Divisors and sandpiles: an introduction to Chip-Firing

341 Pages·2018·2.543 MB·English

by  Corry S., Perkinson D.

Divisors and Sandpiles
Scott Corry
David Perkinson
Department of Mathematics, Lawrence University, Appleton, Wis-
consin 54911
E-mail address: [email protected]
Department of Mathematics, Reed College, Portland, OR 97202
E-mail address: [email protected]
2010 Mathematics Subject Classi(cid:12)cation. Primary 05C25
To Madera and Diane

Contents
Preface 1
Part 1. Divisors
Chapter 1. The dollar game 7
1.1. An initial game 7
1.2. Formal de(cid:12)nitions 9
1.3. The Picard and Jacobian groups 14
Notes 16
Problems for Chapter 1 17
Chapter 2. The Laplacian 19
2.1. The discrete Laplacian 19
2.2. Con(cid:12)gurations and the reduced Laplacian 24
2.3. Complete linear systems and convex polytopes 28
2.4. Structure of the Picard group 31
Notes 39
Problems for Chapter 2 40
Chapter 3. Algorithms for winning 43
3.1. Greed 43
3.2. q-reduced divisors 46
3.3. Superstable con(cid:12)gurations 50
3.4. Dhar’s algorithm and e(cid:14)cient implementation 50
3.5. The Abel-Jacobi map 54
Notes 58
Problems for Chapter 3 59
vii
viii Contents
Chapter 4. Acyclic orientations 61
4.1. Orientations and maximal unwinnables 61
4.2. Dhar’s algorithm revisited 63
Notes 67
Problems for Chapter 4 68
Chapter 5. Riemann-Roch 69
5.1. The rank function 69
5.2. Riemann-Roch for graphs 71
5.3. The analogy with Riemann surfaces 74
5.4. Alive divisors and stability 82
Notes 84
Problems for Chapter 5 86
Part 2. Sandpiles
Chapter 6. The sandpile group 91
6.1. A (cid:12)rst example 91
6.2. Directed graphs 96
6.3. Sandpile graphs 97
6.4. The reduced Laplacian 101
6.5. Recurrent sandpiles 104
6.6. Images of sandpiles on grid graphs 108
Notes 113
Problems for Chapter 6 114
Chapter 7. Burning and duality 119
7.1. Burning sandpiles 120
7.2. Existence and uniqueness 121
7.3. Superstables and recurrents 123
7.4. Forbidden subcon(cid:12)gurations 126
7.5. Dhar’s burning algorithm for recurrents. 127
Notes 128
Problems for Chapter 7 129
Chapter 8. Threshold density 131
8.1. Markov Chains 132
8.2. The (cid:12)xed-energy sandpile 141
8.3. The threshold density theorem 151
Notes 159
Problems for Chapter 8 160
Contents ix
Part 3. Topics
Chapter 9. Trees 165
9.1. The matrix-tree theorem 166
9.2. Consequences of the matrix-tree theorem 172
9.3. Tree bijections 175
Notes 190
Problems for Chapter 9 191
Chapter 10. Harmonic morphisms 195
10.1. Morphisms between graphs 195
10.2. Branched coverings of Riemann surfaces 204
10.3. Household-solutions to the dollar game 206
Notes 211
Problems for Chapter 10 212
Chapter 11. Divisors on complete graphs 215
11.1. Parking functions 215
11.2. Computing ranks on complete graphs 218
Problems for Chapter 11 227
Chapter 12. More about sandpiles 229
12.1. Changing the sink 229
12.2. Minimal number of generators for S(G) 232
12.3. M-matrices 238
12.4. Self-organized criticality 240
Problems for Chapter 12 243
Chapter 13. Cycles and cuts 245
13.1. Cycles, cuts, and the sandpile group 245
13.2. Planar duality 250
Problems for Chapter 13 255
Chapter 14. Matroids and the Tutte polynomial 257
14.1. Matroids 257
14.2. The Tutte polynomial 259
14.3. 2-isomorphisms 262
14.4. Merino’s Theorem 263
14.5. The Tutte polynomials of complete graphs 266
14.6. The h-vector conjecture 269
Notes 273
Problems for Chapter 14 274
x Contents
Chapter 15. Higher dimensions 279
15.1. Simplicial homology 279
15.2. Higher-dimensional critical groups 287
15.3. Simplicial spanning trees 288
15.4. Firing rules for faces 292
Notes 296
Problems for Chapter 15 297
Appendix
Appendix A. 303
A.1. Undirected multigraphs 303
A.2. Directed multigraphs 307
Appendix B. 311
B.1. Monoids, groups, rings, and (cid:12)elds 311
B.2. Modules 312
Glossary of Symbols 317
Bibliography 321
Index 327
Preface
This book discusses the combinatorial theory of chip-(cid:12)ring on (cid:12)nite graphs, a sub-
ject that has its main sources in algebraic and arithmetic geometry on the one
hand and statistical physics and probabilistic models for dispersion on the other.
We have structured the text to re(cid:13)ect these two di(cid:11)erent motivations, with Part 1
devoted to the divisor theory of graphs (a discrete version of the algebraic geome-
try of Riemann surfaces) and Part 2 devoted to the abelian sandpile model (a toy
modelofaslowly-drivendissipativephysicalsystem). Thefactthattheseseemingly
di(cid:11)erent stories are in some sense two sides of the same coin is one of the beautiful
features of the subject.
To provide maximal coherence, each of the (cid:12)rst two parts focuses on a central
result: Part 1 presents a quick, elegant, and self-contained route to M. Baker and
S. Norine’s Riemann-Roch theorem for graphs, while Part 2 does the same for
L.Levine’sthresholddensitytheoremconcerningthe(cid:12)xed-energysandpileMarkov
chain. In the exposition of these theorems there are many tempting tangents, and
we have collected our favorite tangential topics in Part 3. For instructors, the
ability to include these topics should provide (cid:13)exibility in designing a course, and
the reader should feel free to pursue them at any time. As an example, a reader
wanting an introduction to simplicial homology or matroids could turn to those
chapters immediately. Of course, the choice of topics included in Part 3 certainly
re(cid:13)ects the biases and expertise of the authors, and we have not attempted to be
encyclopedic in our coverage. In particular, we have omitted a thorough discussion
of self-organized criticality and statistical physics, the relationship to arithmetic
geometry, pattern formation in the abelian sandpile model, and connections to
tropical curves.
Theaudiencewehadinmindwhilewritingthisbookwasadvancedundergrad-
uatemathematicsmajors. Indeed,bothauthorsteachatundergraduateliberalarts
colleges in the US and have used this material multiple times for courses. In addi-
tion, the second author has used this subject matter in courses at the African In-
stitute for Mathematical Sciences (AIMS) in South Africa, Ghana, and Cameroon.
1
2 Preface
The only prerequisites for reading the text are (cid:12)rst courses in linear and abstract
algebra,althoughChapter8assumesrudimentaryknowledgeofdiscreteprobability
theory, as can be easily obtained online or through a standard text such as [82].
In fact, one of the charms of this subject is that much of it can be meaningfully
and entertainingly presented even to middle school students! On the other hand,
this text is also suitable for graduate students and researchers wanting an intro-
duction to sandpiles or the divisor theory of graphs. We encourage all readers to
supplement their reading with computer experimentation. A good option is the
free open-source mathematics software system SageMath ([33], [77]), which has
extensive built-in support for divisors and sandpiles.
In addition to presenting the combinatorial theory of chip-(cid:12)ring, we have the
ulterior motive of introducing some generally useful mathematics that sits just at
theedgeofthestandardundergraduatecurriculum. Forinstance, mostundergrad-
uate mathematics majors encounter the structure theorem for (cid:12)nitely generated
abelian groups in their abstract algebra course, but it is rare for them to work
with the Smith normal form, which arises naturally in the computation of sandpile
groups. Inasimilarway,thefollowingtopicsallhaveconnectionstochip-(cid:12)ring,and
the reader can (cid:12)nd an introduction to them in the text: the matrix-tree theorem;
Markov chains; simplicial homology; M-matrices; cycle and cut spaces for graphs;
matroid theory; the Tutte polynomial. Further, for students intending to study al-
gebraicgeometry,learningthedivisortheoryofgraphsisafantasticsteppingstone
toward the theory of algebraic curves. In fact, some of the proofs of the corre-
sponding results are the same in both the combinatorial and the algebro-geometric
setting (e.g., Cli(cid:11)ord’s theorem).
We now provide a more detailed description of the text. A reader wanting to
start with sandpile theory instead of divisors should be able to begin with Part 2,
looking back to Part 1 when needed for vocabulary.
Part 1:
(cid:15) Chapter 1 introduces the dollar game in which vertices of a (cid:12)nite graph trade
dollarsacrosstheedgesinane(cid:11)orttoeliminatedebt. Thissimplegameprovides
aconcreteandtactilesettingforthesubjectofdivisorsongraphs,andthechapter
ends with a list of motivating questions.
(cid:15) Chapter2introducestheLaplacianoperator,whichisreallyourcentralobjectof
study. We reinterpret the dollar game in terms of the Laplacian, and introduce
Smith normal form as a computational tool.
(cid:15) Chapter 3 discusses algorithms for winning the dollar game or certifying un-
winnability. The (cid:12)rst section presents a greedy algorithm for winnability, while
later sections describe the more sophisticated Dhar’s algorithm along with the
attendant concepts of q-reduced divisors and superstable con(cid:12)gurations.
(cid:15) Chapter 4 continues the study of Dhar’s algorithm, using it to establish a cru-
cial bijection between acyclic orientations of the graph and maximal unwinnable
divisors.
(cid:15) Chapter 5 draws on all of the previous chapters to establish the Riemann-Roch
theoremforgraphs. Section5.3laysoutthestrikinganalogyofthedivisortheory
of graphs with the corresponding theory for Riemann surfaces.