# Flow-firing processes

## TL;DR

This paper studies a flow-firing process on planar complexes, showing conditions for termination and uniqueness of final configurations, revealing complex behaviors in higher-dimensional chip-firing models.

## Contribution

It introduces a non-deterministic flow-firing process on planar complexes and characterizes its termination and confluence properties under various initial conditions.

## Key findings

- Non-conservative flows never terminate.
- Conservative flows terminate with multiple possible final states.
- Flows around topological holes terminate uniquely.

## Abstract

We consider a discrete non-deterministic flow-firing process for rerouting flow on the edges of a planar complex. The process is an instance of higher-dimensional chip-firing. In the flow-firing process, flow on the edges of a complex is repeatedly diverted across the faces of the complex. For non-conservative initial configurations we show this process never terminates. For conservative initial flows we show the process terminates after a finite number of rerouting steps, but there are many possible final configurations reachable from a single initial state. Finally, for conservative initial flows around a topological hole we show the process terminates at a unique final configuration. In this case the process exhibits global confluence despite not satisfying local confluence.