The Mathematics of Chip-Firing

If a single vertex is designated as a "black hole", meaning that chips sent to it vanish, then the result of the process is the same no matter what order the other vertices are selected.

The abelian sandpile model applies this model to large grid graphs, with the black hole connected to the boundary vertices of the grid; in this formulation, with all eligible vertices selected simultaneously, it can also be interpreted as a cellular automaton.

[1][2] The book includes many illustrations, and ends each chapter with a set of exercises making it suitable as a textbook for a course on this topic.

[3] Another book on the same general topic, published at approximately the same time, is Divisors and Sandpiles: An Introduction to Chip-Firing by Corry and Perkinson (American Mathematical Society, 2018).

It is written at a lower level aimed at undergraduate students, covering mainly the material from the first part of The Mathematics of Chip-Firing, and framed more in terms of algebraic geometry than combinatorics.