Department of Physics Papers

Document Type

Journal Article

Date of this Version

12-1-1987

Publication Source

Physical Review B

Volume

36

Issue

16

Start Page

8752

Last Page

8764

DOI

10.1103/PhysRevB.36.8752

Abstract

The moments τk of typical diffusion times for ‘‘blind’’ and ‘‘myopic’’ ants on an arbitrary cluster are expressed exactly in terms of resistive correlations for the associated resistor network. For a diluted lattice at bond concentration p, we introduce ‘‘diffusive’’ susceptibilities χk(p) as the average over clusters of τk. For p→pc, where pc is the percolation threshold, χk(p) diverges as |pc-p|k. We show that γk=kΔτ-β with Δτ=β+γ+ζ, where β and γ are percolation exponents and ζ is the resistance scaling exponent. Our analysis provides the first analytic demonstration that the leading exponents γk are the same for a wide class of models, including the two types of ants as special cases, although corrections to scaling are larger for the myopic ant than for the blind one. This class of models includes that for dilute spin waves in Heisenberg ferromagnets. Exact enumerations allow us to study universal amplitude ratios (at p=pck+1χk−1k2 as a function of continuous spatial dimension d. For d>6 these ratios assume a constant value which for k=2 agrees with the exact result for the Cayley tree. The χk have the scaling properties predicted by Gefen, Aharony, and Alexander [Phys. Rev. Lett. 50, 77 (1983)] for anomalous diffusion.

Comments

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

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Date Posted: 12 August 2015

This document has been peer reviewed.