Relaxation on critical percolation clusters, self-avoiding random walks, and the quantum Hall effect

Large contour lines in a random landscape constitute a continuum percolation problem. We consider directed walks on these lines at the percolation threshold—self-avoiding by construction—and calculate the density correlation function using a Monte Carlo simulation. It has a scaling structure where all exponents are related to the fractal dimension dh=7/4 of extended contour lines. The corresponding scaling function, however, vanishes in the limit ω→0 with a power law giving rise to another a priori independent exponent ζ. Our data indicate a Cole-Cole structure for the quantity χ|IJ(ω) averaged over extended lines only, which implies ζ=2/7, η=0 and an anomaly in the small frequency asymptotics of the diffusion coefficient D(q,ω)∝ω−ζ/q2. This anomaly is reminiscent of a similar one in the quantum Hall effect, however, with negative ζqhe. We argue that the difference is due to the decay of phase resonances in the latter which come with a broad distribution of decay times