Chip-Firing and Rotor-Routing on Directed Graphs

  title={Chip-Firing and Rotor-Routing on Directed Graphs},
  author={Alexander E. Holroyd and Lionel Levine and Karola M{\'e}sz{\'a}ros and Yuval Peres and James Gary Propp and David Bruce Wilson},
  journal={arXiv: Combinatorics},
We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems. 
Rotor Walks on Transient Graphs and the Wired Spanning Forest
We study rotor walks on transient graphs with initial rotor configuration sampled from the oriented wired uniform spanning forest (OWUSF) measure. We show that the expected number of visits to any ...
Recurrent Rotor-Router Configurations
We prove the existence of recurrent initial configurations for the rotor walk on many graphs, including Z^d, and planar graphs with locally finite embeddings. We also prove that recurrence and
The Rotor-Router Group of Directed Covers of Graphs
Direct covers of graphs are considered, and several quantities related to rotor-router walks on directed covers are studied, including order of the rotor- Router group,Order of the root element in the rotational group and the connection with random walks.
The behavior of Chip Firing Game and related model : A comprehensive survey
In this paper, we give a survey of known results concerning the presence of order structure and lattices in the context of discrete dynamical models derived from studies of Chip Firing Games.
Algorithmic aspects of rotor-routing and the notion of linear equivalence
Sandpile Groups and the Join of Graphs
We introduced the procedure of joining two graphs by identifying an arbitrary pair of their vertices. The main result is that the sanpile group of the join of several finite graphs is the direct
Chip-firing games, potential theory on graphs, and spanning trees
Abelian Sandpile Model on Symmetric Graphs
The abelian sandpile model, or chip-firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of selforganized criticality. Here we present a thorough
A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees
This paper presents an explicit bijection between the recurrent configurations of a hereditary chip-firing model on a graph and its spanning trees.
Rotor-routing orbits in directed graphs and the Picard group
The number of recurrent rotor-router unicycle-orbits equals the order of the Picard group of the graph, defined in the sense of [1], and during a period, the same chip-moves happen, as during firing the period vector in the chip-firing game.


Chip-Firing and the Critical Group of a Graph
A variant of the chip-firing game on a graph is defined. It is shown that the set of configurations that are stable and recurrent for this game can be given the structure of an abelian group, and
Polynomial Bound for a Chip Firing Game on Graphs
  • G. Tardos
  • Computer Science, Mathematics
    SIAM J. Discret. Math.
  • 1988
A polynomial bound on the length of the game in terms of the number of vertices of the graph provided the length is finite is proved.
Spherical asymptotics for the rotor-router model in Zd
The rotor-router model is a deterministic analogue of random walk invented by Jim Propp. It can be used to define a deterministic aggregation model analogous to internal diffusion limited
Chip-Firing Games on Directed Graphs
It is shown that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable.
The rotor-router model on regular trees
On the Sandpile Group of Dual Graphs
It is proved that the sandpile group of planar graph is isomorphic to that of its dual, and a combinatorial point of view on the subject is developed.
Self-organized critical state of sandpile automaton models.
  • Dhar
  • Computer Science, Physics
    Physical review letters
  • 1990
The critical state is characterized, and its entropy for an arbitrary finite lattice in any dimension is determined, and the two-point correlation function is shown to satisfy a linear equation.
Algebraic Potential Theory on Graphs
This paper encompasses a motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip‐firing games,
Algorithmic Aspects of a Chip-Firing Game
  • J. V. D. Heuvel
  • Computer Science
    Combinatorics, Probability and Computing
  • 2001
It is proved that the number of steps needed to reach a critical configuration is polynomial in the number and number of edges of the graph and thenumber of chips in the starting configuration, but not necessarily in the size of the input.
The sandpile group of a tree