The Abelian sandpile model is a simple cellular automata. It is run on a square lattice, at which every site has a number between 1 and 4, called the grain height. Each time step, the height of a randomly chosen site is increased by one. If this gives the site a height greater than 4, it topples, decreasing its height by 4, and increasing each of its neighbors heights by one (and, if it is on an edge, sending a grain off the edge).
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Despite it's name, it's not clear that the model does a particularly good job of modeling actual piles of sand. This model original provoked a great deal of interest in the 1990's, as a model of self-organized criticality; that is, a simple explanation of how power laws appear in nature without any fine tuning of parameters. It's not clear that it's been so successful in that regard either. Currently, it's probably most interesting because it's a very simple model that turns out to be equivalent to a logarithmic conformal field theory.
After the model has ben run for a while, certain states become very unlikely. The possible states are divided into transient states, which can only appear soon after the sandpile model starts running, and recurrent ones, which can occur even after long amounts of time have passed.
The five by five configuration shown above is a transient one, which means that it cannot have occurred after the sandpile has been run for a large amount of time. It is not hard to tell that it's a transient configuration, because it has two adjacent sites with 1's in them. A little thought will show that once two adjacent sites have both had grains dropped on them, there's no way to ever get both sites back to height one simultaneously. So two adjacent 1's form a "forbidden subconfiguration," which means that any configuration with this subconfiguration must be transient. Similar reasoning finds an infinite number of forbidden subconfigurations. The three simplest ones are shown here:
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It turns out that if you eliminate all states that have forbidden subconfigurations, all the remaining states are recurrent, and are, over the course of evolution of the sandpile, equally likely to occur. These recurrent states turn out to be equivalent to spanning trees on the same lattice, which are in turn described by the central charge -2 conformal field theory. While I am not longer doing research on this model, many of my past papers explored the relationship between the sandpile model and this conformal field theory.