2020/10/09 by Todd Mullen, Richard J. Nowakowski, Mullen, Todd +3
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2010.04750
openalex publication_date 2020/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Tn is the number of configurations with period length 2 that can exist on Pn up to isomorphism and n is an integer greater than 4, we conclude that Tn = 3Tn-1 + 2Tn-2 + Tn-3 - Tn-4.