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 $10 \leq L \leq 80$. 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.