Random walk in two-dimensional self-affine random potentials : strong disorder renormalization approach

Abstract

We consider the continuous-time random walk of a particle in a two-dimensional self-affine quenched random potential of Hurst exponent $H>0$. The corresponding master equation is studied via the strong disorder renormalization procedure introduced in Ref. [C. Monthus and T. Garel, J. Phys. A: Math. Theor. 41 (2008) 255002]. We present numerical results on the statistics of the equilibrium time $t_{eq}$ over the disordered samples of a given size $L\times L$ for $10\le L\le 80$. We find an 'Infinite disorder fixed point', where the equilibrium barrier ${\Gamma }_{eq}\equiv \ln t_{eq}$ scales as ${\Gamma }_{eq}=L^{H}u$ where $u$ is a random variable of order O(1). This corresponds to a logarithmically-slow diffusion $|\vec{r}(t)-\vec{r}(0)|\sim (\ln t{)}^{1/H}$ for the position $\vec{r}(t)$ of the particle.

7 pages, 7 figures; v2=final version

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