Planar maps, random walks and circle packing

126 Pages·2020·1.899 MB·English

by  Nachmias A

Lecture Notes in Mathematics 2243
École d'Été de Probabilités de Saint-Flour
Asaf Nachmias
Planar Maps,
Random Walks
and Circle
Packing
École d'Été de Probabilités
de Saint-Flour XLVIII - 2018
Lecture Notes in Mathematics 2243
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryEditors:
KarinBaur,Leeds
MichelBrion,Grenoble
CamilloDeLellis,Princeton
AlessioFigalli,Zurich
AnnetteHuber,Freiburg
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Cambridge
AngelaKunoth,Cologne
ArianeMézard,Paris
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
GabrieleVezzosi,Firenze
AnnaWienhard,Heidelberg
Moreinformationaboutthissubseriesathttp://www.springer.com/series/7098
Saint-Flour Probability Summer School
The Saint-Flour volumes are reflections of the courses given at the
Saint-Flour Probability Summer School. Founded in 1971, this
school is organised every year by the Laboratoire de Mathéma-
tiques (CNRS and Université Clermont Auvergne, Clermont-
Ferrand, France). It is intended for PhD students, teachers and
researchers who are interested in probability theory, statistics, and
in their applications.
The duration of each school is 12 days (it was 17 days up to 2005), and up to 100 participants
can attend it. The aim is to provide, in three high-level courses, a comprehensive study of some
fields in probability theory or Statistics. The lecturers are chosen by an international scientific
board. The participants themselves also have the opportunity to give short lectures about their
research work.
Participants are lodged and work in the same building, a former seminary built in the 18th
century in the city of Saint-Flour, at an altitude of 900 m. The pleasant surroundings facilitate
scientific discussion and exchange.
The Saint-Flour Probability Summer School is supported by:
Laboratoire de Mathématiques Blaise Pascal
Université Clermont Auvergne
Centre National de la Recherche Scientifique (C.N.R.S.)
For more information, see
http://recherche.math.univ-bpclermont.fr/stflour/stflour-en.php
Christophe Bahadoran
[email protected]
Arnaud Guillin
[email protected]
Hacène Djellout
Hacè[email protected]
[email protected]
Université Clermont Auvergne - Aubière cedex, France
Asaf Nachmias
Planar Maps, Random Walks
and Circle Packing
École d’Été de Probabilités de Saint-Flour
XLVIII - 2018
AsafNachmias
DepartmentofMathematicalSciences
TelAvivUniversity
TelAviv,Israel
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISSN0721-5363
Écoled’ÉtédeProbabilitésdeSaint-Flour
ISBN978-3-030-27967-7 ISBN978-3-030-27968-4 (eBook)
https://doi.org/10.1007/978-3-030-27968-4
MathematicsSubjectClassification(2010):Primary:82B41;Secondary:52C26
Thisbookisanopenaccesspublication.
©TheEditor(s)(ifapplicable)andTheAuthor(s)2020
Open Access This bookis licensed under the terms of the Creative Commons Attribution 4.0Inter-
nationalLicense(http://creativecommons.org/licenses/by/4.0/), whichpermitsuse,sharing,adaptation,
distribution andreproduction inanymediumorformat,aslong asyougive appropriate credit tothe
originalauthor(s)andthesource,providealinktotheCreativeCommonslicenceandindicateifchanges
weremade.
Theimages or other third party material in this book are included in the book’s Creative Commons
licence,unlessindicatedotherwiseinacreditlinetothematerial.Ifmaterialisnotincludedinthebook’s
CreativeCommonslicenceandyourintendeduseisnotpermittedbystatutoryregulationorexceedsthe
permitteduse,youwillneedtoobtainpermissiondirectlyfromthecopyrightholder.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG.
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Écoled’ÉtédeProbabilitésdeSaint-FlourXLVIII-2018
Preface
These lecture notesare intendedto accompanya single-semestergraduatecourse.
They are meant to be entirely self-contained.All the theory required to prove the
mainresultsispresentedandonlybasicknowledgeinprobabilitytheoryisassumed.
In Chap.1, we describe the main storyline of this text. It is meant to be light
bedtimereadingexposingthereadertothe mainresultsthatwillbepresentedand
providingsome background.Chapter 2 introduces the theory of electric networks
and discusses their highly useful relations to random walks. It is roughly based
onChap.8ofYuvalPeres’ excellentlecturenotes[69].We thendiscussthecircle
packingtheoremandpresentitsproofinChap.3.Chapter4discussesthebeautiful
theorem of He and Schramm [40], relating the circle packing type of a graph to
recurrenceandtransienceoftherandomwalkonit.Tothebestofourknowledge,
their work is the first to form connections between the circle packing theorem
and probabilitytheory.Next in Chap.5, we presentthe highly influential theorem
of Benjamini and Schramm [11] about the almost sure recurrence of the simple
randomwalkinplanargraphlimitsofboundeddegrees.Thenotionofalocallimit
(also known as distributionallimit or Benjamini-Schrammlimit) of a sequence of
finite graphs was introduced there for the first time to our knowledge (and also
studiedbyAldous–Steele[3]andAldous–Lyons[2]);thisnotionishighlyimportant
in probability theory as well as other mathematical disciplines (see [2] and the
referenceswithin). In Chap.6, we provide a theorem from which one can deduce
thealmostsurerecurrenceofthesimplerandomwalkonmanymodelsofrandom
planar maps. This theoremwas obtained by Ori Gurel-Gurevichand Nachmias in
[31]. Chapter7 discussesuniformspanningforestsonplanarmapsandappealsto
thecirclepackingtheoremtoshowthatthefreeuniformspanningforestonproper
planar maps is almost surely connected,i.e., it is in fact a tree. This theorem was
obtainedbyHutchcroftandNachmiasin[45].WeclosethesenotesinChap.8with
a descriptionof some relatedcontemporarydevelopmentsin thisfield thatarenot
presentedinthistext.
vii
viii Preface
Wehavemadeanefforttoaddvaluebeyondwhatisinthepublishedpapers.Our
proofof thecircle packingtheoremin Chap.3 is inspiredbyThurston’sargument
[82]andBrightwell–Scheinerman[13],butwehavemadewhatwethinkaresome
simplifications;theproofalsoemploysaneatargumentduetoOhadFeldheimand
OriGurel-Gurevich(Theorem3.14)whichmakesthedrawingpartoftheargument
rather straightforward and avoids topological considerations that are used in the
classical proofs. The original proof of the He–Schramm Theorem [40] is based
on the notion of discrete extremal length which is essentially a form of effective
resistance in electric networks (in fact, the edge extremal length is precisely an
effectiveresistance,see[61,Exercise2.78]).We findthatourapproachinChap.4
using electric networksis somewhat more robust and intuitive to probabilists. We
obtainaquantitativeversionoftheHe-SchrammTheoreminChap.4aswellasthe
Benjamini–SchrammTheorem[11]inChap.5(seeTheorem5.8).Thesequantified
versionsarekeytotheproofsofChap.6.Lastly,someaspectsofstationaryrandom
graphsarebetterexplainedhereinChap.6thaninthepublication[31].Thevideoed
lecturesofthiscoursetakenin48thSaint-Floursummerschoolareavailableatthe
author’swebpagehttp://www.math.tau.ac.il/~asafnach/.
Acknowledgments
IwouldliketodeeplythankDanielJerison,PelegMichaeli,andMatanShalevfor
typing, editing, and proofreading most of this text and for the many comments,
corrections,andsuggestions.IamindebtedtoTomHutchcroftforhisassistancein
writingtheintroductionandforsurveyingrelatedtopicsnotincludedinthesenotes
(Chaps.1 and 8). I thank Sébastien Martineau,Pierre Petit, Dominik Schmid, and
MateoWirthfortheircorrectionsandcommentstothistext.Iamalsogratefultothe
participantsofthe48thSaint-FlourSummerSchoolanditsorganizers,Christophe
Bahadoran, Arnaud Guillin, and Hacène Djellout, for a very enjoyable summer
school.
Iambeholdentomycollaboratorsonthesetopics:OmerAngel,MartinBarlow,
Itai Benjamini, Nicolas Curien, Ori Gurel-Gurevich, Tom Hutchcroft, Daniel
Jerison,GourabRay,SteffenRohde,andJuanSouto.Ihavelearnedalotfromour
work and conversations. Special thanks go to Ori Gurel-Gurevich for embarking
together on this research endeavor beginning in 2011 at the University of British
Columbia,Vancouver,Canada. Many of the ideas and methodspresentedin these
noteswereobtainedinourjointwork.
Preface ix
IamalsohighlyindebtedtothelateOdedSchrammwhosemathematicalwork,
originality, and vision, especially in the topics studied in these notes, have been
an enormous source of inspiration. It is no coincidence that his name appears on
almostevery other page here.It has become routinefor my collaboratorsand I to
askourselves“WhatwouldOdeddo?”hopingthatreflectingonthisquestionwould
cutrightto the heartofmatters. SteffenRohde’swonderfulsurvey[71] ofOded’s
workisverymuchrecommended.
Lastly,IthankShiraWilkofandbabyAdaformuseandinspiration.
TelAviv,Israel AsafNachmias1
July2019
1ThisprojecthasreceivedfundingfromtheEuropeanResearchCouncil(ERC)undertheEuropean
Union’s, Horizon 2020 research and innovation programme (grant agreement number 676970,
RANDGEOM).
Contents
1 Introduction .................................................................. 1
1.1 TheCirclePackingTheorem........................................... 1
1.2 ProbabilisticApplications .............................................. 7
2 RandomWalksandElectricNetworks.................................... 11
2.1 HarmonicFunctionsandVoltages...................................... 11
2.2 FlowsandCurrents ..................................................... 14
2.3 TheEffectiveResistanceofaNetwork ................................ 17
2.4 Energy................................................................... 22
2.5 InfiniteGraphs .......................................................... 25
2.6 RandomPaths........................................................... 29
2.7 Exercises................................................................. 30
3 TheCirclePackingTheorem ............................................... 33
3.1 PlanarGraphs,MapsandEmbeddings................................. 33
3.2 ProofoftheCirclePackingTheorem.................................. 36
4 ParabolicandHyperbolicPackings........................................ 47
4.1 InfinitePlanarMaps..................................................... 47
4.2 TheRingLemmaandInfiniteCirclePackings........................ 48
4.3 StatementoftheHe–SchrammTheorem .............................. 50
4.4 ProofoftheHe–SchrammTheorem ................................... 52
4.5 Exercises................................................................. 59
5 PlanarLocalGraphLimits................................................. 61
5.1 LocalConvergenceofGraphsandMaps .............................. 61
5.2 TheMagicLemma...................................................... 64
5.3 RecurrenceofBoundedDegreePlanarGraphLimits................. 67
5.4 Exercises................................................................. 71
xi