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We consider the random walk of a particle in a two-dimensional self-affinerandom 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 thatsatisfy closed backward master equations. We find that there exists azero-velocity phase in a finite region of the external force $0<F<F_c$, wherethe 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 isthus that the dynamics renormalizes onto an effective directed trap model,where the traps are characterized by a typical length $\xi(F)$ along thedirection of the force, and by a typical barrier $1/\mu(F)$. The fact thatthese 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.

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