2024/09/18 by Thomas Selig, Selig, Thomas, Haoyue Zhu +1
Earth and Planetary Sciences · Engineering · #05A19 (Primary) 05A15 #05B50 #60J10 (Secondary) #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Geological Modeling and Analysis #Geological formations and processes #Hydrocarbon exploration and reservoir analysis #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2409.11811
openalex publication_date 2024/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the sandpile model, vertices of a graph are allocated grains of sand. At each unit of time, a grain is added to a randomly chosen vertex. If that causes its number of grains to exceed its degree, that vertex is called unstable, and topples. In the Abelian sandpile model (ASM), topplings are deterministic, whereas in the stochastic sandpile model (SSM) they are random. We study the ASM and SSM on complete bipartite graphs. For the SSM, we provide a stochastic version of Dhar's burning algorithm to check if a given (stable) configuration is recurrent or not, with linear complexity. We also exhibit a bijection between sorted recurrent configurations and pairs of compatible Ferrers diagrams. We then provide a similar bijection for the ASM, and also interpret its recurrent configurations in terms of labelled Motzkin paths.