Chip-firing game and Riemann-Roch theory for graphs

Theory of divisors on graphs is analogous to the classical theory for algebraic curves. The combinatorial language in this setting is “chip-firing game” which has been independently introduced in other fields.  A divisor on a graph is simply a configuration of dollars (integer numbers) on its vertices. In each step of the chip-firing game we are allowed to select a vertex and then lend one dollar to each of its neighbors, or borrow one dollar from each of its neighbors. The goal of the chip-firing game is to get all the vertices out of debt. In this setting, there is a combinatorial analogue of the classical Riemann-Roch theorem. I will explain the mathematical structure arising from this process and how it sits in a more general framework of (graphical) hyperplane arrangements.