Three of these proble ms are of Winkler type that is they are challenges for a clairvoya nt demon brPage 2br Edited by brPage 3br Three problems for the clairvoyant demon 11 Introduction Probability theory has emerged in recent decades as a crossr o ID: 41781
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1ThreeproblemsfortheclairvoyantdemonGeoreyGrimmettAbstractAnumberoftrickyproblemsinprobabilityarediscussed,havingincom-mononeormoreinnitesequencesofcointosses,andarepresentationasaproblemindependentpercolation.Threeoftheseproblemsareof`Winkler'type,thatis,theyarechallengesforaclairvoyantdemon. Editedby Threeproblemsfortheclairvoyantdemon31.1IntroductionProbabilitytheoryhasemergedinrecentdecadesasacrossroadswheremanysub-disciplinesofmathematicalsciencemeetandinteract.Ofthemanyexampleswithinmathematics,wemention(notinorder):analy-sis,partialdierentialequations,mathematicalphysics,measuretheory,discretemathematics,theoreticalcomputerscience,andnumbertheory.TheInternationalMathematicalUnionandtheAbelMemorialFundhaverecentlyaccordedacclaimtoprobabilists.Thisprocessofrecogni-tionbyothershasbeentooslow,andwouldhavebeenslowerwithouttheeortsofdistinguishedmathematiciansincludingJohnKingman.JFCK'sworklookstowardsboththeoryandapplications.Tosingleoutjusttwoofhistheorems:thesubadditiveergodictheorem[21,22]isapieceofmathematicalperfectionwhichhasalsoprovedratherusefulinpractice;his`coalescent'[23,24]isabeautifulpieceofprobability,nowakeystoneofmathematicalgenetics.Johnisalsoaninspiringanddevotedlecturer,whocontinuedtolecturetoundergraduatesevenastheBristolVice-Chancellor,andtheDirectoroftheIsaacNewtonIn-stituteinCambridge.Indeed,thecurrentauthorlearnedhismeasureandprobabilityfrompartialattendanceatJohn'scourseinOxfordin1970/71.TomisquoteFrankSpitzer[34,Sect.8],weturntoaverydown-to-earthproblem:consideraninnitesequenceoflightbulbs.Thebasiccommodityofprobabilityisaninnitesequenceofcointosses.Suchasequencehasbeenstudiedforsolong,andyetthereremain`simpletostate'problemsthatappearveryhard.Wepresentsomeoftheseproblemshere.Sections1.3{1.5aredevotedtothreefamousproblemsfortheso-calledclairvoyantdemon,apresumablynon-humanbeingtowhomisrevealedthe(innite)realizationofthesequence,andwhoispermittedtoplanaccordinglyforthefuture.Eachoftheseproblemsmaybephrasedasageometricalproblemofpercolationtype.Thedierencewithclassicalpercolation[13]liesinthedependenceofthesitevariables.Percolationisreviewedbrie\ryinSection1.2.Thisarticleendswithtwoshortsectionsonrelatedprob-lems,namely:otherformsofdependentpercolation,andthequestionof`percolationofwords'. 4GeoreyGrimmett1.2SitepercolationWesetthescenebyremindingthereaderoftheclassical`sitepercolationmodel'ofBroadbentandHammersley[9].Consideracountablyinnite,connectedgraphG=(V;E).Toeach`site'v2VweassignaBernoullirandomvariable!(v)withdensityp.Thatis,!=f!(v):v2Vgisafamilyofindependent,identicallydistributedrandomvariablestakingthevalues0and1withrespectiveprobabilities1 pandp.Avertexviscalledopenif!(v)=1,andclosedotherwise.Let0beagivenvertex,calledtheorigin,andlet(p)betheproba-bilitythattheoriginliesinaninniteopenself-avoidingpathofG.Itisclearthatisnon-decreasinginp,and(0)=0,(1)=1.Thecriticalprobabilityisgivenaspc=pc(G):=supfp:(p)=0g:Itisastandardexercisetoshowthatthevalueofpcdoesnotdependonthechoiceoforigin,butonlyonthegraphG.Onemayinsteadassociatetherandomvariableswiththeedgesofthegraph,ratherthanthevertices,inwhichcasetheprocessistermed`bondpercolation'.Percolationisrecognisedasafundamentalmodelforaran-dommedium.Itisimportantinprobabilityandstatisticalphysics,anditcontinuestobethesourceofbeautifulandapparentlyhardmathemat-icalproblems,ofwhichthemostoutstandingistoprovethat(pc)=0forthethree-dimensionallatticeZ3.Oftheseveralrecentaccountsofthepercolationmodel,wemention[13,14].MostattentionhasbeenpaidtothecasewhenGisacrystallinelatticeintwoormoredimensions.Thecurrentarticleisentirelyconcernedwithaspectsoftwo-dimensionalpercolation,particularlyonthesquareandtriangularlatticesillustratedinFigure1.1.Sitepercolationonthetriangularlatticehasfeaturedprominentlyinthenewsinrecentyears,owingtotheworkofSmirnov,Lawler{Schramm{Werner,andothersontherelationshipofthismodel(withp=pc=1 2)totheprocessofrandomcurvesinR2termedSchramm{Lownerevolutions(SLE),andparticularlytheprocessdenotedSLE6.See[35].WhenGisadirectedgraph,onemayaskabouttheexistenceofaninniteopendirectedpathfromtheorigin,inwhichcasetheprocessisreferredtoasdirected(ororiented)percolation.Variantsofthepercolationmodelarediscussedinthefollowingsec-tions,withtheemphasisonmodelswithsite/bondvariablesthataredependent. Threeproblemsfortheclairvoyantdemon5 Figure1.1ThesquarelatticeZ2andthetriangularlatticeT,withtheirduallattices.1.3ClairvoyantschedulingLetG=(V;E)beaniteconnectedgraph.AsymmetricrandomwalkonGisaMarkovchainX=(Xk:k=0;1;2;:::)onthestatespaceV,withtransitionmatrixP(Xk+1=wjXk=v)=8:1 vifvw;0ifvw;wherevisthedegreeofvertexv,anddenotestheadjacencyrelationofG.Randomwalksongeneralgraphshaveattractedmuchinterestinrecentyears,see[14,Chap.1]forexample.LetXandYbeindependentrandomwalksonGwithdistinctstartingsitesx0,y0,respectively.WethinkofX(respectively,Y)asdescribingthetrajectoryofaparticlelabelledX(respectively,Y)aroundG.Aclairvoyantdemonissetthetaskofkeepingthewalksapartfromoneanotherforalltime.Tothisend,(s)heispermittedtoschedulethewalksinsuchawaythatexactlyonewalkermovesateachepochoftime.Thus,thewalksmaybedelayed,buttheyarerequiredtofollowtheirprescribedtrajectories.Moreprecisely,ascheduleisdenedasasequenceZ=(Z1;Z2;:::)inthespacefX;YgN,andagivenscheduleZisimplementedinthefollow-ingway.FromtheXandYtrajectories,weconstructtherescheduledwalksZ(X)andZ(Y),where:1.IfZ1=X,theX-particletakesonestepattime1,andtheY-particleremainsstationary.IfZ1=Y,itistheY-particlethatmoves,and 6GeoreyGrimmetttheX-particlethatremainsstationary.Thus,ifZ1=XthenZ(X)1=X1;Z(Y)1=Y0;ifZ1=YthenZ(X)1=X0;Z(Y)1=Y1:2.Assumethat,aftertimek,theX-particlehasmadermovesandtheY-particlek rmoves,sothatZ(X)k=XrandZ(Y)k=Yk r.IfZk+1=XthenZ(X)k+1=Xr+1;Z(Y)k+1=Yk r;ifZk+1=YthenZ(X)k+1=Xr;Z(Y)k+1=Yk r+1:WecallthescheduleZgoodifZ(X)k=Z(Y)kforallk1,andwesaythatthedemonsucceedsifthereexistsagoodscheduleZ=Z(X;Y).(Weoverlookissuesofmeasurabilityhere.)Theprobabilityofsuccessis(G):=P(thereexistsagoodschedule);andweask:forwhichgraphsGisitthecasethat(G)0?ThisquestionwasposedbyPeterWinkler(seethediscussionin[10,11]).Notethattheanswerisindependentofthechoiceof(distinct)startingpointsx0,y0.TheproblemtakesasimplerformwhenGisthecompletegraphonsomenumber,Msay,ofvertices.Inordertosimplifyitstillfurther,weaddalooptoeachvertex.WriteV=f1;2;:::;Mg,and(M):=(G).ArandomwalkonGisnowasequenceofindependent,identicallydistributedpointsinf1;2;:::;Mg,eachwiththeuniformdistribution.Itisexpectedthat(M)isnon-decreasinginM,anditisclearbycouplingthat(kM)(M)fork1.Also,itisnottoohardtoshowthat(3)=0.Question1.1Isitthecasethat(M)0forsucientlylargeM?Perhaps(4)0?Thisproblemhasageometricalformulationofpercolation-type.Con-siderthepositivequadrantZ2+=f(i;j):i;j=0;1;2;:::gofthesquarelatticeZ2.Apathistakentobeaninnitesequence(un;vn),n0,with(u0;v0)=(0;0)suchthat,foralln0,either(un+1;vn+1)=(un+1;vn)or(un+1;vn+1)=(un;vn+1):WithX,Ytherandomwalksasabove,wedeclarethevertex(i;j)tobeopenifXi=Yj.Itmaybeseenthatthedemonsucceedsifandonlyifthereexistsapathallofwhoseverticesareopen.Somediscussionofthisproblemmaybefoundin[11].Thelawofthe Threeproblemsfortheclairvoyantdemon7openverticesis3-wiseindependentbutnot4-wiseindependent,inthesenseoflanguageintroducedinSection1.6.Theproblembecomessignicantlyeasierifpathsareallowedtobeundirected.Forthetotallyundirectedproblem,itisprovedin[3,36]thatthereexistsaninniteopenpathwithstrictlypositiveprobabilityifandonlyifM4.1.4ClairvoyantcompatibilityLetp2(0;1),andletX1;X2;:::andY1;Y2;:::beindependentse-quencesofindependentBernoullivariableswithcommonparameterp.WesaythatacollisionoccursattimenifXn=Yn=1.Thedemonisnowchargedwiththeremovalofcollisions,andtothisend(s)heispermittedtoremove0sfromthesequences.LetN=f1;2;:::gandW=f0;1gN,thesetofsingly-innitese-quencesof0sand1s.Eachw2Wisconsideredasawordinanalphabetoftwoletters,andwegenerallywritewnforitsnthletter.Forw2W,thereexistsasequencei(w)=(i(w)1;i(w)2;:::)ofnon-negativeinte-gerssuchthatw=0i110i21,thatis,thereareexactlyij=i(w)jzerosbetweenthe(j 1)thandjthappearancesof1.Forx;y2W,wewritex!yifi(x)ji(y)jforj1.Thatis,x!yifandonlyifymaybeobtainedfromxbytheremovalof0s.Twoinnitewordsv,waresaidtobecompatibleifthereexistv0andw0suchthatv!v0,w!w0,andv0nw0n=0foralln.Forgivenrealiza-tionsX,Y,wesaythatthedemonsucceedsifXandYarecompatible.Write (p)=Pp(XandYarecompatible):Notethat,byacouplingargument, isanon-increasingfunction.Question1.2Forwhatpisitthecasethat (p)0.Itiseasytoseeasfollowsthat (1 2)=0.Whenp=1 2,thereexistsalmostsurelyanintegerNsuchthatNXi=1Xi1 2N;NXi=1Yi1 2N:WithNchosenthus,itisnotpossibleforthedemontopreventacollisionintherstNvalues.Byworkingmorecarefully,onemayobtainthat (1 2 )=0forsmallpositive,seethediscussionin[12]. 8GeoreyGrimmettGacshasprovedin[12]that (10 400)0,andhehasnotedthatthereisroomforimprovement.1.5ClairvoyantembeddingTheclairvoyantdemon'sthirdproblemstemsfromworkonlong-rangepercolationofwords(seeSection1.7).LetX1;X2;:::andY1;Y2;:::beindependentsequencesofindependentBernoullivariableswithparam-eter1 2.LetM2f2;3;:::g.Thedemon'staskistondamonotonicembeddingoftheXiwithintheYjinsuchawaythatthegapsbetweensuccessivetermsarenogreaterthanM.Letv;w2W.WesaythatvisM-embeddableinw,andwewritevMw,ifthereexistsanincreasingsequence(mi:i1)ofpositiveintegerssuchthatvi=wmiand1mi mi 1Mforalli1.(Wesetm0=0.)AsimilardenitionismadefornitewordsvlyinginoneofthespacesWn=f0;1gn,n1.ThedemonsucceedsintheabovetaskifXMY,andwelet(M)=P(XMY):Itiselementarythat(M)isnon-decreasinginM.Question1.3Isitthecasethat(M)0forsucientlylargeM?Thisquestionisintroducedanddiscussedin[15],andpartialbutlimitedresultsproved.Oneapproachistoestimatethersttwomo-mentsofthenumberNn(w)ofM-embeddingsofthenitewordw=w1w2wn2WnwithintherandomwordY.ItiselementarythatE(Nn(w))=(M=2)nforanysuchw,anditmaybeshownthatE(Nn(X)2) E(Nn(X))2AMcnMasn!1;whereAM0andcM1forM2.ThefactthatE(Nn(w))1whenM=2isstronglysuggestivethat(2)=0,andthisispartofthenexttheorem.Theorem1.4[15]Wehavethat(2)=0.Furthermore,forM=2,P(w2Y)P(an2Y)forallw2Wn;(1.5)wherean=0101isthealternatingwordoflengthn. Threeproblemsfortheclairvoyantdemon9Itisimmediatethat(1.5)implies(2)=0onnotingthat,foranyinniteperiodicword,P(MY)=0forallM2.Onemayestimatesuchprobabilitiesmoreexactlythroughsolvingappropriatedierenceequations.Forexample,vn(M)=P(anMY)satisesvn+1(M)=(+(M 1))vn (M 2)vn 1;n1;(1.6)withboundaryconditionsv0(M)=1,v1(M)=.Here,+=1;=2 M:Thecharacteristicpolynomialassociatedwith(1.6)isaquadraticwithonerootineachofthedisjointintervals(0;M)and(;1).Thelargerrootequals1 (1+o(1))21 2MforlargeM,sothat,inroughtermsvn(M)(1 21 2M)n:Hereinliesahealthwarningforsimulators.Oneknowsthat,almostsurely,an6MYforlargen,butonehastolookonscalesoforder22M 1ifoneistoobserveitsextinctionwithreasonableprobability.Onemayaskaboutthe`best'and`worst'words.Inequality(1.5)as-sertsthatanalternatingwordanisthemosteasilyembeddedwordwhenM=2.ItisnotknownwhichwordisbestwhenM2.Werethisaperiodicword,itwouldfollowthat(M)=0.Unsurprisingly,theworstwordisaconstantwordcn(ofwhichthereareofcoursetwo).Thatis,forallM2,P(wMY)P(cnMY)forallw2Wn;where,fordeniteness,wesetcn=1n2Wn.LetM=2,sothatthemeannumberE(Nn(w))ofembeddingsofanywordoflengthnisexactly1(asremarkedabove).Afurtherargumentisrequiredtodeducethat(2)=0.Peled[32]hasmaderigorousthefollowingalternativetothatusedintheproofofTheorem1.4.Assumethatthewordw2Wnsatisesw2Y.Forsomesmallc0,onemayidentify(formostembeddings,withhighprobability)cnpositionsatwhichtheembeddingmaybealtered,independentlyofeachother.Thisgives2cnpossible`localvariations'oftheembedding.Itmaybededucedthattheprobabilityofembeddingawordw2Wnisexponentiallysmallinn,andalso(2)=0.ThesequencesX,Yhavebeentakenabovewithparameter1 2.LittlechangeswithQuestion1.3inamoregeneralsetting.Letthetwo(re-spective)parametersbepX;pY2(0;1).Itturnsoutthatthevalidityof 10GeoreyGrimmettthestatement\forallM2,P(XMY)=0"isindependentofthevaluesofpX,pY.Ontheotherhand,(1.5)isnotgenerallytrue.See[15].AnumberofeasiervariationsonQuestion1.3springimmediatelytomind,ofwhichtwoarementionedhere.1.SupposethegapbetweentheembeddingsofXi 1andXimustbeboundedabovebysomeMi.HowslowagrowthontheMisucesthattheembeddingprobabilitybestrictlypositive?[AnelementaryboundfollowsbytheBorel{Cantellilemma.]2.Supposethatthedemonisallowedtolookonlyboundedlyintothefuture.Howmuchclairvoyancemay(s)hebeallowedwithouttheembeddingprobabilitybecomingstrictlypositive?Furtherquestions(andvariationsthereof)havebeenproposedbyothers.1.Ina`penalisedembedding'problem,wearepermittedmismatchesbypayinga(multiplicative)penaltybforeach.Whatisthecostofthe`cheapest'penalisedembeddingoftherstnterms,andwhatcanbesaidasb!1?[ErwinBolthausen]2.Whatcanbesaidifwearerequiredtoembedonlythe1s?Thatis,a`1'mustbematchedtoa`1',buta`0'maybematchedtoeither`0'or`1'.[SimonGriths]3.TheaboveproblemsmaybedescribedasembeddingZinZ.Inthislanguage,mightitbepossibletoembedZminZnforsomem;n2?[RonPeled]Question1.3maybeexpressedasageometricalproblemofpercolationtype.WithXandYasabove,wedeclarethevertex(i;j)2N2tobeopenifXi=Yj.Apathisdenedasaninnitesequence(un;vn),n0,ofverticessuchthat:(u0;v0)=(0;0);(un+1;vn+1)=(un+1;vn+dn);forsomednsatisfying1dnM.ItiseasilyseenthatXMYifandonlyifthereexistsapathallofwhoseverticesareopen.(Wedeclare(0;0)tobeopen.)Withthisformulationinmind,theaboveproblemmayberepresentedbytheiconatthetopleftofFigure1.2.Thefurthericonsofthatgurerepresentexamplesofproblemsofsimilartype.Nothingseemstobeknownabouttheseexceptthat:1.theargumentofPeled[32]maybeappliedtoproblem(b)withM=2toobtainthatP(w2Y)=0forallw2W, Threeproblemsfortheclairvoyantdemon11 12Mn1nMM21n21M12(d)(f)(e)(a)(b)(c) Figure1.2Iconsdescribingavarietyofembeddingproblems.2.problem(e)iseasilyseentobetrivial.Itis,asonemightexpect,mucheasiertoembedwordsintwodimen-sionsthaninone,andindeedthismaybedonealongapathofZ2thatisdirectedinthenorth{easterlydirection.Thisstatementismademorepreciseasfollows.LetY=(Yi;j:i;j=1;2;:::)beatwo-dimensionalarrayofindependentBernoullivariableswithparameterp2(0;1),say.Awordv2WissaidtobeM-embeddableinY,writtenvMY,ifthereexiststrictlyincreasingsequences(mi:i1),(ni:i1)ofpositiveintegerssuchthatvi=Ymi;niand1(mi mi 1)+(ni ni 1)M;i1:(Wesetm0=n0=0.)Thefollowinganswersaquestionposedin[29].Noteaddedatrevision:Arelatedresulthasbeendiscoveredindepen-dentlyin[30]. 12GeoreyGrimmettTheorem1.7[14]SupposeR1issuchthat1 pR2 (1 p)R2~pc,thecriticalprobabilityofdirectedsitepercolationonZ2.Withstrictlypositiveprobability,everyinnitewordwsatisesw5RY.Theidenticationofthesetofwordsthatare1-embeddableinthetwo-dimensionalarrayY,withpositiveprobability,ismuchharder.Thisisaproblemofpercolationofwords,andtheresultstodatearesummarisedinSection1.7.ProofWeuseablockargument.LetR2f2;3;:::g.For(i;j)2N2,denetheblockBR(i;j)=((i 1)R;iR]((j 1)R;jR]N2.Onthegraphofblocks,wedenethe(directed)relationBR(i;j)!BR(m;n)if(m;n)iseither(i+1;j+1)or(i+1;j+2).Bydrawingapicture,oneseesthattheensuingdirectedgraphisisomorphictoN2directednorth{easterly.NotethattheL1-distancebetweentwoverticeslyinginadjacentblocksisnomorethan5R.WecallablockBRgoodifitcontainsatleastone0andatleastone1.ItistrivialthatPp(BRisgood)=1 pR2 (1 p)R2:Iftherightsideexceedsthecriticalprobability~pcofdirectedsiteper-colationonZ2,thenthereisastrictlypositiveprobabilityofaninnitedirectedpathofgoodblocksintheblockgraph,beginningatBR(1;1).Suchapathcontains5R-embeddingsofallwords. Theproblemofclairvoyantembeddingisconnectedtoaquestioncon-cerningisometriesofrandommetricspacesdiscussedin[33].Inbroadterms,twometricspaces(Si;i),i=1;2,aresaidtobe`quasi-isometric'(or`roughlyisometric')iftheirmetricstructureisthesameuptomul-tiplicativeandadditiveconstants.Thatis,thereexistsamappingT:S1!S2andpositiveconstantsM,D,Rsuchthat:1 M1(x;y) D2(T(x);T(y))M1(x;y)+D;x;y2S1;and,forx22S2,thereexistsx12S1with2(x2;T(x1))R.IthasbeenaskedwhethertwoPoissonprocessontheline,viewedasrandomsetswithmetricinheritedfromR,arequasi-isometric.Thisquestionisopenatthetimeofwriting.Anumberofrelatedresultsareprovedin[33],whereahistoryoftheproblemmaybefoundalso.Itturnsoutthattheabovequestionisequivalenttothefollowing.LetX=(:::;X 1;X0;X1;:::)beasequenceofindependentBernoullivariableswithcommonparameterpX.ThesequenceXgeneratesarandommetric Threeproblemsfortheclairvoyantdemon13spacewithpointsfi:Xi=1gandmetricinheritedfromZ.IsitthecasethattwoindependentsequencesXandYgeneratequasi-isometricmetricspaces?Apossiblyimportantdierencebetweenthisproblemandclairvoyantembeddingisthatquasi-isometriesofmetricsubspacesofZneednotbemonotone.1.6DependentpercolationWhereasthereisonlyonetypeofindependence,therearemanytypesofdependence,toomanytobesummarisedhere.Wementionjustthreefurthertypesofdependentpercolationinthissection,ofwhichtherst(atleast)arisesinthecontextofprocessesinrandomenvironments.Ineach,thedependencehasinniterange,andinthissensetheseproblemshavesomethingincommonwiththosetreatedinSections1.3{1.5.Forourrstexample,letX=fXi:i2Zgbeindependent,identicallydistributedrandomvariablestakingvaluesin[0;1].ConditionalonX,thevertex(i;j)ofZ2isdeclaredopenwithprobabilityXi,anddierentverticesreceive(conditionally)independentstates.Theensuingmeasurepossessesadependencethatextendswithoutlimitintheverticaldirec-tion.LetpcdenotethecriticalprobabilityofsitepercolationonZ2.IfthelawofX0placesprobabilitybothbelowandabovepc,thereexist(almostsurely)vertically-unboundeddomainsthatconsiderthemselvessubcritical,andothersthatconsiderthemselvessupercritical.Depend-ingonthechoiceof,theprocessmayormaynotpossessinniteopenpaths,andnecessaryandsucientconditionshaveprovedelusive.Themostsuccessfultechniquefordealingwithsuchproblemsseemstobetheso-called`multiscaleanalysis'.Thisleadstosucientconditionsunderwhichtheprocessissubcritical(respectively,supercritical).See[25,26].Thereisavarietyofmodelsofphysicsandappliedprobabilityforwhichthenaturalrandomenvironmentisexactlyoftheabovetype.Consider,forexample,thecontactmodelinddimensionswithrecoveryratesxandinfectionratese,see[27,28].Supposethattheenvironmentisrandomisedthroughtheassumptionthatthex(respectively,e)areindependentandidenticallydistributed.Thegraphicalrepresentationofthismodelmaybeviewedasa`verticallydirected'percolationmodelonZd[0;1),inwhichtheintensitiesofinfectionsandrecoveriesaredependentintheverticaldirection.See[1,8,31]forfurtherdiscussion.Verticaldependencearisesnaturallyincertainmodelsofstatisticalphysicsalso,ofwhichwepresentoneexample.The`quantumIsing 14GeoreyGrimmettmodel'onagraphGmaybeformulatedasaprobleminstochasticge-ometryonaproductspaceoftheformG[0;],whereistheinversetemperature.Afairbitofworkhasbeendoneonthequantummodelinarandomenvironment,thatis,whenitsparametersvaryrandomlyarounddierentvertices/edgesofG.ThecorrespondingstochasticmodelonG[0;]has`verticaldependence'ofinniterange.See[7,16].Itiseasytoadapttheabovestructuretoprovidedependenciesinbothhorizontalandverticaldirections,althoughtheensuingproblemsmaybeconsidered(sofar)tohavegreatermathematicalthanphysicalinterest.Forexample,considerbondpercolationonZ2,inwhichthestatesofhorizontaledgesarecorrelatedthus,andsimilarlythoseofverticaledges.Arelatedthree-dimensionalsystemhasbeenstudiedbyJonasson,Mossel,andPeres[18].DrawplanesinR3orthogonaltothex-axis,suchthattheyintersectthex-axisatpointsofaPoissonprocesswithgivenintensity.Similarly,drawindependentfamiliesofplanesorthogonaltothey-andz-axes.Thesethreefamiliesdenea`stretched'copyofZ3.Anedgeofthisstretchedlattice,oflengthl,isdeclaredtobeopenwithprobabilitye l,independentlyofthestatesofotheredges.Itisprovedin[18]that,forsucientlylarge,thereexists(a.s.)aninniteopendirectedpercolationclusterthatistransientforsimplerandomwalk.Themethodofproofisinteresting,proceedingasitdoesbythemethodof`exponentialintersectiontails'(EIT)of[5].WhencombinedwithanearlierargumentofHaggstrom,thisprovestheexistenceofapercolationphasetransitionforthemodel.ThemethodofEITisinvalidintwodimensions,becauserandomwalkisrecurrentonZ2.ThecorrespondingpercolationquestionintwodimensionswasansweredusingdierentmeansbyHoman[17].Inournalexample,thedependencecomeswithoutgeometricalin-formation.Letk2,andcallafamilyofrandomvariablesk-wiseindependentifanyk-subsetisindependent.Notethatthevertex-statesarisingintheclairvoyantschedulingproblemofSection1.3are3-wiseindependentbutnot4-wiseindependent.Benjamini,Gurel-Gurevich,andPeled[6]haveinvestigatedvariouspropertiesofk-wiseindependentBernoullifamilies,andinparticularthefollowingpercolationquestion.Considerthen-boxBn=[1;n]dinZdwithd2,inwhichthemeasuregoverningthesitevariablesf!(v):v2Bnghaslocaldensitypandisk-wiseindependent.LetLnbetheeventthattwogivenoppositefacesareconnectedbyanopenpathinthebox.Thus,forlargen,theprobabilityofLnundertheproductmeasurePphasasharpthresholdaroundp=pc(Zd).Theproblemistond Threeproblemsfortheclairvoyantdemon15boundsonthesmallestvalueofksuchthattheprobabilityofLnisclosetoitsvaluePp(Ln)underproductmeasure.Thisquestionmaybeformalisedasfollows.Let=(n;k;p)bethesetofprobabilitymeasuresonf0;1gBnthathavedensitypandarek-wiseindependent.Letn(p;k)=maxP2P(Ln) minP2P(Ln);andKn(p)=minfk:n(p;k)g;wherefordenitenesswemaytake=0:01asin[6].Thus,roughlyspeaking,Kn(p)isaquanticationoftheamountofindependencere-quiredinorderthat,forallP2,P(Ln)diersfromPp(Ln)byatmost.Benjamini,Gurel-Gurevich,andPeledhaveproved,inanongoingproject[6],thatKn(p)clognwhend=2andp=pc(andwhend2andppc),forsomeconstantc=c(p;d).TheyhaveinadditionalowerboundforKn(p)thatdependsonp,d,andn,andgoesto1asn!1.1.7PercolationofwordsRecallthesetW=f0;1gNofwordsinthealphabetcomprisingthetwoletters0,1.ConsiderthesitepercolationprocessofSection1.2onacountablyinniteconnectedgraphG=(V;E),andwrite!=f!(v):v2Vgfortheensuingconguration.Letv2VandletSvbethesetofallself-avoidingwalksstartingatv.Each2Svisapathv0;v1;v2;:::withv0=v.Withthepathweassociatethewordw()=!(v1)!(v2),andwewriteWv=fw():2Svgforthesetofwords`visiblefromv'.ThecentralquestionofsitepercolationconcernstheprobabilitythatWv311,where11denotestheinniteword111.Theso-calledAB-percolationproblemconcernstheexistenceinWvoftheinnitealternatingword01010,see[2].Moregenerally,forgivenp,weaskwhichwordslieintherandomsetWv.Partialanswerstothisquestionmaybefoundinthreepapers[4,19,20]ofKestenandco-authorsBenjamini,Sidoravicius,andZhang,andtheirresultsaresummarisedhereasfollows.ForZd,withp=1 2anddsucientlylarge,wehavefrom[4]thatP1 2(W0=W)0; 16GeoreyGrimmettandindeedthereexists(a.s.)somevertexvforwhichWv=W.PartialresultsareobtainedforZdwithedge-orientationsinincreasingcoordi-natedirections.ForthetriangularlatticeTandp=1 2,wehavefrom[19]thatP1 2 Sv2VWvcontainsalmosteveryword=1;(1.8)wherethesetofwordsseenincludesallperiodicwordsapartfrom01and11.ThemeasureonWcanbetakenin(1.8)asanynon-trivialproductmeasure.ThisextendstheobservationthatAB-percolationtakesplaceatp=1 2,whereasthereisnoinniteclusterintheusualsitepercolationmodel.Finally,forthe`close-packed'latticeZ2cpobtainedfromZ2byaddingbothdiagonalstoeachface,Pp(W0=W)0for1 pcppc,withpc=pc(Z2).Moreover,everywordis(a.s.)seenalongsomeself-avoidingpathinthelattice.See[20].AcknowledgementsTheauthoracknowledgesthecontributionsofhisco-authorsTomLiggettandThomasRichthammer.HeprotedfromdiscussionswithAlexanderHolroydwhileattheDepartmentofMathematicsattheUniversityofBritishColumbia,andwithRonPeledandVladasSidoraviciuswhilevis-itingtheInstitutHenriPoincare{CentreEmileBorel,bothduring2008.ThisarticlewaswrittenduringavisittotheSectiondeMathematiquesattheUniversityofGeneva,supportedbytheSwissNationalScienceFoundation.TheauthorthanksRonPeledforhiscommentsonadraft.References[1]Andjel,E.1992.Survivalofmultidimensionalcontactprocessinrandomenvironments.BulletinoftheBrazilianMathematicalSociety,23,109{119.[2]Appel,M.J.B.,andWierman,J.C.1993.ABpercolationonbond-decoratedgraphs.JournalofAppliedProbability,30,153{166.[3]Balister,P.N.,Bollobas,B.,andStacey,A.M.2000.Dependentper-colationintwodimensions.ProbabilityTheoryandRelatedFields,117,495{513. 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