The a.c. conductivity Σ(ω,θ) and the a.c. dielectric constant ε(ω,θ) near the percolation threshold of carbon black-epoxy bulk composites were measured in the frequence range from 100Hz to 10GHz. The power-law dependence of the a.c. conductivity, $Σ(ω,θ_c)$~ω^x$, was observed at the percolation threshold in the frequency range of test up to 10GHz, but that of the a,c. dielectric constant, ε(ω,θ _c)$ω~^{-y}$, was valid in the bound frequency range of 2KHz-1.5MHz and the three stages variation of the dielectric constant with frequency is apparent at the percolation threshold. The critical exponents x=0.65 ± 0.05 and y=0.26 ± 0.01 obtained in the power-law frequency range are close to the theoretical predictions of intercluster polarization. The critical volume fraction $θ_c$=0.18 obtained at the divergence of dielectric constant is close to that of continuous percolation system.
By the transfer matrix simulation we observed the scaling law near the percolation threshold in two dimensional bond percolation. We were rigorously studied the finite size scaling of the conductivity and the frequency dependence of R-C lattice percolation model. Our results are good agreement to the prediction of the scaling theory of the electrical percolation. In greater concentration than the percolation threshold we simulate the d.c. conductivity of two dimensional site percolation by diffusion method.
The random walk on the two dimensional gasket fractal has been studied with the various forms of constant bias fields applied using Monte Carlo method. The crossover from anomalous diffusion to drift was observed in the horizontal and upward bias fields while the crossover from anomalous to normal diffusion was found in the downward bias field. The scaling relation $R(B_s,t)~,t^kf(B_st^k)$ was confirmed and the scaling exponent was obtained as β=1.175±0.004 for the horizontal bias field and β=0.319±0.002 for the downward bias field. The random walk in the percolation cluster near the percolation threshold under the application of both the time dependent external field and the static field $B(t)=B_0,+,B_1$ sin ωt shows the nonlinear response at crossover field $B_1^*$. The phase shift of the random walk particle depends on the a.c. field amplitude at low frequency. The strong d.c. external field $B_0$ changes the functional form of the response.
