Random walks on bond percolation clusters: ac hopping conductivity below the threshold

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The frequency dependence of the ac hopping conductivity in two and three dimensional lattices with random interruptions is calculated by Monte Carlo simulation of random walks on bond percolation clusters. At low frequencies the real and imaginary parts of the ac conductivity vanish linearly and quadratically with the frequency, respectively. The critical behaviour of the imaginary part of the ac conductivity below the percolation threshold is shown to depend on the long time limit of the mean square displacement of random walksR ∞ 2 , while the real part depends on the time constant of the system as well.R ∞ 2 is found to diverge with an exponentu=2ν-β according to the conjecture of Stauffer.