How hard is it to predict sandpiles on lattices? A survey

TL;DR

This survey examines the computational complexity of predicting the behavior of sandpile models on lattices, highlighting the open problem of determining the difficulty of predicting two-dimensional sandpile dynamics.

Contribution

It formalizes the prediction problem for sandpiles and reviews existing results, emphasizing the unresolved challenge of understanding the complexity of 2D sandpile prediction.

Findings

Prediction complexity varies across different models.

The problem remains open for the classic 2D sandpile model.

Survey summarizes key results and open questions in the field.

Abstract

Since their introduction in the 80s, sandpile models have raised interest for their simple definition and their surprising dynamical properties. In this survey we focus on the computational complexity of the prediction problem, namely, the complexity of knowing, given a finite configuration $c$ and a cell $x$ in $c$, if cell $x$ will eventually become unstable. This is an attempt to formalize the intuitive notion of "behavioral complexity" that one easily observes in simulations. However, despite many efforts and nice results, the original question remains open: how hard is it to predict the two-dimensional sandpile model of Bak, Tang and Wiesenfeld?