Random walk in a two-dimensional self-affine random potential: Properties of the anomalous diffusion phase at small external force

We study the dynamical response to an external force F for a particle performing a random walk in a two-dimensional quenched random potential of Hurst exponent H = 1/2. We present numerical results on the statistics of first-passage times that satisfy closed backward master equations. We find that there exists a zero-velocity phase in a finite region of the external force 0 < F < F-c, where the dynamics follows the anomalous diffusion law x(t) similar to xi(F)t(mu(F)). The anomalous exponent 0 < mu(F) < 1 and the correlation length xi(F) vary continuously with F. In the limit of vanishing force F -> 0, we measure the following power laws: the anomalous exponent vanishes as mu(F) proportional to F-a with a similar or equal to 0.6 (instead of a = 1 in dimension d = 1), and the correlation length diverges as xi(F) proportional to F-nu with nu similar or equal to 1.29 (instead of nu = 2 in dimension d = 1). Our main conclusion is thus that the dynamics renormalizes onto an effective directed trap model, where the traps are characterized by a typical length xi(F) along the direction of the force, and by a typical barrier 1/mu(F). The fact that these traps are "smaller" in linear size and in depth than in dimension d = 1, means that the particle uses the transverse direction to find lower barriers.