A Three-Regime Theorem for Flow-Firing
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.
Cite
@article{arxiv.2303.02526,
title = {A Three-Regime Theorem for Flow-Firing},
author = {Sarah Brauner and Galen Dorpalen-Barry and Selvi Kara and Caroline Klivans and Lisa Schneider},
journal= {arXiv preprint arXiv:2303.02526},
year = {2025}
}