A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees

TL;DR

This paper establishes a direct bijection between recurrent configurations in hereditary chip-firing models and spanning trees, extending classical results to a broader family of chip-firing games.

Contribution

It introduces an explicit bijection linking recurrent configurations of hereditary chip-firing models to spanning trees, generalizing known correspondences in classical models.

Findings

Bijection applies to a wide class of hereditary chip-firing models.

Recurrent configurations correspond uniquely to spanning trees.

The approach preserves key properties of the classical Abelian sandpile model.

Abstract

Hereditary chip-firing models generalize the Abelian sandpile model and the cluster firing model to an exponential family of games induced by covers of the vertex set. This generalization retains some desirable properties, e.g. stabilization is independent of firings chosen and each chip-firing equivalence class contains a unique recurrent configuration. In this paper we present an explicit bijection between the recurrent configurations of a hereditary chip-firing model on a graph and its spanning trees.