# Diffusion on graphs is eventually periodic

## TL;DR

This paper proves that the diffusion process on graphs, a variant of chip-firing, always becomes periodic after some steps, with the period being either 1 or 2, confirming a longstanding conjecture.

## Contribution

The paper proves the conjecture that diffusion on graphs is always eventually periodic with period 1 or 2, resolving a question posed in 2016.

## Key findings

- Diffusion on graphs is always eventually periodic.
- The period of diffusion is either 1 or 2.
- The conjecture from 2016 is confirmed.

## Abstract

We study a variant of the chip-firing game called \emph{diffusion}. In diffusion on a graph, each vertex of the graph is initially labelled with an integer interpreted as the number of chips at that vertex, and at each subsequent step, each vertex simultaneously fires one chip to each of its neighbours with fewer chips. Since this firing rule may result in negative labels, diffusion, unlike the parallel chip-firing game, is not obviously periodic. In 2016, Duffy, Lidbetter, Messinger and Nowakowski nevertheless conjectured that diffusion is always eventually periodic, and moreover, that the process eventually has period either 1 or 2. Here, we establish this conjecture.