TY - JOUR

T1 - Diffusion on percolating clusters

AU - Harris, A. Brooks

AU - Meir, Yigal

AU - Aharony, Amnon

PY - 1987

Y1 - 1987

N2 - The moments k 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 k(p) as the average over clusters of k. For ppc, where pc is the percolation threshold, k(p) diverges as &. We show that k=k- with =++, where and are percolation exponents and is the resistance scaling exponent. Our analysis provides the first analytic demonstration that the leading exponents k 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=pc)k+1k-1/k2 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 k have the scaling properties predicted by Gefen, Aharony, and Alexander [Phys. Rev. Lett. 50, 77 (1983)] for anomalous diffusion.

AB - The moments k 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 k(p) as the average over clusters of k. For ppc, where pc is the percolation threshold, k(p) diverges as &. We show that k=k- with =++, where and are percolation exponents and is the resistance scaling exponent. Our analysis provides the first analytic demonstration that the leading exponents k 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=pc)k+1k-1/k2 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 k have the scaling properties predicted by Gefen, Aharony, and Alexander [Phys. Rev. Lett. 50, 77 (1983)] for anomalous diffusion.

UR - http://www.scopus.com/inward/record.url?scp=0040250220&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.36.8752

DO - 10.1103/PhysRevB.36.8752

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AN - SCOPUS:0040250220

SN - 0163-1829

VL - 36

SP - 8752

EP - 8764

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

IS - 16

ER -