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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.
The probabilistic abacus
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Ten Lectures on the Probabilistic Method
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V. Anantharam and P. Tsoucas · 1989
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D. J. Lorenzini · 1989
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Self-organized critical state of sandpile automaton models
D. Dhar · 1990
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Geometry and dynamics of deterministic sand piles
S. H. Liu, T. Kaplan, and L. J. Gray · 1990
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Chip-firing games on graphs
A. Björner, L. Lovász, and P. W. Shor · 1991
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A growth model, a game, an algebra, Lagrange inversion, and characteristic classes
P. Diaconis and W. Fulton · 1991
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A finite group attached to the Laplacian of a graph
D. J. Lorenzini · 1991
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Chip-firing games on directed graphs
A. Björner and L. Lovász · 1992
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Internal diffusion limited aggregation
G. F. Lawler, M. Bramson, and D. Griffeath · 1992
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Asymmetric abelian sandpile models
E. R. Speer · 1993
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The electrical resistance of a graph captures its commute and cover times
A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky, and P. Tiwari · 1996
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Eulerian walkers as a model of self-organised criticality
V. B. Priezzhev, D. Dhar, A. Dhar, and S. Krishnamurthy · 1996
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Iterating the branching operation on a directed graph
C. A. Athanasiadis · 1997
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Algebraic potential theory on graphs
N. Biggs · 1997
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Algorithmic aspects of a chip-firing game
J. van den Heuvel · 2001
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On the identity of the sandpile group
Y. Le Borgne and D. Rossin · 2002
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The Rotor-Router Model, 2002
L. Levine · 2002
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On playing golf with two balls
I. Dumitriu, P. Tetali, and P. Winkler · 2003
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Height arrow model
A. Dartois and D. Rossin · 2004
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The rotor-router shape is spherical
L. Levine and Y. Peres · 2005
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Simulating a random walk with constant error
J. N. Cooper and J. Spencer · 2006
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Dynamics of Eulerian walkers
A. M. Povolotsky, V. B. Priezzhev, and R. R. Shcherbakov · 1998
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How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph
J. G. Propp and D. B. Wilson · 1998
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Local divergence of markov chains and the analysis of iterative load-balancing schemes
Y. Rabani, A. Sinclair, and R. Wanka · 1998
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Chip-firing and the critical group of a graph
N. L. Biggs · 1999
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Essential spanning forests and electrical networks on groups
R. Solomyak · 1999
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Limiting shapes for deterministic centrally seeded growth models, 2007
A. Fey-den Boer and F. Redig · 2007
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The sandpile group of a tree, 2007
L. Levine · 2007
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The rotor-router model on regular trees, 2007
I. Landau and L. Levine · 2007
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Spherical asymptotics for the rotor-router model in ℤ d {\mathbb{Z}}^{d}
L. Levine and Y. Peres · 2007
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Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, 2007
L. Levine and Y. Peres · 2007
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Abelian sandpile models and sampling of trees and forests
G. Paoletti · 2007
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Explicit characterization of the identity configuration in an abelian sandpile model
Sergio Caracciolo, Guglielmo Paoletti, and Andrea Sportiello · 2008
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Rotor-router walks, 2008
A. E. Holroyd and J. Propp · 2008
Closest in time.