Avalanche dynamics of the Abelian sandpile model on the expanded cactus graph

Abstract: We investigate the avalanche dynamics of the abelian sandpile model on arbitrarily large balls of the expanded cactus graph (the Cayley graph of the free product $\mathbb{Z}_3 * \mathbb{Z}_2$). We follow the approach of Dhar and Majumdar (1990) to enumerate the number of recurrent configurations. We also propose the filling method of enumerating all the recurrent configurations in which adding a grain to a designated origin vertex (far enough away from the boundary vertices) causes topplings to occur in a specific cluster (a connected subgraph that is the union of cells, or copies of the 3-cycle) within the first wave of an avalanche. This filling method lends itself to combinatorial evaluation of the number of positions in which a certain number of cells topple in an avalanche starting at the origin, which are amenable to analysis using well-known recurrences and corresponding generating functions. We show that, when counting cells that topple in the avalanche, the cell-wise first-wave critical exponent of the Abelian sandpile model on the expanded cactus is 3/2.