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

We consider the random walk of a particle in a two-dimensional self-affine random potential of Hurst exponent $H=1\mathrm{/}2$ 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)} $\mathrm{.}Theanomalousexponent$0<\mu(F)<1$andthecorrelationlength$\xi(F)$varycontinuouslywith$F$\mathrm{.}Inthelimitofvanishingforce$F \to 0$, 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 $\nu \simeq 1.29$(insteadof$\nu=2$indimension$d=1$). 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 $1\mathrm{/}\mu (F)$. 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.