Non-linear response in percolation systems

The authors study diffusion in d=2 percolation clusters at criticality under the influence of an external time-dependent field E(t)=E₀sin( omega t). They find that the mean displacement (x(t)) of a random walker is a periodic function with a phase shift phi approximately= pi /3 independent of omega and E₀. The amplitude A(E₀, omega ) of (x(t)) is linear in E₀ below a crossover field E*₀( omega ). Above E*₀( omega ), A(E₀, omega ) is strongly non-linear, showing a maximum as a function of E₀. They find that both the linear and the non-linear regime can be described by a single scaling function: A(E₀, omega ) approximately E₀ omega ^-2d/F(E₀/E*₀( omega )) where d approximately=2.87 is the diffusion exponent and E*₀( omega ) approximately omega ^beta with beta approximately=0.3. The linear response regime is reached for E₀/ omega ^beta<<1.