The mathematics of chip-firing

308 Pages·2019·2.348 MB·English

by  Klivans, Caroline J

The Mathematics of
Chip-firing
Discrete Mathematics and Its Applications
Series Editors
Miklos Bona
Donald L. Kreher
Patrice Ossona de Mendez
Douglas West

Representation Theory of Symmetric Groups
Pierre-Loïc Méliot
Advanced Number Theory with Applications
Richard A. Mollin
Handbook of Linear Algebra, Second Edition
Leslie Hogben
Combinatorics, Second Edition
Nicholas A. Loehr
Handbook of Discrete and Computational Geometry, Third Edition
C. Toth, Jacob E. Goodman and Joseph O’Rourke
Handbook of Discrete and Combinatorial Mathematics, Second Edition
Kenneth H. Rosen
Crossing Numbers of Graphs
Marcus Schaefer
Graph Searching Games and Probabilistic Methods
Anthony Bonato and Paweł Prałat
Handbook of Geometric Constraint Systems Principles
Meera Sitharam, Audrey St. John, and Jessica Sidman,
Additive Combinatorics
Béla Bajnok
Algorithmics of Nonuniformity:
Tools and Paradigms
Micha Hofri and Hosam Mahmoud
Extremal Finite Set Theory
Daniel Gerbner and Balazs Patkos
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The Mathematics of
Chip-firing
Caroline J. Klivans
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Klivans, Caroline J., 1977- author.
Title: The mathematics of chip-firing / Caroline J. Klivans.
Description: Boca Raton : CRC Press, Taylor & Francis Group, 2018. | Includes
bibliographical references.
Identifiers: LCCN 2018034117 | ISBN 9781138634091
Subjects: LCSH: Graph theory. | Combinatorial analysis. | Abelian groups. |
Sequences (Mathematics)
Classification: LCC QA166 .K544 2018 | DDC 511/.5--dc23
LC record available at https://lccn.loc.gov/2018034117
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To Audrey, Aaron, and Pedro.

Contents
Preface xi
I Fundamentals 1
1 Introduction 3
1.1 A brief introduction . . . . . . . . . . . . . . . . . . . . 3
1.2 Origins and History . . . . . . . . . . . . . . . . . . . . 7
1.2.1 The abelian sandpile model . . . . . . . . . . . . 7
1.2.2 A combinatorial game . . . . . . . . . . . . . . . 9
1.2.3 Abstract rewriting systems . . . . . . . . . . . . 10
2 Chip-(cid:12)ring on Finite Graphs 11
2.1 The chip-(cid:12)ring process . . . . . . . . . . . . . . . . . . 11
2.1.1 The graph Laplacian . . . . . . . . . . . . . . . . 13
2.1.2 Cluster-(cid:12)res . . . . . . . . . . . . . . . . . . . . . 16
2.2 Con(cid:13)uence . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Toppling time . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Stabilization with a sink . . . . . . . . . . . . . . . . . 25
2.6 Long-term stable con(cid:12)gurations . . . . . . . . . . . . . 28
2.6.1 Criticality . . . . . . . . . . . . . . . . . . . . . . 29
2.6.2 Firing equivalence . . . . . . . . . . . . . . . . . 31
2.6.3 Superstability . . . . . . . . . . . . . . . . . . . . 33
2.6.4 Energy minimization . . . . . . . . . . . . . . . . 35
2.6.5 Duality . . . . . . . . . . . . . . . . . . . . . . . 38
2.6.6 Structure . . . . . . . . . . . . . . . . . . . . . . 41
2.6.7 Burning . . . . . . . . . . . . . . . . . . . . . . . 44
2.7 The sandpile Markov chain . . . . . . . . . . . . . . . . 47
2.7.1 Avalanche operators . . . . . . . . . . . . . . . . 47
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 50
vii
viii Contents
3 Spanning Trees 53
3.1 Spanning trees . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Statistics on trees . . . . . . . . . . . . . . . . . . . . . 57
3.2.1 Level. . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.2 Activity . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 The Tutte polynomial . . . . . . . . . . . . . . . 61
3.3 Merino’s theorem . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 The O-conjecture . . . . . . . . . . . . . . . . . . 66
3.4 Cori{Le Borgne bijection . . . . . . . . . . . . . . . . . 71
3.5 Acyclic orientations . . . . . . . . . . . . . . . . . . . . 73
3.5.1 Hyperplane arrangements . . . . . . . . . . . . . 76
3.6 Parking functions . . . . . . . . . . . . . . . . . . . . . 77
3.7 Dominoes . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.8 Avalanche polynomials . . . . . . . . . . . . . . . . . . 85
3.8.1 Avalanche polynomials of trees . . . . . . . . . . 88
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Sandpile Groups 93
4.1 Toppling dynamics . . . . . . . . . . . . . . . . . . . . . 94
4.2 Group of chip-(cid:12)ring equivalence . . . . . . . . . . . . . 95
4.3 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4 Combinatorial invariance . . . . . . . . . . . . . . . . . 102
4.5 Sandpile groups and invariant factors . . . . . . . . . . 104
4.5.1 Explicit forms of the sandpile group . . . . . . . 106
4.5.2 Sandpile groups of random graphs . . . . . . . . 108
4.6 Discriminant groups . . . . . . . . . . . . . . . . . . . . 111
4.7 Sandpile torsors . . . . . . . . . . . . . . . . . . . . . . 115
4.7.1 Rotor-routing . . . . . . . . . . . . . . . . . . . . 116
4.7.2 Bernardi process . . . . . . . . . . . . . . . . . . 118
4.7.3 Cycle{cocycle reversal . . . . . . . . . . . . . . . 121
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 Pattern Formation 129
5.1 Compelling visualizations . . . . . . . . . . . . . . . . . 129
5.2 In(cid:12)nite graphs . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 The one-dimensional grid . . . . . . . . . . . . . . . . . 134
5.4 Labeled chip-(cid:12)ring . . . . . . . . . . . . . . . . . . . . . 138
5.5 Two and more dimensional grids . . . . . . . . . . . . . 144
5.5.1 Odometer . . . . . . . . . . . . . . . . . . . . . . 145
5.5.2 Support . . . . . . . . . . . . . . . . . . . . . . . 146
Contents ix
5.5.3 Backgrounds . . . . . . . . . . . . . . . . . . . . 149
5.5.3.1 Higher dimensions . . . . . . . . . . . . 153
5.5.4 Scaling limits . . . . . . . . . . . . . . . . . . . . 153
5.6 Other lattices . . . . . . . . . . . . . . . . . . . . . . . 156
5.7 The identity element . . . . . . . . . . . . . . . . . . . 160
5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 163
II Extensions 167
6 Avalanche Finite Systems 169
6.1 M-matrices . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.2 Chip-(cid:12)ring on M-matrices . . . . . . . . . . . . . . . . 172
6.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.3.1 Superstability . . . . . . . . . . . . . . . . . . . . 173
6.3.2 Criticality . . . . . . . . . . . . . . . . . . . . . . 174
6.3.3 Energy minimization . . . . . . . . . . . . . . . . 175
6.3.4 Uniqueness . . . . . . . . . . . . . . . . . . . . . 176
6.4 Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.5 Directed graphs . . . . . . . . . . . . . . . . . . . . . . 179
6.5.1 Digraphs . . . . . . . . . . . . . . . . . . . . . . 180
6.5.2 Stabilization . . . . . . . . . . . . . . . . . . . . 183
6.5.3 Toppling time . . . . . . . . . . . . . . . . . . . . 185
6.5.4 Oriented spanning trees . . . . . . . . . . . . . . 186
6.6 Cartan matrices as M-matrices . . . . . . . . . . . . . . 187
6.7 M-pairings . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7 Higher Dimensions 197
7.1 Illustrative examples . . . . . . . . . . . . . . . . . . . . 197
7.2 Cell complexes . . . . . . . . . . . . . . . . . . . . . . . 201
7.3 Combinatorial Laplacians . . . . . . . . . . . . . . . . . 204
7.4 Chip-(cid:12)ring in higher dimensions . . . . . . . . . . . . . 208
7.5 The sandpile group . . . . . . . . . . . . . . . . . . . . 211
7.6 Higher-dimensional trees . . . . . . . . . . . . . . . . . 211
7.6.1 Enumeration of trees . . . . . . . . . . . . . . . . 215
7.7 Sandpile groups . . . . . . . . . . . . . . . . . . . . . . 217
7.7.1 Precise forms of sandpile groups . . . . . . . . . 220
7.8 Cuts and (cid:13)ows . . . . . . . . . . . . . . . . . . . . . . . 220
7.9 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.9.1 M-pairings . . . . . . . . . . . . . . . . . . . . . 224
7.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 227