# Sandpile models

@article{Jarai2014SandpileM, title={Sandpile models}, author={Antal A. J'arai}, journal={arXiv: Probability}, year={2014} }

This survey is an extended version of lectures given at the Cornell Probability Summer School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and its connections to related models are presented. We discuss exactly computable results via Majumdar and Dhar's method. The main ideas of Priezzhev's computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice. We also…

## 32 Citations

### Abelian sandpiles on Sierpinski gasket graphs

- Mathematics
- 2022

The aim of the current work is to investigate structural properties of the sandpile group of a special class of self-similar graphs. More precisely, we consider Abelian sandpiles on Sierpin´ski…

### The distribution of sandpile groups of random regular graphs

- Mathematics
- 2018

We study the distribution of the sandpile group of random d-regular graphs. For the directed model, we prove that it follows the Cohen-Lenstra heuristics, that is, the limiting probability that the…

### Dynamic Dimensional Reduction in the Abelian Sandpile

- MathematicsCommunications in Mathematical Physics
- 2022

We prove the dimensional reduction conjecture of Fey, Levine, and Peres (2010) on the hypercube. The proof shows that dimensional reduction, symmetry, and regularity of the Abelian sandpile persist…

### Tropical curves in sandpile models

- Mathematics
- 2015

A sandpile is a cellular automata on a subgraph $\Omega_h$ of ${h}\mathbb Z^2$ which evolves by the toppling rule: if the number of grains at a vertex is at least four, then it sends one grain to…

### The Looping Rate and Sandpile Density of Planar Graphs

- MathematicsAm. Math. Mon.
- 2016

The looping rate formula is well suited to taking limits where the graph tends to an infinite lattice, and is used to give an elementary derivation of the (previously computed)looping rate and sandpile densities of the square, triangular, and honeycomb lattices.

### Non-criticality criteria for Abelian sandpile models with sources and sinks

- MathematicsJournal of Mathematical Physics
- 2018

We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in \cite{rrs}, is not critical for all branching probabilities $p<1$; by estimating the tail of the…

### Essential enhancements in Abelian networks: continuity and uniform strict monotonicity

- Mathematics
- 2020

We prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope which is uniform with…

### The Abelian Sandpile Model on Fractal Graphs

- Mathematics
- 2016

We study the Abelian sandpile model (ASM), a process where grains of sand are placed on a graph's vertices. When the number of grains on a vertex is at least its degree, one grain is distributed to…

### Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions

- Mathematics
- 2018

We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner…

### Scaling limit of the odometer in divisible sandpiles

- MathematicsProbability theory and related fields
- 2018

This work shows that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.

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