Chip-firing on general invertible matrices

TL;DR

This paper generalizes the classical chip-firing model by allowing any invertible integer matrix to govern redistribution, preserving key dynamical properties like criticality and energy minimization.

Contribution

It introduces a broad framework extending chip-firing dynamics beyond graphs to general invertible matrices, maintaining essential properties.

Findings

Preserves critical and superstable states.

Maintains energy minimizing behavior.

Generalizes the model to arbitrary invertible matrices.

Abstract

We propose a generalization of the graphical chip-firing model allowing for the redistribution dynamics to be governed by any invertible integer matrix while maintaining the long term critical, superstable, and energy minimizing behavior of the classical model.