Criticality of the Abelian sandpile model on converging sequences of graphs

We consider the question of the criticality of the Abelian sandpile model on converging sequences of graphs. Classically, one studies the model on a sequence of finite graphs exhausting some infinite regular lattice. The dynamics of the model is encoded by so-called avalanches and its criticality is reflected, asymptotically, by power-law decays of various statistics related to these avalanches. After considering few examples, we extend the study of the criticality of the ASM to graphs converging in a more general sense to some infinite limits.