A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group
Keywords: Chip-firing, Simplicial complexes
AbstractCartwright (2015) introduced the notion of a weak tropical complex in order to generalize the theory of divisors on graphs from Baker and Norine (2007). A weak tropical complex $\Gamma$ is a $\Delta$-complex equipped with algebraic data that allows it to be viewed as the dual complex to a certain kind of degeneration over a discrete valuation ring. Every graph has a unique tropical complex structure (which is the same structure studied by Baker and Norine) in which divisors correspond to states in the chip-firing game on that graph. Let $G$ and $H$ be graphs, and let $\Gamma$ be a triangulation of $G\times H$ obtained by adding in one diagonal of each resulting square. There is a particular weak tropical complex structure on $\Gamma$ that Cartwright conjectured was closely related to the weak tropical complex structures on $G$ and $H$. The main result of this paper is a proof of Cartwright's conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.