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Abstract
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.
Original language  English 

Type  ArXiv preprint 
Media of output  ArXiv 
Number of pages  19 
Publication status  Published  19 Sept 2024 
Keywords
 Sandpile model
 Complete bipartite graphs
 Recurrent configurations
 Ferrers diagrams
 Motzkin paths
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Dive into the research topics of 'Abelian and stochastic sandpile models on complete bipartite graphs'. Together they form a unique fingerprint.Projects
 2 Active

Towards a combinatorial theory of sandpile models
1/01/23 → 31/12/25
Project: Internal Research Project

Stochastic variants of the Abelian sandpile model
1/01/22 → 31/12/24
Project: Governmental Research Project