We start with the applet for Model 2 on 30×4 lattice. The sand transport in this model is isotropic, i.e. an unstable sand column loses one grain of sand to each of its four nearest neighbors. Boundary conditions are open in x-direction and periodic in y-direction.  The description of different buttons and checkboxes on the canvas of these applets can be found here.   Repetitively adding sand at the central column often causes system-wide avalanches (this way the stationary sandpile gets rid of the constant influx of sand),  but rarely changes the configuration.  To see this for yourself push the "Continue" button in the applet below and start clicking (adding sand) in the central column. You can see how avalanches roll through the lattice rarely affecting sites that are far to the right from your driving site.

Model 2:

Model 2A is a slight modification of Model 2, defined on 30×2 lattice. The sandpile dynamics is still isotropic, but boundary conditions in y-direction are closed, i.e. only one grain of sand goes to the nearest neighbor in the column, compared to two grains transported for periodic boundary conditions. Now stable columns can only have heights 0, 1, or 2.  The advantage of this model is that we have a better understanding of how avalanche affects the underlying configuration.

Model 2A:

There are two important facts about the dynamics of this model:

1. Adding a grain of sand causes a trough (a column with both heights smaller than 2) to move around the system. As we have found out in our paper, if you add a grain of sand at x₀ and the trough is located at x₁ in the resulting configuration trough will move to x₂ = (x₀+ x₁) mod (L+1). If this formula gives x₂=0 that simply means that the trough would disappear. Motion of the trough does not destroy the memory of the configuration and, therefore, can be disregarded for our purposes.

2. Similar to directed model 1A, irreversible changes in configuration happen extremely rarely.  In fact one can show that in order for the region of changed sites to propagate by one step to the right (left), the configuration to the right (left) from the driving site has to be (2,1),(2,1) ... (2,1),(2,0) (by this (a,b) notation I mean a vertical column with a above the b). Since this requires the sandpile to be in a unique state among ~3.732^x possible SOC states, such change is exponentially unlikely for large x.