Dynamic diffusion in thed-dimensional termite model

The coherent-medium approximation is used to study the frequency-dependent diffusion constant of a random walker in the d-dimensional termite model, where two types of jump rates, ‘‘normal’’ and ‘‘superconducting,’’ are distributed randomly. From the static diffusion constant, a critical exponent u, which characterizes the divergence of the static diffusion constant at the percolation threshold, is found to be (1/2), and the superconducting exponent s and the crossover exponent φ are found to be 1 and (1/2), respectively. The imaginary part of the dynamic diffusion constant below the percolation threshold at the static limit vanishes linearly in frequency ω when d>2, as ω lnω when d=2, and as ${\omega }^{d/2\mathrm{}}$ when 1≤d<2. The coefficient of these leading terms diverges at the percolation threshold, indicating an enhancement of the dielectric constant. The critical exponent of this divergence is shown to be 1 when d≥2 and 2-d/2 when 1≤d≤2. The real part of the dynamic diffusion constant below the percolation threshold approaches the static limit quadratically in frequency when d>4, as ${\omega }^2$lnω when d=4 and as ${\omega }^{d/2\mathrm{}}$ when 1≤d<4. The coefficient for d<4 diverges at the threshold with critical exponent 2-d/2. The dynamic diffusion constant near the termite limit shows a crossover from a linear dependence on iω to a linear dependence on 1/iω as the frequency is increased. The low-frequency diffusion constant near the termite limit is also discussed at and above the percolation threshold.