A Three-Regime Theorem for Flow-Firing

TL;DR

The paper extends the classical three-regime theory of graphical chip-firing to flow-firing on a two-dimensional grid, focusing on termination and confluence for the pulse configurations $\mathsf{K}(n,r)$ and the Aztec diamond $\mathrm{Az}(n)$. It develops a path-firing framework and leverages a discrete-derivative connection to reduce higher-dimensional dynamics to one-dimensional chip-firing, enabling precise regime analysis. The main result, a Three-Regime Theorem for flow-firing, characterizes when termination is unique, when non-unique termination occurs but Aztec-termination is possible, and when termination in the Aztec diamond is impossible for large enough radius, with explicit bounds $r\le1$, $2\le r\le\lceil n/2\rceil$, and $r\ge\lceil n/\sqrt{3}\rceil+1$. The work further establishes confluence for subsystems, including certain path-firings, and discusses improvements to regime bounds, contributing to a deeper understanding of termination phenomena in higher-dimensional discrete dynamical systems.

Abstract

Graphical chip-firing is a discrete dynamical system where chips are placed on the vertices of a graph and exchanged via simple firing moves. Recent work has sought to generalize chip-firing on graphs to higher dimensions, wherein graphs are replaced by cellular complexes and chip firing becomes flow-rerouting along the faces of the complex. Given such a system, it is natural to ask (1) whether this firing process terminates and (2) if it terminates uniquely (e.g. is confluent). In the graphical case, these questions were definitively answered by Bjorner--Lovasz--Shor, who developed three regimes which completely determine if a given system will terminate. Building on the work of Duval--Klivans--Martin and Felzenszwalb-Klivans, we answer these questions in a context called flow-firing, where the cellular complexes are 2-dimensional.

Figures (5)

• Figure 1: An example of flow-firing on the grid
• Figure 2: Two distinct ways to decompose the grid into quadrants.
• Figure 3: The face labels of the grid, face weights in $Q_1$ and $Q_2$ (respectively)
• Figure 4: The face weights in $Q_1$
• Figure :

Theorems & Definitions (26)

• Theorem 1: Three-Regime Theorem for Flow-Firing
• Theorem 2: flowfiringprocesses
• Example 3
• Example 4
• Theorem 5: flowfiringprocesses
• Theorem 6: bjorner-lovasz-shor: Three Regimes for Graphical Chip-Firing
• Proposition 7
• Example 8
• Corollary 9: BallsWalls
• Lemma 10