Figure 1

Main panel: Data collapse [$\mathcal{P}(s;L,p_r,p_l)s^{\tau }$ vs $s/L^D$] of the normalized and binned distributions for the two extreme cases ($p_r=1$ with $p_l=0$ and $p_l=1$) for $L=1280,2560,5120$. The rescaling was done using $\tau =1.333\dots$ and $D=1.5$ for the AOM and $\tau =1.555\dots$ and $D=2.25$ for the OOM. Inset: Distributions for three choices of $p_r>p_l$, namely $(p_r,p_l)=(1.0,0.95),(1.0,0.25),(0.75,0.25)$, and two choices of $p_r=p_l$ ($0.75$ and $0.25$), for $L=640,1280,2560$. The data collapse of each tuple $(p_r,p_l)$ would form a single line, thereby fixing $\tau$ and $D$. By tuning $a$ and $b$ in Eq. (1), as done in the inset, the result collapses with one of the extreme cases shown in both panels.Reuse & Permissions

Figure 2

Scaling of $⟨s^2⟩/L^{2.5}$ for different anisotropies. The solid circles are results for $p_r-p_l$ as indicated and $p_r=1$. Open circles show other parameters $(p_r,p_l)$. The dashed lines are the two extreme cases OOM ($p_r=p_l=1$) and the solvable model ($p_r=1,p_l=0$). The short arrows mark the approximate crossover.Reuse & Permissions

Figure 3

Behavior of $g_3$ [see Eq. (3)] for different anisotropies. The solid circles are results for $p_r-p_l$ in the same vertical order as shown in Fig. 2 and $p_r=1$. Open circles show other parameters $(p_r,p_l)$ as indicated. The dashed lines are the two extreme cases OOM ($p_r=p_l=1$) and the solvable model ($p_r=1,p_l=0$). The short arrows mark the crossover points in Fig. 2.Reuse & Permissions

Figure 4

A stylized toppling of a single grain: If an elongated grain of width $b$ and height $a$ topples from site $i$ to site $i+1$, it reduces the height at $i$ by $b$ and increases the height at $i+1$ by $a$, thereby increasing the slope at site $i-1$ by $b$, decreasing the slope at $i$ by $a+b$ and increasing the slope at $i+1$ by $a$. The net flux of slope is $a-b$ to the right.Reuse & Permissions