Table Of ContentProbability on Trees
and Networks
Russell Lyons
with Yuval Peres
A love and respect of trees has been characteristic of mankind since
the beginning of human evolution. Instinctively, we understood the
importance of trees to our lives before we were able to ascribe reasons
for our dependence on them.
| America’s Garden Book, James and Louise Bush-Brown, rev.
ed. by The New York Botanical Garden, Charles Scribner’s
Sons, New York, 1980, p. 142.
The cover shows a sample from the wired uniform spanning forest on the edges
of the (2;3;7)-triangle tessellation of the hyperbolic plane.
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Table of Contents
Preface vii
Chapter 1: Some Highlights 1
1. Graph Terminology 1
2. Branching Number 3
3. Electric Current 6
4. Random Walks 6
5. Percolation 8
6. Branching Processes 9
7. Random Spanning Trees 10
8. Hausdorff Dimension 14
9. Capacity 16
10. Embedding Trees into Euclidean Space 17
11. Notes 19
12. Collected In-Text Exercises 19
13. Additional Exercises 20
Chapter 2: Random Walks and Electric Networks 21
1. Circuit Basics and Harmonic Functions 21
2. More Probabilistic Interpretations 28
3. Network Reduction 32
4. Energy 37
5. Transience and Recurrence 43
6. Rough Isometries and Hyperbolic Graphs 51
7. Hitting and Cover Times 56
8. The Canonical Gaussian Field 58
9. Notes 62
10. Collected In-Text Exercises 66
11. Additional Exercises 68
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Chapter 3: Special Networks 81
1. Flows, Cutsets, and Random Paths 81
2. Trees 87
3. Growth of Trees 89
4. Cayley Graphs 95
5. Notes 99
6. Collected In-Text Exercises 99
7. Additional Exercises 100
Chapter 4: Uniform Spanning Trees 104
1. Generating Uniform Spanning Trees 104
2. Electrical Interpretations 115
3. The Square Lattice Z2 122
4. Notes 128
5. Collected In-Text Exercises 136
6. Additional Exercises 138
Chapter 5: Branching Processes, Second Moments, and Percolation 144
1. Galton-Watson Branching Processes 145
2. The First-Moment Method 151
3. The Weighted Second-Moment Method 154
4. Quasi-Independent Percolation 159
5. Transience of Percolation Clusters in Zd 161
6. Reversing the Second-Moment Inequality 165
7. Surviving Galton-Watson Trees 170
8. Harris’s Inequality 178
9. Galton-Watson Networks 180
10. Notes 183
11. Collected In-Text Exercises 185
12. Additional Exercises 186
Chapter 6: Isoperimetric Inequalities 194
1. Flows and Submodularity 194
2. Spectral Radius 202
3. Non-Backtracking Paths and Cogrowth 208
4. Relative Mixing Rate, Spectral Gap and Expansion in Finite Networks 212
5. Planar Graphs 219
6. Pro(cid:12)les and Transience 225
7. Anchored Expansion and Percolation 228
8. Euclidean Lattices and Entropy 238
9. Notes 242
10. Collected In-Text Exercises 244
11. Additional Exercises 245
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Chapter 7: Percolation on Transitive Graphs 253
1. Groups and Amenability 255
2. Tolerance and Ergodicity 260
3. The Number of In(cid:12)nite Clusters 262
4. Inequalities for p 268
c
5. Merging In(cid:12)nite Clusters and Invasion Percolation 275
6. Upper Bounds for p 281
u
7. Lower Bounds for p 283
u
8. Bootstrap Percolation on Regular Trees 290
9. Notes 292
10. Collected In-Text Exercises 297
11. Additional Exercises 298
Chapter 8: The Mass-Transport Technique and Percolation 301
1. The Mass-Transport Principle for Cayley Graphs 302
2. Beyond Cayley Graphs: Unimodularity 305
3. In(cid:12)nite Clusters in Invariant Percolation 313
4. Critical Percolation on Non-Amenable Transitive Unimodular Graphs 317
5. Bernoulli Percolation on Planar Quasi-Transitive Graphs 318
6. Properties of In(cid:12)nite Clusters 323
7. Invariant Percolation on Amenable Graphs 326
8. Appendix: Unimodularity of Planar Quasi-Transitive Graphs 329
9. Notes 333
10. Collected In-Text Exercises 337
11. Additional Exercises 338
Chapter 9: In(cid:12)nite Electrical Networks and Dirichlet Functions 343
1. Free and Wired Electrical Currents 343
2. Planar Duality 346
3. Harmonic Dirichlet Functions 348
4. Planar Graphs and Hyperbolic Graphs 356
5. Random Walk Traces 363
6. Notes 368
7. Collected In-Text Exercises 372
8. Additional Exercises 373
Chapter 10: Uniform Spanning Forests 377
1. Limits Over Exhaustions 377
2. Coupling, Harmonic Dirichlet Functions, and Expected Degree 382
3. Planar Networks and Euclidean Lattices 390
4. Tail Triviality 393
5. The Number of Trees 396
6. The Size of the Trees 406
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7. Loop-Erased Random Walk and Harmonic Measure From In(cid:12)nity 419
8. Appendix: Von Neumann Dimension and ℓ2-Betti Numbers 421
9. Notes 427
10. Collected In-Text Exercises 428
11. Additional Exercises 430
Chapter 11: Minimal Spanning Forests 434
1. Minimal Spanning Trees 434
2. Deterministic Results 437
3. Basic Probabilistic Results 442
4. Tree Sizes 444
5. Planar Graphs 451
6. Non-Treeable Groups 453
7. Notes 455
8. Collected In-Text Exercises 456
9. Additional Exercises 458
Chapter 12: Limit Theorems for Galton-Watson Processes 460
1. Size-Biased Trees and Immigration 460
2. Supercritical Processes: Proof of the Kesten-Stigum Theorem 465
3. Subcritical Processes 467
4. Critical Processes 469
5. Notes 472
6. Collected In-Text Exercises 473
7. Additional Exercises 474
Chapter 13: Speed of Random Walks 476
1. Basic Examples 476
2. The Varopoulos-Carne Bound 483
3. An Application to Mixing Time 485
4. Branching Number of a Graph 491
5. Markov Type of Metric Spaces 494
6. Embeddings of Finite Metric Spaces 498
7. Appendix: Ergodic Theorems 506
8. Notes 510
9. Collected In-Text Exercises 514
10. Additional Exercises 515
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Chapter 14: Hausdorff Dimension 520
1. Basics 520
2. Coding by Trees 524
3. Galton-Watson Fractals 530
4. H(cid:127)older Exponent 534
5. Derived Trees 537
6. Notes 541
7. Collected In-Text Exercises 541
8. Additional Exercises 542
Chapter 15: Capacity 545
1. De(cid:12)nitions 545
2. Percolation on Trees 548
3. Euclidean Space 549
4. Fractal Percolation and Brownian Intersections 553
5. Generalized Diameters and Average Meeting Height on Trees 561
6. Notes 564
7. Collected In-Text Exercises 566
8. Additional Exercises 567
Chapter 16: Random Walks on Galton-Watson Trees 570
1. Markov Chains and Ergodic Theory 570
2. Stationary Measures on Trees 573
3. Speed on Galton-Watson Trees 580
4. Harmonic Measure: The Goal 586
5. Flow Rules and Markov Chains on the Space of Trees 587
6. The H(cid:127)older Exponent of Limit Uniform Measure 590
7. Dimension Drop for Other Flow Rules 594
8. Harmonic-Stationary Measure 596
9. Con(cid:12)nement of Simple Random Walk 599
10. Numerical Calculations 603
11. Notes 608
12. Collected In-Text Exercises 609
13. Additional Exercises 610
Comments on Exercises 614
Bibliography 661
Index 701
Glossary of Notation 710
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Preface
This book is concerned with certain aspects of discrete probability on in(cid:12)nite graphs
that are currently in vigorous development. Of course, (cid:12)nite graphs are analyzed as well,
but usually with the aim of understanding in(cid:12)nite graphs and networks. These areas
of discrete probability are full of interesting, beautiful, and surprising results, many of
which connect to other areas of mathematics and theoretical computer science. Numerous
fascinating questions are still open|some are profound mysteries.
Our major topics include random walks and their intimate connection to electrical
networks; uniform spanning trees, their limiting forests, and their marvelous relationships
with random walks and electrical networks; branching processes; percolation and the pow-
erful, elegant mass-transport technique; isoperimetric inequalities and how they relate to
both random walks and percolation; minimal spanning trees and forests, and their connec-
tions to percolation; Hausdorff dimension, capacity, and how to understand them via trees;
and random walks on Galton-Watson trees. Connections among our topics are pervasive
and rich, making for surprising and enjoyable proofs.
There are three main classes of graphs on which discrete probability is most interest-
ing, namely, trees, Cayley graphs of groups (or more generally, transitive, or even quasi-
transitive, graphs), and planar graphs. More classical discrete probability has tended to
focus on the special and important case of the Euclidean lattices Zd, which are prototypical
Cayley graphs. This book develops the general theory of various probabilistic processes on
graphs and then specializes to the three broad classes listed, always seeing what we can
say in the case of Zd.
Besides their intrinsic interest, there are several reasons for a special study of trees.
Since in most cases, analysis is easier on trees, analysis can be carried further. Then one
can often either apply the results from trees to other situations or can transfer to other
situations the techniques developed by working on trees. Trees also occur naturally in
many situations, either combinatorially or as descriptions of compact sets in Euclidean
space Rd.
In choosing our topics, we have been swayed by those results we (cid:12)nd most striking,
as well as those that do not require extensive background. Thus, the only prerequisite is
basic knowledge of Markov chains and conditional expectation with respect to a (cid:27)-algebra.
For Chapter 16, basic knowledge of ergodic theory is also required, though we review it
there. Of course, we are highly biased by our own research interests and knowledge. We
include the best proofs available of recent as well as classic results.
Most exercises that appear in the text, as opposed to those at the ends of the chapters,
areonesthatwillbeparticularlyhelpfultodowhentheyarereached. Theyeitherfacilitate
one’s understanding or will be used later in the text. These in-text exercises are also
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collected at the end of each chapter for easy reference, just before additional exercises are
presented.
Some general notation we use is ⟨(cid:1)(cid:1)(cid:1)⟩ for a sequence (or, sometimes, more general
function),↾fortherestrictionofafunctionormeasuretoaset,E[X; A]fortheexpectation
of X on the event A, and j(cid:15)j for the cardinality of a set. Also, \decreasing" will mean
\non-increasing" unless we say \strictly decreasing", and likewise for \increasing" and
\non-decreasing". De(cid:12)ned terms are in bold italics. Some de(cid:12)nitions are repeated in
different chapters to enable more selective reading.
A question labelled as Question m.n is one to which the answer is unknown, where
m and n are numbers. Unattributed results are usually not due to us. Items such as
theorems are numbered in this book as C:n, where C is the chapter number and n is the
item number in that chapter.
Major chapter dependencies are indicated in the (cid:12)gure below.
1: Intro
2: RW&EN
4: UST 9: Dirich 3: SpecNet
10: USF 5: GW,2ndMom,Perc
6: Isop 12: GWLim 13: Speed 14: Hdim
7: PercTrans 15: Cap 16: RWGW
8: MTP
11: MSF
Differentinformationisgiveninthenext(cid:12)gure,wherethethicknessofanedgebetween
two chapters is proportional to the number of hyperlinked cross-references between those
two chapters, omitting all pairs where that number is at most 3.
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1 SomeHighlights
1 2 RandomWalksandElectricalNetworks
3 SpecialNetworks
2
4 UniformSpanningTrees
5 BranchingProcesses,SecondMoments,andPercolation
3 4
6 IsoperimetricInequalities
5 9 7 PercolationonTransitiveGraphs
8 TheMass-TransportTechniqueandPercolation
12 14 6
9 InfiniteElectricalNetworksandDirichletFunctions
10 UniformSpanningForests
15 13 7
11 MinimalSpanningForests
16 8 12 LimitTheoremsforGalton-WatsonProcesses
13 SpeedofRandomWalks
10
14 HausdorffDimension
15 Capacity
11
16 RandomWalksonGalton-WatsonTrees
It is possible to choose only small parts of various chapters to make a coherent course
on speci(cid:12)c subjects. For example, a judicious choice of material from the following sections
can be used for a one-semester course on relationships of probability to geometric group
theory: 3.4, 7.1, 6.1, 6.2, 6.3, 5.1, 6.7, 7.2{7.7, 8.1, 8.3, 8.4, 11.1{11.4, 11.6, 2.1{2.5, 6.6,
4.1, 4.2, 9.1, 9.3, 9.4, 10.1, 10.2, 10.9.
In the electronic version of this book, most symbols that are used with a (cid:12)xed meaning
arehyperlinkedtotheirde(cid:12)nitions, althoughthefactthatsuchhyperlinksexistisnotmade
visible.
This book began as lecture notes for an advanced graduate course called \Probability
on Trees" that Lyons gave in Spring 1993. We are grateful to Rabi Bhattacharya for
having suggested that he teach such a course. We have attempted to preserve the informal
(cid:13)avor of lectures. Many exercises at varying levels of difficulty are included, with many
comments, hints, or solutions in the back of the book. A few of the authors’ results and
proofs appear here for the (cid:12)rst time. At this point, almost all of the actual writing was
done by Lyons. We hope to have a more balanced co-authorship eventually.
Lyons is grateful to the Institute for Advanced Studies and the Institute of Mathe-
matics, both at the Hebrew University of Jerusalem, and to Microsoft Research for sup-
port during some of the writing. We are grateful to Brian Barker, Jochen Geiger, Janko
Gravner, Svante Janson, Tri Minh Lai, Steve Morrow, Peter M(cid:127)orters, Perla Sousi, Jason
(cid:19)
Schweinsberg, Jeff Steif, and Ad(cid:19)am Tim(cid:19)ar for noting several corrections to the manuscript.
In addition, G(cid:19)abor Pete helped with editing a few sections and provided a careful reading
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