AbstractWe study numerically directed polymers in random potential fields for one-dimensional and fractal substrates. For fractal substrates the time evolution of the mean transverse fluctuations depends besides on the randomness of the potential also on the fractal nature of the substrate. The two effects enter in a subordinated way, i.e. the corresponding characteristic exponents due to the potential and the substrate combine multiplicatively. For a one-dimensional substrate the propagator P(x, t), the probability distribution of the transverse displacement x(t), follows the scaling form P(x, t) ∼ 〈x2(t)〉-1/2f (ξ), where ξ is the scaling variable ξ = x/〈x2(t)〉1/2. The numerical results support the scaling function f (ξ) ∼ exp (-cξδ) with δ > 2 which indicates an “enhanced” Gaussian behavior. These results are compared with those of a related “toy model”.
Statistical Properties of One-Dimensional Directed Polymers in a Random Potential