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Abelian and stochastic sandpile models on complete bipartite graphs

Thomas Selig, Haoyue Zhu

TL;DR

The paper analyzes Abelian and stochastic sandpile dynamics on the complete bipartite graph $K_{m,n}^0$, providing a linear-time criterion and algorithm for stochastic recurrence via a stochastic burning process. It establishes a precise bijection between sorted stochastically recurrent configurations and pairs of compatible Ferrers diagrams, with the configuration level equaling a Ferrers-area difference, and extends these ideas to deterministic recurrence, yielding a Ferrers-diagram interpretation and a strong compatibility condition. The authors further connect recurrent configurations to parallelogram polyominoes and to labelled Motzkin paths, yielding complementary combinatorial encodings and DAG representations that illuminate the structure and enable exact counting and efficient testing. Collectively, the work deepens the link between sandpile dynamics and classical combinatorial objects, offering linear-time algorithms and rich bijective correspondences that bridge stochastic and deterministic frameworks.

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.

Abelian and stochastic sandpile models on complete bipartite graphs

TL;DR

The paper analyzes Abelian and stochastic sandpile dynamics on the complete bipartite graph , providing a linear-time criterion and algorithm for stochastic recurrence via a stochastic burning process. It establishes a precise bijection between sorted stochastically recurrent configurations and pairs of compatible Ferrers diagrams, with the configuration level equaling a Ferrers-area difference, and extends these ideas to deterministic recurrence, yielding a Ferrers-diagram interpretation and a strong compatibility condition. The authors further connect recurrent configurations to parallelogram polyominoes and to labelled Motzkin paths, yielding complementary combinatorial encodings and DAG representations that illuminate the structure and enable exact counting and efficient testing. Collectively, the work deepens the link between sandpile dynamics and classical combinatorial objects, offering linear-time algorithms and rich bijective correspondences that bridge stochastic and deterministic frameworks.

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.
Paper Structure (10 sections, 13 theorems, 11 equations, 9 figures, 4 algorithms)

This paper contains 10 sections, 13 theorems, 11 equations, 9 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Stochastically recurrent configurations on complete bipartite graphs
  3. Characterisation of $\mathrm{StoRec}_{m,n}$
  4. A stochastic burning algorithm for complete bipartite graphs
  5. Recurrent configurations as pairs of Ferrers diagrams
  6. Deterministically recurrent configurations on complete bipartite graphs
  7. Characterisation of $\mathrm{DetRec}_{m,n}$
  8. Ferrers diagram interpretation
  9. Connection to parallelogram polyominoes
  10. Connection to Motzkin paths

Key Result

Theorem 2.1

Let $c = (c^t;c^b) \in \mathrm{Stable}_{m,n}$ be a stable configuration on $K_{m,n}^0$. Then $c \in \mathrm{StoRec}_{m,n}$ if, and only if, for all subsets $A \subseteq [m], B \subseteq [n]$, we have: If $A,B$ do not satisfy Inequality eq:forbidden_subconfig, we say that $(A,B)$ is a forbidden subconfiguration.

Figures (9)

  • Figure 1: Illustrating the deterministic stabilisation for $c = (2,1; 0,2) \in \mathrm{Config}_{2, 2}$. Vertices under the arrows represent the vertex being toppled in that phase.
  • Figure 2: Illustrating a possible stochastic stabilisation for $c = (2,1; 0,2) \in \mathrm{Config}_{2, 2}$. Vertices under the arrows represent the vertex being toppled in that phase.
  • Figure 3: Illustrating a legal sequence from $F(1,1,3)$ to $F(2,2,2)$, showing that the configuration $c = (0,2,2;2,2,2)$ is stochastically recurrent.
  • Figure 4: The graph $\mathrm{DAG}^{\mathrm{SR}}_{3, 3}$.
  • Figure 5: Illustrating a legal sequence from $F(1,1,3)$ to $F(2,2,3)$, showing that the configuration $c = (0,2,2;2,2,3)$ is deterministically recurrent.
  • ...and 4 more figures

Theorems & Definitions (34)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Example 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Definition 3.1
  • Theorem 3.2
  • ...and 24 more