Driving Sandpiles to Criticality and Beyond

A popular theory of self-organized criticality relates driven dissipative systems to systems with conservation. This theory predicts that the stationary density of the Abelian sandpile model equals the threshold density of the fixed-energy sandpile. We refute this prediction for a wide variety of underlying graphs, including the square grid. Driven dissipative sandpiles continue to evolve even after reaching criticality. This result casts doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the Abelian sandpile model at stationarity.

Figure 1

The data of Table from $n=64$ to $n=16384$ are well approximated by ${\zeta }_c({\mathbb{Z}}_n^2)=2.1252881\pm 3\times {10}^{-7}-(0.390\pm 0.001)n^{-1.7}$. (The error bars are too small to be visible, so the data are shown as points.) We conclude that the asymptotic threshold density ${\zeta }_c({\mathbb{Z}}_n^2)$ is $2.125288$ to six decimals. In contrast, the stationary density ${\zeta }_s({\mathbb{Z}}_n^2)$ is $2.125000000000$ to twelve decimals.

Figure 2

The graphs on which we compare ${\zeta }_s$ and ${\zeta }_c$: the grid (upper left), bracelet graph (upper right), flower graph (2nd row left), complete graph (2nd row right), Cayley trees (Bethe lattices) of degree $d=3$, 4, 5 (3rd row), and ladder graph (bottom).

Figure 3

Density $\rho (\lambda )$ of the final stable configuration as a function of initial density $\lambda$ on the bracelet graph (top row) and flower graph (bottom row) as the graph size tends to infinity. A phase transition occurs at $\lambda ={\zeta }_c$. At first glance (left panels) it appears that the driven sandpile reaches its stationary density ${\zeta }_s$ at this point, but closer inspection (right panels) reveals that the final density $\rho (\lambda )$ continues to change as $\lambda$ increases beyond ${\zeta }_c$. These graphs are exact.