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Preface
¢ñ Fundamentals
1 Introduction
1.1 A brief introduction
1.2 Origins and History
1.2.1 The abelian sandpile model
1.2.2 A combinatorial game
1.2.3 Abstract rewriting systems
2 Chip-firing on Finite Graphs
2.1 The chip-firing process
2.1.1 The graph Laplacian
2.1.2 Cluster-fires
2.2 Confluence
2.3 Stabilization
2.4 Toppling time
2.5 Stabilization with a sink
2.6 Long-term stable configurations
2.6.1 Criticality
2.6.2 Firing equivalence
2.6.3 Superstability
2.6.4 Energy minimization
2.6.5 Duality
2.6.6 Structure
2.6.7 Burning
2.7 The sandpile Markov chain
2.7.1 Avalanche operators
2.8 Exercises
3 Spanning Trees
3.1 Spanning trees
3.2 Statistics on trees
3.2.1 Level
3.2.2 Activity
3.2.3 The Tutte polynomial
3.3 Merino's theorem
3.3.1 The O-conjecture
3.4 Cori Le Bcrgne bijection
3.5 Acyclic orientations
3.5.1 Hyperplane arrangements
3.6 Parking functions
3.7 Dominoes
3.8 Avalanche polynomials
3.8.1 Avalanche polynomials of trees
3.9 Exercises
4 Sandpile Groups
4.1 Toppling dynamics
4.2 Group of chip-firing equivalence
4.3 Identity
4.4 Combinatorial invariance
4.5 Sandpile groups and invariant factors
4.5.1 Explicit forms of the sandpile group
4.5.2 Sandpile groups of random graphs
4.6 Discriminant groups
4.7 Sandpile tcrsors
4.7.1 Rotor-routing
4.7.2 Bernardi process
4.7.3 Cycle-cocycle reversal
4.8 Exercises
5 Pattern Formation
5.1 Compelling visualizations
5.2 Infinite graphs
5.3 The one-dimensional grid
5.4 Labeled chip-firing
5.5 Two and more dimensional grids
5.5.1 Odometer
5.5.2 Support
5.5.3 Backgrounds
5.5.3.1 Higher dimensions
5.5.4 Scaling limits
5.6 Other lattices
5.7 Tile identity element
5.8 Exercises
¢ò Extensions
6 Avalanche Finite Systems
6.1 M-matrices
6.2 Chip-firing on M-matrices
6.3 Stability
6.3.1 Superstability
6.3.2 Criticality
6.3.3 Energy minimization
6.3.4 Uniqueness
6.4 Burning
6.5 Directed graphs
6.5.1 Digraphs
6.5.2 Stabilization
6.5.3 Toppling time
6.5.4 Oriented spanning trees
6.6 Cartan matrices as M-matrices
6.7 M-pairings
6.8 Exercises
7 Higher Dimensions
7.1 Illustrative examples
7.2 Cell complexes
7.3 Combinatorial Laplacians
7.4 Chip-firing in higher dimensions
7.5 The sandpile group
7.6 Higher-dimensional trees
7.6.1 Enumeration of trees
7.7 Sandpile groups
7.7.1 Precise forms of sandpile groups
7.8 Cuts and flows
7.9 Stability
7.9.1 M-pairings
7.10 Exercises
8 Divisors
8.1 Divisors on curves
8.2 The Picard group and Abel-Jacobi theory
8.3 Riemann-Roch Theorems
8.3.1 The rank function
8.3.2 Proof
8.4 Torelli's theorem
8.5 The Picg(G) torus
8.6 Metric graphs and tropical geometry
8.7 Arithmetic geometry
8.8 Riemann-Roch for lattices
8.9 Two-variable zeta-functions
8.10 Enumerating arithmetical structures
8.11 Exercises
9 Ideals
9.1 Toppling ideals
9.2 Tree ideals
9.3 Resolutions
9.3.1 Cellular resolutions
9.3.2 Betti numbers
9.4 Critical ideals
9.5 Riemann Roch for monomial ideals
9.6 Exercises
List of Figures
Bibliography
Index
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