The Abelian Sandpile and its Avalanche Polynomials

Imagine yourself on a beach, playing in the sand. You begin to make a sandpile by adding handfuls of sand. Now you consider dropping another grain of sand onto the pile but you don’t know what will happen. It may cause nothing to happen or it may cause the entire pile to collapse in a massive slide. This is the idea behind the Abelian sandpile model. We accomplish this task by using uses directed graphs where we denote one vertex as the sink and at all other vertices we have a nonnegative integer. These integers represent the number of grains of sand placed in that sandpile. When the pile gets too big, meaning the number gets too large, an avalanche occurs sending grains along each edge adjacent to the toppling vertex. We measure the sizes of these topplings and build what is called the avalanche polynomial. This topic is based on graph theory, group theory, and enumerative combinatorics.