Progress in High-Dimensional Percolation and Random Graphs

285 Pages·2017·2.511 MB·English

by  Markus Heydenreich, Remco van der Hofstad

CRM Short Courses
Markus Heydenreich
Remco van der Hofstad
Progress in High-
Dimensional
Percolation and
Random Graphs
CRM Short Courses
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Markus Heydenreich (cid:2) Remco van der Hofstad
Progress in High-Dimensional
Percolation and Random
Graphs
MarkusHeydenreich RemcovanderHofstad
MathematischesInstitut DepartmentofMathematicsand
Ludwig-Maximilians-UniversitätMünchen ComputerScience
Munich,Bayern EindhovenUniversityofTechnology
Germany Eindhoven
TheNetherlands
CRMShortCourses
ISBN978-3-319-62472-3 ISBN978-3-319-62473-0(eBook)
DOI10.1007/978-3-319-62473-0
LibraryofCongressControlNumber:2017945258
MathematicsSubjectClassification(2010):60K35,60K37,82B43
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Preface
Thisbookfocusesonpercolationonhigh-dimensionallattices.Wegiveageneralintroduction
topercolation,statingthemainresultsanddefiningthecentralobjects.Weassumenoprior
knowledge about percolation. This text is aimed at graduate students and researchers who
wishtoenterthewondrousworldofhigh-dimensionalpercolation,withtheaimtodemystify
thelace-expansionmethodologythathasbeenthekeytechniqueinhighdimensions.Thistext
canbeusedforreadingseminarsoradvancedcoursesaswellasforreferenceandindividual
study.Theexpositioniscomplementedwithmanyexercises,andweinvitereaderstotrythem
outandgaindeeperunderstandingofthetechniquespresentedhere.Letusnowsummarize
thecontentinmoredetail.
We describe mean-field results in high-dimensional percolation that make the intuition
that“farawaycriticalpercolationclustersareclosetobeingindependent”precise.Wehave
two main purposes. The first main purpose is to give a self-contained proof of mean-field
behavior for high-dimensional percolation, by proving that percolation in high dimensions
hasmean-field criticalexponents ˇ D (cid:2) D 1,ı D 2and(cid:3) D 0,asforpercolation onthe
tree. This proof is obtained by combining the Aizenman–Newman and Barsky–Aizenman
differential inequalities, that rely on the triangle condition, with the lace-expansion proof
ofHaraandSladeoftheinfraredboundthat,inturn,verifiesthetrianglecondition.While
thereareexpositorytextsdiscussinglace-expansionmethodology,suchasSlade’sexcellent
Saint-Flourlecturenotes,anintroductiontohigh-dimensionalpercolationdidnotyetexist.
Asidefromtheseclassicalresults,thatarenowover25yearsold,oursecondmainpurpose
is to discuss recent extensions and additions. We focus on (1) the recent proof that mean-
field critical behavior holds for percolation in d (cid:3) 11; (2) the proof of existence of arm
exponents; (3) results on finite-size scaling and percolation on high-dimensional tori and
theirrelationshiptotheErdo˝s–Rényirandomgraph;(4)extensionsofthesefinite-sizescaling
results to hypercube percolation; (5) the existence of the incipient infinite cluster and its
scaling properties, as well as the proof of the Alexander–Orbach conjecture for random
walksonthehigh-dimensionalincipientinfinitecluster;(6)thenovellaceexpansionforthe
two-point function with a fixed number of pivotals; and (7) super-process limits of critical
percolationclusters.Thetextisenrichedwithnumerousopenproblems,which,wehope,will
stimulatefurtherresearchinthefield.
v
vi Preface
Thistextisorganizedasfollows.InPartI,consistingofChaps.1–3,weintroduceperco-
lationandproveitsmainpropertiessuchasthesharpnessofthephasetransition.InPartII,
consisting of Chaps. 4–9, we discuss mean-field critical behavior by describing the two
main techniques used, namely, differential inequalities and the lace expansion. In Parts I
and Part II, all results are proved, making this the first self-contained text discussing high-
dimensionalpercolation.InPartIII,consistingofChaps.10–13,wedescriberecentprogress
inhigh-dimensionalpercolation.Weprovidepartialproofsandgivesubstantialoverviewand
heuristics about how the proofs are obtained. In many of these results, the lace expansion
anddifferentialinequalitiesortheirdiscreteanaloguesarecentral.InPartIV,consistingof
Chaps.14–16,wediscussrelatedmodelsandfurtheropenproblems.Hereweonlyprovide
heuristicsandfewdetailsoftheproofs,thusfocussingontheoverviewandbigpicture.
Thistextcouldnothavebeenwrittenwithouthelpfrommany.WearegratefultoKilian
Matzke, Andrea Schmidbauer, Gordon Slade, Si Tang, Sebastian Ziesche, as well as the
readinggroupsinGenevaandSapporo,forvaluablecommentsandpointingouttyposand
omissionsinanearlierversionofthemanuscript.SpecialthanksgotoRobertFitzner,who
kindlypreparedthegraphicsinthistext.
Thisworkwouldnothavebeenpossiblewithoutthegeneroussupportofvariousinstitu-
tions.TheworkofMHissupportedbytheNetherlandsOrganisationforScientificResearch
(NWO) through VENI grant 639.031.035. The work of RvdH is supported by the Nether-
landsOrganisationforScientificResearch(NWO)throughVICIgrant639.033.806andthe
GravitationNetworksgrant024.002.003.
ThecontentofthisbookhasbeenpresentedbyRvdHattheCRM–PIMSSummerSchoolin
Probability2015inMontréal.HewarmlythankstheorganizersLouigiAddario-Berry,Omer
Angel,Louis-PierreArguin,MartinBarlow,EdPerkinsandLeaPopovicforthisopportunity,
aswellastheCRMforgeneroussupport.
Bothofushavebeenworkingformanyyearsonpercolationandthelaceexpansion,and
wethankourcolleaguesforjoyfulandinspiringcollaborationsthatledtomanyjointarticles.
Mostnotably,wethankChristianBorgs,JenniferChayes,RobertFitzner,TakashiHara,Frank
denHollander,MarkHolmes,TimHulshof,AsafNachmias,AkiraSakai,GordonSladeand
JoelSpencer.Thankyouforparticipatinginthewonderfuljourneysinthisbeautifulbranch
ofmathematicalresearch!
Munich,Germany MarkusHeydenreich
Eindhoven,TheNetherlands RemcovanderHofstad
March2017
Contents
PartI IntroductiontoPercolation
1 IntroductionandMotivation ........................................... 3
1.1 IntroductionoftheModel........................................... 3
1.2 CriticalBehavior .................................................. 7
1.3 Russo,FKG,andBKR ............................................. 12
1.4 AimofThisBookandWhatIsNew?................................. 15
1.5 OrganizationandNotation .......................................... 17
2 FixingIdeas:PercolationonaTreeandBranchingRandomWalk .......... 19
2.1 PercolationonaTree .............................................. 19
2.2 BranchingRandomWalkasthePercolationMean-FieldModel ........... 25
3 UniquenessofthePhaseTransition...................................... 31
3.1 MainResult ...................................................... 31
3.2 TheDuminil-CopinandTassionUniquenessProof...................... 32
3.3 TheAizenman–BarskyUniquenessProof ............................. 36
3.4 TheMagnetization ................................................ 37
3.5 TheLowerBoundontheMagnetization............................... 38
3.6 Aizenman–BarskyProofofMean-FieldLowerBoundon(cid:4).p/ ........... 39
3.7 ProofoftheAizenman–BarskyDifferentialInequalities ................. 40
PartII Mean-FieldBehavior:DifferentialInequalitiesandtheLaceExpansion
4 CriticalExponentsandtheTriangleCondition ........................... 47
4.1 DefinitionoftheTriangleCondition .................................. 47
4.2 TheSusceptibilityCriticalExponent(cid:2) ................................ 48
4.3 Mean-FieldBoundsontheCriticalExponentsıandˇandOverview ...... 54
5 ProofofTriangleCondition:TheInfraredBound......................... 55
5.1 TheInfraredBound................................................ 55
5.2 Spread-OutModels................................................ 59
vii
viii Contents
5.3 OverviewofProof:ALace-ExpansionAnalysis........................ 60
5.4 TheRandomWalkTriangle ......................................... 62
6 TheDerivationoftheLaceExpansionviaInclusion–Exclusion ............. 65
6.1 TheInclusion–ExclusionLaceExpansion ............................. 65
6.2 DerivationoftheInclusion–ExclusionLaceExpansion .................. 67
6.3 FullExpansion:HowtoDealwiththeErrorTerm ...................... 74
7 DiagrammaticEstimatesfortheLaceExpansion ......................... 77
7.1 OverviewoftheBounds............................................ 77
7.2 AWarmUp:Boundson˘.0/and˘.1/ ............................... 81
7.3 BoundsontheLaceExpansion:BoundingEvents ...................... 87
7.4 BoundsontheLaceExpansion:DiagrammaticEstimates ................ 90
7.5 BoundsontheLaceExpansion:ReductiontoSimpleDiagrams........... 93
7.6 OutlookontheRemainderoftheArgument............................ 99
8 BootstrapAnalysisoftheLaceExpansion ............................... 101
8.1 OverviewoftheBootstrapArgument ................................. 101
8.2 TheBootstrapFunctions............................................ 104
8.3 ConsequencesoftheBootstrapBound ................................ 109
8.4 TheBootstrapArgumentCompleted.................................. 116
8.5 ConsequencesoftheCompletedBootstrap ............................ 123
9 Proofthatı D2andˇ D1undertheTriangleCondition.................. 125
9.1 ADifferentialInequalityInvolvingtheTriangle ........................ 125
9.2 TheClusterTailCriticalExponentı.................................. 126
9.3 ThePercolation-FunctionCriticalExponentˇ ......................... 128
9.4 ProofoftheDifferentialInequalityInvolvingtheTriangle................ 131
PartIII Mean-FieldBehavior:RecentResults
10 TheNonbacktrackingLaceExpansion................................... 141
10.1 Mean-FieldBehaviorford (cid:3)11..................................... 141
10.2 NonbacktrackingWalk ............................................. 143
10.3 TheNonbacktrackingLaceExpansion(NoBLE) ....................... 147
10.4 TheNoBLEBootstrapArgument .................................... 151
10.5 TheNumericalAnalysis............................................ 152
11 FurtherCriticalExponents............................................. 155
11.1 Correlation-LengthExponents(cid:5) and(cid:5)2andGapExponent(cid:6) ............ 155
11.2 TheTwo-Point-FunctionCriticalExponent(cid:3) .......................... 157
11.3 ArmExponents(cid:7) and(cid:7) .......................................... 160
in ex
11.4 OnthePercolationUpperCriticalDimension .......................... 166
12 Kesten’sIncipientInfiniteCluster....................................... 169
12.1 MotivationfortheIncipientInfiniteCluster............................ 169
12.2 TheIncipientInfiniteClusterinHighDimensions ...................... 170
Contents ix
13 Finite-SizeScalingandRandomGraphs ................................. 175
13.1 Inspiration:TheErdo˝s–RényiRandomGraph.......................... 176
13.2 CriticalHigh-DimensionalTori...................................... 184
13.3 GeneralHigh-DimensionalTori...................................... 186
13.4 HypercubePercolation ............................................. 195
13.5 ScalingLimitsofCriticalRandomGraphs............................. 206
13.6 TheRoleofBoundaryConditions.................................... 213
PartIV RelatedandOpenProblems
14 RandomWalksonPercolationClusters.................................. 221
14.1 RandomWalksontheInfiniteCluster................................. 221
14.2 RandomWalksonFiniteCriticalClusters ............................. 223
14.3 RandomWalkontheIncipientInfiniteCluster ......................... 226
15 RelatedResults ....................................................... 231
15.1 Super-ProcessLimitsofPercolationClusters .......................... 231
15.2 OrientedPercolation ............................................... 240
15.3 ScalingLimitofPercolationBackbones............................... 242
15.4 Long-RangePercolation............................................ 245
15.5 TheAsymptoticExpansionoftheCriticalValue........................ 248
15.6 PercolationonNonamenableGraphs ................................. 251
16 FurtherOpenProblems................................................ 257
16.1 InvasionPercolation ............................................... 257
16.2 RandomWalkPercolationandInterlacements.......................... 264
16.3 Scale-FreePercolation ............................................. 267
16.4 FKPercolation.................................................... 270
Bibliography ............................................................ 275
Symbols
EıF disjointoccurrenceofeventsE andF;cf.(1.3.3)
x !y verticesxandy areconnected;cf.p.3
x ()y Dfx !ygıfx !yg,verticesxandy aredoublyconnected;
cf.Def.6.3(a)
C.x/ Dfy Wx !yg,clusterofvertexx;cf.p.3
Ceb.x/ restrictedclusterwithoutusingbondb;cf.Def.6.3(b)
(cid:4).p/ DPp.jC.x/jD1/,clusterdensity/percolationfunction;
cf.(1.1.1)
p Dinffp W(cid:4).p/>0g,percolationcriticalthreshold;cf.(1.1.2)
c
(cid:8).p/ DEpŒ(cid:2)jC.x/j(cid:9),expectedc(cid:3)lustersize/susceptibility;cf.(1.1.3)
(cid:8)f.p/ DEp jC.x/j1fjC.x/j<1g ,truncatedexpectedclustersize;
cf.(1.1.5)
(cid:10)p.x;y/ DPp.x !y/,two-pointfunction;cf.(1.1.6)
(cid:10)pf.x;y/ DPp.x !y;jC.x/j<1/,truncatedtwo-pointfunction;
cf.(1.1.7)
(cid:11).p/ D(cid:4)qlimn!1n=log(cid:10)pf.ne1/,correlationlength;cf.(1.1.11)
P
(cid:11)2.p/ D .1=(cid:8)f.p// x2Zdjxj2(cid:10)pf.x/,averageradiusofgyration;
cf.(1.1.12)
ˇ percolationfunctioncrit.exponent;cf.(1.2.1)
(cid:2) susceptibilitycrit.exponent;cf.(1.2.5)
(cid:5) correlationlengthcrit.exponent;cf.(1.2.7)
(cid:5)2 av.radiusofgyrationcrit.exp;cf.(1.2.8)
(cid:6) gapcrit.exponent;cf.(1.2.9)
ı clustertailcrit.exponent;cf.(1.2.10)
(cid:7) extrinsicone-armcrit.exponent;cf.(1.2.11)
ex
(cid:7) intrinsicone-armcrit.exponent;cf.(1.2.12)
in
(cid:3) two-pointfunctioncrit.exponent;cf.(1.2.13)
D.x/ D1PfjxjD1g=.2d/,simplerandomwalkstepdistribution;cf.(1.2.18)
C(cid:2).x/ D n(cid:2)0(cid:12)nD?n.x/,randomwalkGreen’sfunction;cf.(2.2.8)
J.x/ Dp1fjxjD1g D2dpD.x/;cf.(6.2.1)
A
x !y connectionthroughA;cf.Def.6.2
xi