Prime graphical parking functions and strongly recurrent configurations of the Abelian sandpile model

TL;DR

This paper explores the relationship between prime G-parking functions and strongly recurrent configurations in the Abelian sandpile model, extending classical concepts and analyzing their correspondence on various graph families.

Contribution

It introduces prime G-parking functions and establishes their correspondence with strongly recurrent configurations in the ASM, extending previous notions of primeness.

Findings

Prime G-parking functions are in bijection with strongly recurrent ASM configurations.

The study covers various graph families including wheel, complete, and multipartite graphs.

New connections between parking functions and ASM configurations are established.

Abstract

This work investigates the duality between two discrete dynamical processes: parking functions, and the Abelian sandpile model (ASM). Specifically, we are interested in the extension of classical parking functions, called $G$-parking functions, introduced by Postnikov and Shapiro in 2004. $G$-parking functions are in bijection with recurrent configurations of the ASM on $G$. In this work, we define a notion of prime $G$-parking functions. These are parking functions that are in a sense "indecomposable". Our notion extends the concept of primeness for classical parking functions, as well as the notion of prime $(p,q)$-parking functions introduced by Armon et al. in recent work. We show that from the ASM perspective, prime $G$-parking functions correspond to certain configurations of the ASM, which we call strongly recurrent. We study this new connection on a number of graph families, including wheel graphs, complete graphs, complete multi-partite graphs, and complete split graphs.