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 Monthus and Garel [J. Phys. A: Math. Theor. 41, 255002 (2008)]. We present numerical results on the statistics of the equilibrium time $t_{\text{eq}}$ 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 ${\Gamma }_{\text{eq}}\equiv \text{ln}{t}_{\text{eq}}$ scales as ${\Gamma }_{\text{eq}}=L^{H}u$ where $u$ is a random variable of order $O\left(1\right)$. This corresponds to a logarithmically slow diffusion $\left|\vec{r}\left(t\right)-\vec{r}\left(0\right)\right|\sim {\left(\text{ln}t\right)}^{1/H}$ for the position $\vec{r}\left(t\right)$ of the particle.

Figure 1

Random self-affine random potential $U\left(x,y\right)$ on a square of size $100\times 100$, obtained via the method of Eq. (6) for the value $H=0.5$ of the Hurst exponent: (a) example of one realization of the random potential $U\left(x,y\right)$. (b) Log-log plot of the correlation function $C\left(r\right)\equiv \overline{{\left[U\left({\vec{r}}_1\right)-U\left({\vec{r}}_2\right)\right]}^2}$ as a function of the distance $r\equiv \left|{\vec{r}}_1-{\vec{r}}_2\right|$ after averaging over angles and over disorder realizations: the slope is here $2H=1$, as it should for $H=0.5$ [see Eq. (3)].

Figure 2

(Color online) Statistics of the equilibrium time $t_{\text{eq}}$ over the disordered samples of sizes $L^2$: the appropriate variable is ${\Gamma }_{\text{eq}}=\text{ln}{t}_{\text{eq}}$ [Eq. (19)] (a) the disorder-averaged value $\overline{{\Gamma }_{\text{eq}}}\left(L\right)$ and the width $\Delta \left(L\right)$ shown here for $H=0.8$ scale with the same exponent $\psi$ [see Eq. (20)]. (b) the exponent $\psi$ of the width $\Delta \left(L\right)$ coincides with the Hurst exponent $H$, as shown here for $H=0.3$, 0.4, 0.5, 0.6, 0.7, and 0.8.

Figure 3

(Color online) Statistics of the equilibrium time $t_{\text{eq}}$ over the disordered samples of sizes $L^2$ for the case $H=0.8$: (a) probability distribution $Q_L\left({\Gamma }_{\text{eq}}=\text{ln}{t}_{\text{eq}}\right)$ for $L=10$, 20, 30, 40, and 50 and (b) the same data after the rescaling of Eq. (22) and in log scale to see the tails.