Chip-firing on a graph: vertices hold chips, a firing move sends chips to neighbors. The theory connects to algebraic geometry (through divisors), probability (through sandpile models), and combinatorics (through parking functions). Standard chip-firing transfers integer numbers of chips.
Backman, Loehr, and Warrington quantize it. In their model, vertices fire by sending a/b chips to each non-sink neighbor, where a and b are coprime integers. Since chip counts must be integers, the total chips at each vertex are rounded down to the nearest integer after each firing. The rounding is the quantization — continuous rational transfers producing discrete integer states.
On complete graphs with c = 1, the superstable configurations — chip states that cannot fire any further — correspond exactly to rational parking functions. This extends the classical correspondence between ordinary chip-firing and standard parking functions to the rational setting, where a/b-proportional parking rules replace unit-step rules.
The algebraic structure is clean: the group of configurations modulo firing moves is isomorphic to a product of (b-1) copies of Z/aZ. Each coset of this group contains a unique k-skeletal representative — a minimal configuration that encodes the coset's combinatorial properties through a connection to rational lattice paths.
Integer chip-firing is a quotient of the integers. Rational chip-firing is a quotient of the rationals, discretized by rounding. The quantization connects two combinatorial worlds — lattice paths and parking functions — through the floor function.