Counting Path Configurations in Parallel Diffusion

TL;DR

This paper studies the periodic behavior of Parallel Diffusion processes on paths, providing a recurrence relation to count configurations with period 2 up to isomorphism, advancing understanding of chip-firing dynamics.

Contribution

It proves the eventual periodicity of Parallel Diffusion and derives a recurrence relation for counting period-2 configurations on paths.

Findings

Periodic behavior occurs after a pre-period in Parallel Diffusion.

A recurrence relation for counting period-2 configurations on paths is established.

The number of such configurations can be computed recursively for any path length.

Abstract

Parallel Diffusion is a variant of Chip-Firing introduced in 2018 by Duffy et al. In Parallel Diffusion, chips move from places of high concentration to places of low concentration through a discrete-time process. At each time step, every vertex sends a chip to each of its poorer neighbours, allowing for some vertices to perhaps fall into debt (represented by negative stack sizes). In their recent paper, Long and Narayanan proved a conjecture from the original paper by Duffy et al. that every Parallel Diffusion process eventually, after some pre-period, exhibits periodic behaviour. With this result, we are now able to count the number of these periods that exist up to a definition of isomorphism. We determine a recurrence relation for calculating this number for a path of any length. If $T_n$ is the number of configurations with period length 2 that can exist on $P_n$ up to isomorphism and $n$ is an integer greater than 4, we conclude that $T_n = 3T_{n-1} + 2T_{n-2} + T_{n-3} - T_{n-4}$.