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Advances
in
Mathematics
www.elsevier.com/locate/aim
Quenched
Voronoi
percolation
✩Daniel Ahlberga,b, Simon Griffithsc, Robert Morrisa,∗, Vincent Tassiond
a
IMPA,EstradaDonaCastorina110,JardimBotânico,RiodeJaneiro,RJ,Brazil
b
DepartmentofMathematics,UppsalaUniversity,SE-75106Uppsala,Sweden
cDepartmentofStatistics,UniversityofOxford,Oxford,UnitedKingdom dDépartementdeMathématiques, UniversitédeGenève,Genève,Switzerland
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received16January2015 Accepted14September2015 CommunicatedbytheManaging EditorsofAIM
Keywords: Voronoipercolation Noisesensitivity
Quenchedcrossingprobabilities
Weprove that theprobability ofcrossing a large squarein quenchedVoronoipercolationconvergesto1/2 atcriticality,
confirming a conjecture of Benjamini, Kalai and Schramm
from1999.Themainnewtoolsareaquenchedversionofthe box-crossing property for Voronoi percolation at criticality,
and an Efron–Stein type bound on the variance of the
probabilityofthecrossingevent intermsof thesumof the squares of the influences. As a corollary of the proof, we moreoverobtainthatthequenchedcrossingeventatcriticality isalmostsurelynoisesensitive.
© 2015TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).
✩ Researchsupportedinpartbypostdoctoral grant 637-2013-7302fromthe SwedishResearchCouncil
(D.A.),EPSRCgrantEP/J019496/1(S.G.),CNPq(Proc. 479032/2012-2andProc. 303275/2013-8)(R.M.), andANRgrantMAC2(ANR-10-BLAN-0123),theSwissNSFandNCCRSwissmap(V.T.).
* Correspondingauthor.
E-mailaddresses:ahlberg@impa.br(D. Ahlberg),simon.griffiths@stats.ox.ac.uk(S. Griffiths),
rob@impa.br(R. Morris),Vincent.Tassion@unige.ch(V. Tassion).
http://dx.doi.org/10.1016/j.aim.2015.09.005
0001-8708/© 2015TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Thenoisesensitivity ofaBooleanfunctionwasintroducedin1999inaseminalpaper
ofBenjamini,KalaiandSchramm[5],andhassincedevelopedintoanimportantareaof probabilitytheory(see,e.g.,[13,14,22]),linkingdiscreteFourieranalysiswithpercolation theory and combinatorics.Oneof themain resultsof [5]gave asufficientconditionfor asequenceof functionsfn:{0,1}n → {0,1} tobe sensitivetosmall amountsofrandom
noiseinthefollowingprecisesense:ifω∈ {0,1}n ischosenuniformlyatrandom,andωε is obtainedfromω byresamplingeachvariable withsomefixedprobabilityε> 0,then fn(ω) andfn(ωε) areasymptoticallyindependent.Theyusedthistheoremtoshowthat
the sequenceoffunctionswhichencodescrossingsof n× n squares inbondpercolation on Z2 is noisesensitive. Thus, even if oneknows all buta random o(1)-proportion of
theedges,onestill(withhighprobability)hasverylittleinformationaboutthecrossing event.
Theauthorsof[5]furthermoremadeanumberofconjecturesregardingmoreprecise notionsof sensitivityand sensitivityto different typesofnoise.Severalofthese conjec-tures have since played an important role inthe subsequent development of the area, mostspectacularlyin[22]and[13],whereextremelypreciseresultswereobtainedabout the Fourierspectrumofthe crossingevent,andabout the‘dynamical percolation’ pro-cess introduced byHäggström,Peres andSteif[15] and (independently)byBenjamini, see[25].Togiveanotherexample,theymadethefollowingconjectureforBernoullibond percolationonthesquarelattice:evenifyouaretoldthestatusofall theverticaledges, youstillhaveverylittleinformationaboutthecrossingevent.Thisconjecturewasproved byGarban,PeteandSchramm[13,Theorem 1.3],asaconsequenceoftheirveryprecise boundsontheFourierspectrum.Notethatinthistheoremwearegivenadeterministic set ofedges(ofdensity1/2),ratherthanarandom setof edges(ofdensity1− o(1))as intheresultstatedabove.
Inthis paper,wewill proveasimilarresult(alsoconjecturedin[5])inthesettingof
Voronoi percolation: thatknowingthe pointset (but notthe coloursof thecells) gives
asymptoticallynoinformationaboutthecrossingevent.Inordertostateourmainresult precisely, wewillneedafewbasicdefinitions.
Consideraset η ofn pointsinthesquareS = [0,1]2,eachchosenindependentlyand
uniformly atrandom.Foreachu∈ η,define theVoronoi(or Dirichlet)cell1 ofu tobe
C(u) =x∈ [0, 1]2 : u − x2 v − x2 for every v∈ η
,
and letω:η → {−1,1} be auniformlyrandom two-colouringof thepointsof η;wewill
call the points u (and the associated cells C(u)) with ω(u) = 1 ‘red’ and those with
ω(u) =−1 ‘blue’. Wesay thatthere isared horizontalcrossing of S if there isapath
1 ThestudyoftheseobjectsdatesbackatleasttoDirichlet[10]in1850,whousedtheminhisworkon
quadraticforms,althoughtheyappeartohavebeenintroducedevenearlier,byKeplerand(independently) Descartes,see[17].Thenaturalgeneralisationtod dimensionswasfirststudiedbyVoronoi[27]in1908.
fromtheleft- totheright-handsideofS thatonlyintersectsredcells,andwriteHS for
theeventthatthere existssucharedhorizontalcrossingofS.Note thatP(HS)= 1/2,
bysymmetry.Wereferthereader whoisunfamiliarwith Voronoipercolationto [8]for amoreextensiveintroduction.
Thefollowing theorem confirms (in astrong form) aconjecture ofBenjamini, Kalai andSchramm[5].
Theorem1.1. Thereexistsc> 0 suchthat P 1 2− n −c PH S| η 1 2+ n −c 1 − n−c
forallsufficientlylarge n∈ N.
Letfη:{−1,1}η → {0,1} bethefunctionsuchthatfη(ω)= 1 ifandonlyifH
S holds.
Thekeynew ideaoftheproof ofTheorem 1.1is thefollowing Efron–Steintype bound (seeTheorem 2.1,below)onthevarianceoftheprobabilityofthecrossingeventinterms oftheinfluencesoffη,whichcanbe viewedasarandomBooleanfunction:
Var PHS| η n m=1 E Infm(fη)2 . (1)
Recall that the influence Infm(fn) of the m-th variable of a Boolean function
fn:{−1,1}n → {0,1} is defined to be the expected absolute change in fn when the
signofthem-th variableisflipped,i.e.,
Infm(fn) =P
fn(ω)= fn(ω)
,
where ω is chosenuniformly, and ω is obtained from ω by flipping the m-th variable. Benjamini,KalaiandSchramm[5]provedthat
n
m=1
Infm(fn)2→ 0 as n → ∞ ⇒ (fn)n∈N is noise sensitive, (2)
and moreover introduced a technique (the ‘algorithm method’, see below) which can often be used to bound nm=1Infm(fn)2 when fn encodes crossing events in
percola-tion models. We will use this method (or, more precisely, the ‘randomised’version of it developed by Schramm and Steif [22]), together with anew ‘box-crossing property’ for quenched Voronoi percolation (see below),to bound2 n
m=1Infm(fη)2, and hence
deduceTheorem 1.1.
2
Moreprecisely,sincefηisarandomfunctionwewillprovethatourboundon nm=1Infm(fη)2 holds
As animmediate consequenceoftheproofoutlinedabove,togetherwith(2),wealso obtain the following theorem. Let us say that quenched Voronoi percolation is almost
surely noisesensitiveatcriticality if
E fη(ω)fη(ωε)| η − E fη(ω)| η E fη(ωε)| η → 0 (3)
as n→ ∞ withprobability 1 foreveryε∈ (0,1),whereω andωε areasdefinedabove.
Theorem1.2.QuenchedVoronoipercolationisalmostsurelynoisesensitiveatcriticality. Infact,as aconsequenceof theSchramm–Steifmethod,weobtainastrongerresult: that the noise sensitivity exponent for quenched Voronoi percolation is positive. This meansthatthereexists aconstantc> 0 such that(3)holdseven ifε= n−c.
Remark 1.3. The word ‘quenched’ refers to the fact that we are proving a statement whichholds foralmostallchoicesofη.Thephrase‘atcriticality’refersto thefactthat ω is chosenuniformly at random.We remindthereaderthatthecritical probabilityof Voronoipercolationintheplaneis1/2,as wasprovedbyBollobásandRiordan[9].
We remark that Theorem 1.2 is not the first result of this type for a continuum percolationmodel.Indeed,asimilartheoremforthePoissonBooleanmodel3wasproved bythefirstthree authorswithBroman[2],andthetechniquesintroducedinthatpaper have recently been extended by the first three authors with Balister and Bollobás [1] to thesettings of (annealed) Voronoi percolationand the Poisson Booleanmodel with randomradii.(Ineachcasethepointsetη isperturbed,togetherwiththecolours/radii.) We emphasise, however, that the techniques introduced in this paper are completely different from those used in [1,2], where the method involved choosing the point set in two stages, and applying thealgorithm methodinthe non-uniformsetting. Indeed, noneofthepreviously-introducedtechniquesseemtohaveanychanceofworkinginthe setting ofquenchedVoronoipercolation.
As mentioned above, inorder to usethe algorithm method we will need to provea 1-armestimatethatwillfollowfromaquenchedversionofthebox-crossingpropertyfor Voronoipercolationatcriticality.Thisresultgivesboundsontheprobabilitythata rect-angle(offixedaspectratio)iscrossedatcriticality,andisananalogueofthecelebrated resultsforbondpercolationonZ2ofRusso[21]andSeymourandWelsh[23].
Correspond-ingresultshavebeenobtainedinvariousrelatedsettings, andobtainingsuchboundsis frequently akeystepintheproof ofvarious importantapplications,see e.g.[3,9,11,18, 20,26].Inparticular,animportantbreakthroughwasmadebyBollobásandRiordan[9], who provedan RSW-typetheorem for(annealed) Voronoi percolation,4 and usedit to
3 Inthismodel,apairu,v∈ η isconsideredtobeadjacentifthedistancebetweenthemisatmost 1. 4
Moreprecisely,theyprovedthatthereexistsaninfinitesequenceofvaluesofL suchthattheprobability ofcrossinganL× λL rectangleisboundedawayfrom 0.
deducethatthecriticalprobabilityforpercolationis1/2.Thefullbox-crossingproperty intheannealed setting wasobtainedonly very recently,bythe fourthauthor [26]. We remarkthatthisresultwillplayanimportantrole inourproofofTheorem 1.4, below.
Asabove,wewriteHRfortheeventthatthere isaredhorizontalcrossingofR.
Theorem 1.4 (The quenched box-crossing property for Voronoi percolation). For every
rectangleR⊂ R2,thereexistsaconstantc> 0 suchthatthefollowingholds.Letn∈ N,
letη⊂ R beasetofn points, each chosenuniformly atrandom,andletω:η→ {−1,1}
beauniform colouring.Then
Pc <PHR| η
< 1− c → 1
asn→ ∞.
We remark that, moreover, for every γ > 0 there exists c = c(γ,R) > 0 such that PHR|η
/
∈ (c,1−c) hasprobabilityatmostn−γ.Ananalogoustheoremifη isaPoisson
pointprocessintheplane(orinthehalf-plane) followsbyexactlythesameproof. We will prove Theorem 1.4 in three steps. First, we will prove a weaker result for Voronoipercolationintheplane(seeTheorem 3.1):thissaysthatthereexistsaconstant
c> 0 such that P PHR| η 1 2k (1 − c)k (4)
for all sufficiently large k. We will then deduce an analogous statement for Voronoi percolationinahalf-plane; somewhatsurprisingly, thededuction is nottrivial,and we will haveto dosomework to dealwith theboundaryeffects (see Section3.2). Finally, we will use these results, together with the algorithm method(see Section 4) and our Efron–Steintypeinequality(1),provedinSection2,to show(seeTheorem 4.1)that
PHR| η
→ E PHR| η
inprobability, as n → ∞. This resultwill imply both Theorem 1.1 and Theorem 1.4, usingthebox-crossingpropertyforannealedVoronoiprovedin[26].
The organisation of the rest of the paper is as follows. First, in Section 2, we will boundthe varianceoftheprobability ofthecrossingeventby theexpectedsumofthe squaresof theinfluencesof fη.Wewill doso byintroducingamartingale, whosesteps
correspondtochoosingthepointsofη one-by-one,andboundingthevarianceofstepm intermsofthe expectationofthesquare of theinfluenceofthem-th element ofη, see Lemma 2.4.Armedwith thislemma,theclaimedbound(1)follows easily.
Second, in Section 3, we will prove weak bounds for the crossing probabilities in quenchedVoronoipercolation(4)inboththeplane,andthehalf-plane.The keytoolsin
our(surprisingly simple)proofwillbe the‘box-crossingproperty’forannealedVoronoi percolation, provedin[26], togetherwith colour-switching.Inparticular,we wouldlike to highlightLemma 3.2, whichstatesthat
PHR| η
=E 2−X| η ,
where X is the random variable which counts the number of vertex-disjoint vertical monochromatic crossingsof R.Although this lemma, once stated, is easyto prove, we havefounditto beextremelyuseful,and expectittohavemanyotherapplications.
Finally, inSection4, wewill complete theproof ofthe main theorems, byusing the algorithm method of Benjamini, Kalai and Schramm [5] and Schramm and Steif [22] to bound the sum of the squares of the influences of fη. (Indeed, once we are armed
withtheresultsofSection3,therequiredboundfollowsbysimplyrepeatingthemethod of [22].) Combining this bound with the results of Section2, we obtainTheorem 1.1. By(2), weobtainTheorem 1.2,andbythebox-crossingproperty forannealedVoronoi percolation[9,26], weobtainTheorem 1.4.
2. Varianceandinfluence
Inthissection,wewillproveasomewhatsurprisingboundonthe(typical)dependence ofthecrossingeventonthepointsetη intermsofthe(expected)influencesofthecolours. Since wewillneedtousetheresultsofthissectionintheproofofTheorem 1.4, aswell asthatofTheorem 1.1,wewillworkinthemoregeneralset-upofanarbitraryrectangle
R ⊂ R2, so letη be aset of n pointsinR,each of whichis chosen independentlyand
uniformly atrandom. Wewill writefRη:{−1,1}η → {0,1} for thefunctionthatencodes
whetherornotthereisaredhorizontalcrossingofR inthecorrespondingVoronoitiling. Recall that
Infm(fRη) :=P
fRη(ω)= fRη(ω)η, where ω equalsω exceptonthem-th elementofη.
Themainresultofthissectionisthefollowinginequality,whichishighlyreminiscent of thewell-knowninequalityofEfronandStein[12].
Theorem 2.1.Forevery rectangle R⊂ R2,
VarPHR| η n m=1 E Infm(fRη)2 .
Note thatthefollowingcorollary isanimmediate consequenceoftheabovetheorem and Chebyshev’s inequality.
Corollary2.2. Leta(n)=E nm=1Infm(fRη)2 .Then PP(HR| η) − P(HR) a(n)1/3 a(n)1/3.
TheproofofTheorem 2.1usesthefollowingsimplemartingale(qm)nm=0.Letuschoose
theelementsofη one-by-one,andletηm denotethem-th element.Nowwrite
qη=PHRη
fortheprobabilityofsuchacrossinggivenη,anddefine
qm:=E
qη| Fm
, whereFmdenotes theσ-algebragenerated byη1,. . . ,ηm.
Observation2.3. Varqη= n m=1 Varqm− qm−1 .
Proof. Sinceqη−E[qη]= n
m=1(qm−qm−1),itwillsufficetoshowthatCov(qi−qi−1,qj−
qj−1)= 0 for every1 i< j n.Toseethis, wesimplyconditiononFi,whichgives
Cov(qi− qi−1, qj− qj−1) =E E qi− qi−1 qj− qj−1 Fi = 0, sinceE qj− qj−1|Fi
= 0,and qi− qi−1 isdeterminedbyη1,. . . ,ηi. 2
ByObservation 2.3, thefollowing lemmacompletestheproofofTheorem 2.1. Lemma2.4. Forevery 1 m n,
Varqm− qm−1 E Infm fRη2 almostsurely.
Proof. Firstobserve that,sinceE qm− qm−1|Fm−1
= 0 almost surely, bythe condi-tionalvarianceformula5 wehave
Varqm− qm−1=E VarqmFm−1 . (5)
Now,letη− beobtainedfromη bydeletingηm.Sinceqη
−
doesnotdependonthem-th elementofη,itfollows that
5
VarqmFm−1 = Var E qη| Fm − E qη−| Fm−1 Fm−1 = Var E qη− qη−| Fm Fm−1 .
Now, sinceVar(X) E[X2] foreveryrandomvariableX,thisisatmost
EE qη− qη−| Fm
2F m−1
, whichisinturnat most
Eqη− qη−2Fm−1
,
byJensen’sinequality.Wemake thefollowing claim,whichcompletestheproof. Claim. |qη− qη−| Inf
m(fRη),almostsurely.
Proof. Letuswriteω+(resp.ω−)fortheelementof{−1,1}nobtainedfromω bysetting
the m-th coordinateequal to 1 (resp.−1). Alsodefine fRη−(ω) for ω ∈ {−1,1}n in the
obvious way,byignoringthem-th coordinateofω.Wehave qη− qη− PfRη(ω) > fRη−(ω)| η
PfRη(ω+) > fRη(ω−)| η= Infm(fRη),
sincefRη ismonotone(asafunctionon{−1,0,1}η).6Anidenticalcalculationshowsthat
qη−− qη Inf
m(fRη),andsotheclaimfollows. 2
Thelemma nowfollowssince,by(5),wehave
Varqm− qm−1 =E VarqmFm−1 EE qη− qη−2Fm−1 EE Infm fRη2Fm−1 =E Infm fRη2 , as required. 2
As noted above, Theorem 2.1 follows immediately from Observation 2.3 and Lem-ma 2.4.
6 Indeed,abusingnotationslightly,wecandefineafunctionfη
R:{−1,0,1}
η→ {0,1} bysettingfη
R(ω)= 1
ifthereisaredhorizontalcrossingintheVoronoitilingdefinedbythesetη={u∈ η : ω(u)= 0} with colouringω= ω|η.ThefunctionfRηismonotoneincreasingsince(fromthepointofviewofthecrossing
3. CrossingprobabilitiesforquenchedVoronoipercolation:weakbounds
Inthis section we will proveaweaker versionof Theorem 1.4 forquenched Voronoi percolationintheplane,anddeducethecorrespondingtheoremwhenη isasubsetofthe half-plane.Crucially,theseresultswillbesufficienttodeduceaboundontheprobability of the one-arm event that is strong enough for our application of the Schramm–Steif methodinSection4.
3.1. Crossing probabilitiesin theplane
Inthissection,η willdenoteaPoissonprocessintheplaneofintensity1,andω:η→
{−1,1} willbe a(uniform) random two-colouring of η.7 Recall that,given arectangle
R withsides parallel to the axes, HR denotes theevent that there is a redhorizontal
crossingofR intheVoronoitilinggivenbyη,colouredbyω.
Theorem3.1. Foreveryrectangle R⊂ R2,thereexistsaconstant c> 0,depending only
ontheaspectratio of R,suchthat
P PHR| η 1 2k (1 − c)k
forallsufficientlylarge k∈ N.
We begin by defining the following random variable,whose value depends on both
η and ω,and which counts themaximum number of vertex-disjoint8 vertical crossings
of R:
X = X(η, ω) := maxm∈ N0 : there exist m vertex-disjoint, monochromatic
vertical crossings{γ1, . . . , γm} of R
.
Thefollowinglemmawill beakey toolinourproofof Theorem 3.1. Lemma3.2. Foralmosteveryη,
PHR| η
=E 2−X| η .
As noted in the Introduction, we have found this lemma to be surprisingly power-ful, and expect it to have many other applications. Despite this, the lemma is very
7 Hereω isacolouringofη,whichiscountable.Foreaseofnotationweusethisconvention,butremark
thatweareonlyinterestedintherestrictionofω toηR,thesubsetofpointswhosecellsintersectR,which
isalmostsurelyfinite.WealsocontinuetouseP todenotetheprobabilitymeasureassociatedwithchoosing thepair(η,ω),andtrustthatthiswillcausethereadernoconfusion.
8
easyto prove;indeed,itfollowsalmostimmediately fromabasic(andwell-known)fact about‘colour-switching’,see(6)below.Forthosereaderswhoareunfamiliarwith colour-switching,wewillbegin bygivingabriefintroduction.
ConsidertheeventthatX = k,i.e.,thatthere existk,butnotk + 1,vertex-disjoint, monochromatic vertical crossings of R. The following standard algorithm provides a method offinding suchpaths.First,we discover theleft-mostmonochromatic path(in the coloured Voronoi tiling of R given by the pair (η,ω)) starting from the left-most cell whichintersects thelower side ofthe rectangle. Ifit reachesthe topof the rectan-gle, thenwe add this monochromaticpath to ourcollection; otherwisewe discover the wholemonochromaticcomponentofourstarting-point.Ineithercase,wethen discover the left-most monochromatic path entirely to the right of the already-discovered cells, starting fromthenextavailablecellonthelower sideofR (if itexists).Repeating this process untilwe reachtheright-side ofR,weobtainacollection(γ1,. . . ,γk) ofdisjoint
monochromaticcrossings.
An important feature of the algorithm above is that it allows us to use so-called ‘colour-switching’arguments,seee.g.[4,24].Thecrucialobservationisthat,foragiven η, andagivencollectionofpaths(γ1,. . . ,γk) obtainedviathealgorithm,thereisabijection
between the set ofcolourings inwhich γj is red, andthose inwhich itis blue.Indeed,
if we swap thecolours ofall cellsthatare on or to theright of γj, thenthe algorithm
produces exactly the same set of paths. More generally, writing Π ∈ {−1,1}k for the sequenceofcoloursofthepaths(γ1,. . . ,γk),wehavethefollowingsimplefact.Forevery
σ∈ {−1,1}k,wehave
PΠ = σX = k, η= 1
2k (6)
almost surely.Lemma 3.2isanalmost immediateconsequenceof(6).
ProofofLemma 3.2. ObservethattheeventHR holdsifandonlyifallmonochromatic
vertical pathsarered.By(6),itfollowsthat
PHR| η = ∞ k=0 PΠ = (1, . . . , 1)X = k, η PX = kη = ∞ k=0 P(X = k | η) 2k =E 2−X| η , as required. 2
InordertodeduceTheorem 3.1,weonlyneedtoshowthatX cannotbetoolarge.This will beaconsequenceofthreepropertiesofannealedVoronoipercolation:theFKGand BKinequalities,andtheboxcrossingproperty,provedrecentlybythefourthauthor[26]. An event A that depends on the pair (η,ω) is said to be red-increasing if removing a
bluepoint or adding ared point cannot cause the status of A to changefrom trueto false.Foraproofof thefollowing lemma,see[8,Lemma 8.14].
Lemma 3.3 (The FKG inequality for annealed Voronoi percolation). Let A and B be
red-increasingevents. Then
P(A ∩ B) P(A) · P(B),
andmoreover thereverseinequalityholdsif B isreplacedby itscomplementBc.
The following lemma was proved by van den Berg [6]; his proof, sketched below, can also be found in the PhD thesis of Joosten [16, Section 4.3] (who also refers to van den Berg).Thecorrespondinginequalityinthediscretesettingisacelebratedresult ofvan den BergandKesten[7].
Lemma3.4 (The BK inequalityforannealed Voronoipercolation). LetA and B be
red-increasing events.Then
P(A ◦ B) P(A) · P(B),
whereA◦ B denotestheevent thatA and B occurdisjointly.9
Proof. LetB denote theeventcorrespondingto B withcoloursreversed.Wehave P(A ◦ B) = E P(A ◦ B | η) E P(A ∩ B | η)
=P(A ∩ B) P(A)P(B) = P(A)P(B),
where the first inequality follows from Reimer’s inequality [19, Theorem 1.2], and the secondinequalityfollowsfrom Lemma 3.3. 2
Ourfinaltoolisthefollowing resultof Tassion[26].
Lemma3.5(Thebox-crossing propertyforannealedVoronoipercolation). Thereexistsa
constant c0> 0,depending only ontheaspectratio of R,suchthat
PHR
> c0.
We are now ready to prove the weak box-crossing property for quenched Voronoi percolationintheplane.
9
Forgeneraleventsonemustbealittlecarefulindefiningdisjointoccurrence,butforeventsinvolving crossingsthedefinitionisstraightforward:thecrossingsmustbevertex-disjoint.
Proof ofTheorem 3.1. Giventhe pair(η,ω), defineX+ (resp.X−)to be themaximal number of disjoint red (resp. blue) paths from top to bottom in R, so in particular
X = X++ X−.ByLemma 3.5andtheBKinequalityforannealedVoronoipercolation,
there existsaconstantc0> 0,dependingonlyontheaspectratioofR,suchthat
PX+ k (1 − c0)k
for all k 0, and similarly P(X− k) (1− c0)k. Since the events
X+ i and
X−< jarebothred-increasing,itfollows bytheFKGinequalitythat
PX k i+jk PX+ i∩X− j i+jk (1− c0)i+j (1− c)k 2 (7)
forsomeconstantc> 0,ifk is sufficientlylarge.Now,byLemma 3.2wehave
PHR| η =E 2−X| η 1 2 k−1 PX < k| η, almost surely.Thus,byMarkov’sinequalityand(7),we obtain
P PHR| η 1 2 k P PX k | η1 2 (1 − c)k, as required. 2
3.2. Quenchedcrossing probabilities inthehalf-plane
Inorder toboundone-arm eventsstartingat pointsneartheboundaryofR,wewill require aresult analogous to Theorem 3.1 for aPoisson point process η of intensity 1 restrictedto thehalf-plane
H :=(x, y)∈ R2: x 0.
For eachrectangleR ⊂ H withsides parallelto theaxes, letHR∗ denote theeventthat thereisaredhorizontalcrossingofR intheVoronoitilingofH givenbyH∩ η,coloured byω.
Theorem 3.6. Forevery rectangle R⊂ H,there exists aconstant c> 0, depending only
P PHR∗| η 1 2k (1 − c)k,
forallsufficientlylarge k∈ N.
TheproofofTheorem 3.6 isidenticaltothatofTheorem 3.1,exceptwewillneedto replaceLemma 3.5with thefollowing boundintheannealedsetting.
Lemma3.7.ForeveryrectangleR,thereexistsaconstantc1> 0,depending onlyonthe
aspectratio ofR, suchthat
PHR∗
> c1.
WhentherectangleR issufficiently farfrom theboundary,Lemma 3.7followsfrom Lemma 3.5, since the Voronoitilings of R is (with high probability) the sameinboth cases.Webegin withaneasylemmathatmakesthisstatementprecise.
Lemma3.8.Givenλ> 0,letL> 0 besufficientlylarge.Letη beaPoissonpointprocess
intheplane of intensity 1,andletR be aλL× L rectanglewith
min{x : (x, y) ∈ R} (log L)2/3.
ThentheVoronoitilingsofR inducedbyη andbyH∩η arenon-identicalwithprobability
atmost1/L3.
Proof. IftheVoronoitilingsofR inducedbyη andbyH∩η arenon-identical,thenthere mustbe apointu∈ R suchthattheclosestpoint ofη tou liesoutsideH.This implies thatthere is anempty ball of radius (log L)2/3 centredin R,the probability of which
(bystandardpropertiesofPoissonprocesses)issuper-polynomiallysmallin L. 2 Wewill assumefrom now on thatR isaλL× L rectangle withL sufficiently large, and suchthat theleft-hand side of R is onthe line x= 0. (Note that,by choosing c1
sufficientlysmall,thismaybeassumedwithoutloss ofgenerality.)Theideaoftheproof of Lemma 3.7 is as follows. Set = (log L)2/3, and partition10 the set {(x,y) ∈ R :
0 x } intoL/ squares S1,. . . ,SL/.Now observe thatifthere isaredhorizontal
crossingoftherectangle R ={(x,y)∈ R : x },butnoredhorizontalcrossingofR, thentheremustexistasquareSj suchthatthefollowing‘3-armevent’holds(seeFig. 1,
below).
Definition 3.9. For each 1 j L/, let A(3)(j) denote the event thatthe following
hold:
10
Fig. 1. IfthereexistsaredhorizontalcrossinginR,butnoredhorizontalcrossingofR,thena‘3-armevent’ mustholdfromasquareSj.(Redandbluepathsarerepresentedbysolidanddottedpaths,respectively.)
(a) thereisaredpathfromSj totheright-handsideofR thatiscontainedinR;
(b) there are blue paths from Sj to the top and the bottom of R that are contained
in R.
ByLemma 3.8andtheobservationsabove,wehave
PHR∗ PHR −L13− L/ j=1 PA(3)(j). (8)
Thus, byLemma 3.5,itwill sufficetoprovethefollowing lemma. Lemma 3.10.Thereexistsc> 0 such that
L/
j=1
PA(3)(j) L−c
forallsufficiently largeL> 0.
IntheproofofLemma 3.10,wewillusethefollowingtwo events:
• A(1)(j) denotes theeventthatthere isaredpath fromSj totheright-handsideof
R thatiscompletelycontainedinsideR.
• A(2)(j) denotestheeventthatthereisaredpathfromS
jtothetopofR,andablue
pathfromSj tothebottomofR,bothofwhicharecompletely containedinside R.
Wearenow readytoproveLemma 3.10.
ProofofLemma 3.10. We firstclaimthat,foreachj∈ [L/],wehave
To prove (9), we apply colour-switching and the BK inequality for annealed Voronoi percolation (Lemma 3.4). Indeed, the three paths in Definition 3.9 are of alternating colours, so must be vertex-disjoint. Moreover, if such vertex-disjoint, monochromatic paths exist, then by colour-switching theyare eachequally likely to be red or blue.11
Thus,lettingB(2)(j) denotetheeventthatthere arevertex-disjoint redpathsfrom S j
tothetopandbottomofR,bothofwhicharecompletelycontainedinsideR,wehave PA(3)(j)| η=PA(1)(j)◦ B(2)(j)| η and PA(2)(j)| η=PB(2)(j)| η foralmostallη.Takingexpectationoverη,andnotingthatA(1)(j) andB(2)(j) areboth
red-increasingevents,wehave
PA(3)(j) PA(1)(j)· PB(2)(j)=PA(1)(j)· PA(2)(j) bytheBKinequality,as claimed.
By(9), thelemma isanimmediatelyconsequenceofthefollowingtwoclaims. Claim1.PA(1)(j) L−2c forsomeconstant c> 0.
Proof. ByLemma 3.8,theprobabilitythattheVoronoitilingofR byH∩ η differsfrom thatbyη isatmost1/L3.Theclaimthereforefollowsfromthecorrespondingstatement intheplane,whichisastandardconsequenceofLemma 3.5,see[26, Theorem 3]. 2
Claim2.
L/
j=1
PA(2)(j) 2C forsomeconstant C > 0.
Proof. Foreachj ∈ [L/],choosea(distinct)cellu(j)∈ η whichintersectsbothSj and
theleft-handsideofR.ConsidertheeventE(j),illustratedinFig. 2,thatthereisared
pathfrom u(j) tothe topofR,and abluepath fromthe cellu(j) immediately below
u(j) tothebottomofR,bothofwhicharecompletely containedinsideR.
Weclaimthat
PA(2)(j) 2C· PE(j) (10)
foreachj∈ [L/].Toprovethis,wesimplyusebruteforcetotunnelfromtheboundary
ofSj tou(j). Indeed,as longas,foreverypairofpoints{v,w} ontheboundaryofSj,
therearevertex-disjointpaths(intheVoronoitilingofSj)fromu(j) tov andfromu(j)
tow,eachoflengthat mostC/2,thenwecanconnectu(j) andu(j) totheendpoints
11 Tobeslightly moreprecise,ifsuch(vertex-disjoint,monochromatic)pathsexist thenwemay choose
the‘left-most’suchtriple(γ1,γ2,γ3),andprovearesultcorrespondingto(6)byswitchingthecoloursof
allcellsthatareonortotherightofγj foreachj∈ {1,2,3}.Theleft-mosttripleisobtainedbychoosing
γj tobetheleft-mostmonochromaticpathtothetop/right/bottomofR thatis entirelytotherightof
Fig. 2. Illustration oftheevent E(j):there existaredpathfromacellu(j) intersectingSj andtheleft
boundaryofR tothetopofR,andabluepathfromthecellu(j) immediatelybelowu(j) tothebottom of R.
of the paths guaranteed by theevent A(2)(j) withprobability at least 2−C. But such
pathsexist unlessthePoissonprocessismuchdenserinSj thanonewouldexpect,and
this occurswithprobabilitythatissuper-polynomiallysmallinL.
Finally,simplynote that,sinceatmostoneoftheeventsE(j) canoccur,wehave
L/ j=1 PA(2)(j) 2C L/ j=1 PE(j) 2C as claimed. 2
Combining Claims 1 and 2with(9),and recallingthat= (log L)2/3,weobtain L/ j=1 PA(3)(j) L/ j=1 PA(1)(j)· PA(2)(j) L−2c· 2C L−c ifL is sufficientlylarge,as required. 2
Forcompleteness,letusspellouthowtodeduceLemma 3.7fromLemmas 3.5 and 3.10. ProofofLemma 3.7. By(8),andLemmas 3.5 and 3.10,wehave
PHR∗ PHR − 1 L3 − L/ j=1 PA(3)(j) c0− 1 L3− L−c c1
ifL is sufficientlylarge,as required. 2
As notedabove, Theorem 3.6 follows by repeating the proof of Theorem 3.1, using Lemma 3.7inplaceofLemma 3.5.
3.3. A boundontheprobability ofthe1-arm eventin arectangle
To finish the section, let us deduce the following proposition from Theorems 3.1 and 3.6. Let R ⊂ R2 be a rectangle of area n, and let η ⊂ R be a set of n points, each chosen uniformly at random, and let ω be a uniform colouring of η. (Thus, the distributionofη insideR isveryclosetothatofaPoissonprocessofintensity1.)Given
u∈ R andd> 0,wewriteM (u,d) fortheevent(dependingonη andω)thatthereisa
monochromaticpathfromu tosomepointofR at2-distanced from u.
WewillusethefollowingresultinSection4inordertoboundthe‘revealment’ofour randomised algorithm,andhenceto deduceourmain theorems. Sincethededuction of this result from the box-crossing property is standard, we will be fairly brief with the details.
Proposition 3.11. For every γ > 0, there exists ε > 0 such that the following holds.
Supposethat d= d(n)→ ∞ as n→ ∞ and letu∈ R. Then
PPM (u, d)| η d−ε
d−γ
forallsufficientlylarge n∈ N.
Proof. Fixapointu∈ R,andforeachj∈ N,letAj denotethesquareannulus,centred
onu,withinnerside-length7j andouterside-length3· 7j.LetOj denotetheeventthat
thereisabluecircuitaroundthe(partial)annulusAj∩ R.(Thiscircuitmusteitherbe
closed,orhavebothitsendpointsontheboundaryofR;ineithercase,itmustseparate
u fromtheexteriorofAj.)Let
J =j∈ N : √d 7j+1 d,
andconsider thecollectionofannuli C = {Aj : j∈ J}.Roughlyspeaking, wewill show
thateitherat leasthalftheannuli inC contain an‘unusual’collectionofpoints,or the probabilitythatnoneoftheeventsOj occursis atmostd−ε.
Let c be the constant in Theorems 3.1 and 3.6, and fix a large constant k ∈ N (dependingon c and γ).Foreachj∈ J,letD(1)j denotetheevent(dependingonη)that POj|η
> 2−4k,andletD(2)j denote theeventthatforeveryz∈ Aj,thereexistssome
pointx∈ η atdistance atmostlog d fromz.Define
Dj := D
(1) j ∩ D
(2) j ,
andobserve thattheeventsDj areindependent.Weclaimthat
P(Dj) 1 − 5(1 − c)k (11)
PDj(2)→ 1
as d→ ∞.Next,observe that,bytheFKGinequality12 andTheorems 3.1 and 3.6,we
have
PPOj| η
2−4k 4(1 − c)k
forallsufficientlylargek∈ N.ThusP(Dc
j) 4(1− c)k+ o(1) 5(1− c)k,as claimed.
Now, let D∗ denote the eventthatDj holds for at least halfof theelements j ∈ J,
and observethat
P(D∗) 1 − 2|J|5(1− c)k|J|/2 1 − d−γ,
since |J| = Ωlog d, and k was chosen sufficiently largein terms of γ and c. Butfor those η suchthatD∗ holds,wehave
PM (u, d)| η P j∈J Ojcη = j∈J POjc| η1− 2−4k|J|/2 d−ε,
forsomeε> 0,asrequired. 2
4. Theproofofthemain theorems
InthissectionwewillusethealgorithmmethodofBenjamini,KalaiandSchramm[5] andSchrammandSteif[22],togetherwiththeresultsprovedintheprevioussections,in order toboundthesumofthesquaresoftheinfluences,andhencededucethefollowing theorem.Throughoutthis section,letusfixarectangleR⊂ R2.
Theorem 4.1. Thereexists c = c(R)> 0 such that the followingholds. Letη ⊂ R be a
set of n points, each chosen uniformlyat random,and letω:η→ {−1,1} be auniform
colouring. Then
PP(HR| η) − P(HR) n−c
n−c
forallsufficiently largen∈ N.
Theorem 1.1followsimmediatelyfromTheorem 4.1bytakingR = [0,1]2,andrecalling
that inthis caseP(HR)= 1/2. Notethat Theorem 1.4also follows from Theorem 4.1,
together with the RSW theorem for annealed Voronoi percolation, which implies that P(HR) isboundedawayfrom0and 1uniformly in n.
12
Notethat weare applying thisin thequenched world,so the usualFKGinequality (also knownas Harris’lemma)issufficient.
Therandomisedalgorithmmethodwasintroducedin[22]andusedtheretoshowthat theareexceptionaltimesindynamicalsitepercolationonthetriangularlattice.Sincethis methodiswell-known,andourapplicationisratherstandard,weshallbesomewhatless carefulwiththedetailsthaninearliersections,focusinginsteadonconveyingthemain ideas.Givenarandomised algorithmA thatdeterminesafunctionfn:{−1,1}n → {0,1},
definetherevealmentofA to be δA(fn) := max
i∈[n]P
i is queried byA.
Schramm and Steif [22] proved a very powerful bound on the Fourier coefficients of a real-valued function on the hypercube {−1,1}n in terms of the revealment of any
randomised algorithm thatdetermines f . For monotone Booleanfunctions theirresult easilyimpliesthefollowingtheorem,whichwillbesufficientforourpurposes.
Theorem 4.2 (Schramm and Steif, 2010). Given a monotone function f :{−1,1}n →
{0,1},and arandomised algorithmA thatdetermines f ,wehave
n
m=1
Infm(f )2 δA(f ).
By Theorems 2.1 and 4.2, it only remains to show that, with probability at least 1− n−c, there exists an algorithm that determines fRη with revealment that is poly-nomially small inn. (Recallthat fRη:{−1,1}η → {0,1} denotes thefunction suchthat
fRη(ω)= 1 if and onlyifHR occurs.) Thealgorithm we will useisessentially thesame
as thatintroducedin[22], so weshalldescribe itinanintuitive(and thereforeslightly non-rigorous)fashion,andreferthereadertotheoriginalpaperforthedetails.
TheSchramm–Steifrandomisedalgorithm.LetA bethealgorithmthat,givenη,queries bitsofω asfollows:
1. Chooseapointx inthemiddlethirdoftheleft-handsideofR uniformlyatrandom. 2. Explore the boundary between red and blue cells, with red on the left, starting from x.Here weplaceboundaryconditionsas follows:theleft-handsideofR isred above x, andbluebelow,andthebottomofR isalsoblue.Ifthispath:
(a) reachestheright-handsideofR,thenfRη(ω)= 1;
(b) reachesthebottomofR,andendsatthetop,thenfRη(ω)= 0; (c) endsatthetopofR withoutreachingthebottom,thengotostep 3.
3. Explore the boundary between red and blue cells, with red on the right, starting fromx.Herewereversetheboundaryconditions,i.e.,theleft-handsideofR isblue abovex,andredbelow,andthetopofR is alsoblue.Ifthis path:
(a) reachestheright-handsideofR,thenfRη(ω)= 1; (b) otherwise fRη(ω)= 0.
Note thatwe only query bits as needed, i.e., we query those vertices whose cellwe meet along one of our paths. Still following [22], this allows us to immediately bound therevealmentofA asfollows.Recallthat,givenu∈ R andd> 0,wewriteM (u,d) for theeventthatthereisamonochromaticpathfrom u tosomepoint ofR at2-distance
d from u.
Lemma 4.3.LetA be theSchramm–Steifrandomised algorithm.Then δA(fRη) max u∈η P Mu, n−1/4 η + On−1/4 almostsurely.
Sketch of the proof. Let u ∈ η, and consider the probability thatu is queried by A. First, notethatthe probabilitythattherandomstart-point x iswithindistance n−1/4 of u is O(n−1/4). Butif the distance between u andx is greater than n−1/4, and u is nonetheless queriedbyA,thentheeventMu,n−1/4holds. 2
ToboundtherevealmentofA,itwillthereforesufficetoboundtheprobabilityofthe eventMu,n−1/4. ThisisanimmediateconsequenceofProposition 3.11.
Lemma 4.4.Forevery γ > 0,there existsc> 0 such that Pmax u∈η P Mu, n−1/4 η n−c 1 nγ
forallsufficiently largen∈ N.
Proof. Renormalising R tohavearean,wefindthatdistancesaremultipliedbyΘ√n, andsowemayapplyProposition 3.11withd= Θn1/4.Theclaimedboundnowfollows from theproposition,usingtheunionboundoverpointsu∈ η. 2
We arenow ready todeduce ourbound onthesumof thesquaresof theinfluences, and hence(byTheorem 2.1 andtheresultsof[5]and[26])ourmain theorems.
Lemma 4.5.Forevery γ > 0,there existsc> 0 such that
P n m=1 Infm fRη2 n−c 1 nγ
forallsufficiently largen∈ N.
Proof. Thisfollowsimmediately fromTheorem 4.2,togetherwithLemmas 4.3 and 4.4. Indeed,we have
n m=1 Infm(fRη) 2 δ A(fRη) n−c
withprobabilityatleast1− n−γ,as required. 2 WecannowdeduceTheorem 4.1.
Proofof Theorem 4.1. ByCorollary 2.2 andLemma 4.5 (applied with γ = 1,say), we have
PP(HR| η) − P(HR) n−c
n−c
forsomec> 0,andallsufficientlylargen∈ N,asrequired. 2
Finally,letus notethatthetheorems statedintheIntroductionallfolloweasily. ProofofTheorem 1.1. Asnotedabove, thisis animmediate corollaryof Theorem 4.1. Indeed,simplysetR = [0,1]2and observethatP(H
R)= 1/2. 2
NextwededucethatquenchedVoronoipercolationisnoisesensitiveatcriticality.We remark that,although wedo notgive thedetails here,it is astandardconsequence of themethodof [22]thatastrongerstatement holds:That(3) holds evenifε= n−c for some(sufficientlysmall)constantc> 0.
ProofofTheorem 1.2. ThetheoremfollowsimmediatelyfromLemma 4.5,togetherwith theBenjamini–Kalai–SchrammTheorem(see(2)). 2
Finally,weprovethequenchedbox-crossingpropertyforVoronoipercolation. ProofofTheorem 1.4. ThisfollowsfromTheorem 4.1,togetherwiththeRSWtheorem forannealedVoronoipercolation,whichwasprovedin[26].Indeed,foreveryrectangle R, thereexists c0> 0 suchthat
c0<P(HR) < 1− c0,
see[26,Theorem 3]. Hence,byTheorem 4.1,wehave,forsomec> 0, Pc <PHR| η
< 1− c 1 − n−c (12)
forallsufficientlylargen∈ N. 2
Notethat,bypartitioningR intoaboundednumberofstripsandtakingc sufficiently small,inequality(12)impliesthattheprobabilityoftheeventPHR|η
/
∈ (c,1− c) can
Acknowledgments
TheauthorswouldliketothankRobvandenBergforallowingustoincludehisproof of theBK inequalityforannealedVoronoipercolation.The firstauthor wouldalsolike to thankElchananMossel andGábor Peteforencouraging discussions, andthesecond and thirdauthorswouldlike to thankPaulBalisterandBéla Bollobásforanumberof veryinterestingconversations aboutquenchedVoronoipercolation.
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