networks is given which shows that previous calculations (which indicate that these systems having continuous symmetry have the same crossover exponents as the Ising model) are in error. By studying the limit wherein the energy gap goes to zero, we exhibit the mathematical mechanism which leads to qualitatively different results for *xy*-like as contrasted to Ising-like systems. A distinctive feature of the results is that there is an *infinite* sequence of crossover exponents needed to completely describe the probability distribution for R(x,x’), the resistance between sites *x* and x’. Because of the difference in symmetry between the *xy* model and the resistor network, the former has an infinite sequence of crossover exponents in addition to those of the resistor network. The first crossover exponent φ_{1}=1+ε/42 governs the scaling behavior of R(x,x’) with ‖x-x’‖≡r: [R(x,x’)]_{c}~x^{φ1/ν}, where [ ]_{c} indicates a conditional average, subject to *x* and x’ being in the same cluster, *ν* is the correlation length exponent for percolation, and *ε=6-d*, where *d* is the spatial dimensionality. We give a detailed analysis of the scaling properties of the bulk conductivity and the anomalous diffusion constant introduced by Gefen *et al*. Our results show conclusively that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, at least in high spatial dimension. We also evaluate various amplitude ratios associated with susceptibilities, χ_{n} involving the *n*th power of the resistance R(x,x’), e.g., lim_{p→pc}χ_{2}χ_{0}/χ_{1}^{2}=2[1(19ε/420)]. In an appendix we outline how the calculation can be extended to treat the diluted *m*-component spin model for *m>2*. As expected, the results for φ_{1} remain valid for *m>2*. The techniques described here have led to several recent calculations of various infinite families of exponents.

#### Recommended Citation

Harris, A.,
&
Lubensky, T. C.
(1987).
Randomly Diluted xy and Resistor Networks Near the Percolation Threshold.
*Physical Review B,*
*35*
(13),
6964-6986.
http://dx.doi.org/10.1103/PhysRevB.35.6964

**Date Posted:** 12 August 2015

This document has been peer reviewed.