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

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 ${\mathrm{\Gamma }}_{eq}\equiv \ln t_{eq}$ scales as ${\mathrm{\Gamma }}_{eq}=L^{H}u$ 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}$fortheposition$\vec r(t)$ of the particle.