Chip-firing games, potential theory on graphs, and spanning trees

TL;DR

This paper explores the relationship between chip-firing games, potential theory, and spanning trees on graphs, introducing algorithms for computing reduced divisors and generating random spanning trees with improved efficiency.

Contribution

It provides a novel characterization of reduced divisors as energy minimizers and introduces efficient algorithms for spanning tree enumeration and sampling.

Findings

Algorithms for reduced divisors with improved running times

A bijective proof of Kirchhoff's matrix-tree theorem

New methods for generating random spanning trees

Abstract

We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors ($G$-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm to efficiently compute reduced divisors. Applications include an "efficient bijective" proof of Kirchhoff's matrix-tree theorem and a new algorithm for finding random spanning trees. The running times of our algorithms are analyzed using potential theory, and we show that the bounds thus obtained generalize and improve upon several previous results in the literature. We also extend some of these considerations to metric graphs.