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

Abstract

We consider the random walk of a particle in a two-dimensional self-affine random potential of Hurst exponent $H=1/2$ in the presence of an external force $F$. 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) \sim \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 $F \to 0$, we measure the following power-laws : the anomalous exponent vanishes as $\mu(F) \propto F^a$ with $a \simeq 0.6$ (instead of $a=1$ in dimension $d=1$), and the correlation length diverges as $\xi(F) \propto F^{-\nu}$ with $\nu \simeq 1.29$ (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.