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 -