The abelian sandpile model introduced by Per Bak, Chao Tang and Kurt Wiesenfeld was the first discovered example of a dynamical system exhibiting self-organized criticality. Since its introduction in 1987, the model has seen widespread research interest from mathematicians and physicists alike, with a focus on explaining the complex global behaviour that emerges from the interplay of the local toppling rules.

In my talk, I will introduce the abelian sandpile model, its toppling rules and how we use these dynamics to define the abelian sandpile Markov chain. We will cover the most important aspects of the sandpile Markov chain, and discuss how these apply to modern research questions on the abelian sandpile model about the distribution of particles and avalanche sizes, with a focus on how the model behaves on fractal state spaces.