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

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.