do the same for the ows independently fr om the columns point of will be called blocked if its ow and column have the same color say that this random con57346guration per colates if ther is path in starting at the origin consisting of rightwar and u ID: 67310
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THECLAIRVOYANTDEMONHASAHARDTASKPETERG´ACSABSTRACT.ConsidertheintegerlatticeL=Z2.Forsomem4,letuscoloreachcolumnofthislatticeindependentlyanduniformlyintooneofmcolors.Wedothesamefortherows,independentlyfromthecolumns.ApointofLwillbecalledblockedifitsrowandcolumnhavethesamecolor.WesaythatthisrandomcongurationpercolatesifthereisapathinLstartingattheorigin,consistingofrightwardandupwardunitsteps,andavoidingtheblockedpoints.Asaproblemarisingindistributedcomputing,ithasbeenconjecturedthatform4,thecongurationpercolateswithpositiveprobability.Thishasnowbeenproved(inalaterpaper)forlargem.Weprovethattheprobabilitythatthereispercolationtodistancenbutnottoinnityisnotexponentiallysmallinn.Thisnarrowstherangeofmethodsavailableforprovingtheconjecture.1.STATEMENTOFTHERESULT1.1.Introduction.Letx=(x(0),x(1),...)beaninnitesequenceandu=(u(0),u(1),...)beabinarysequencewithelementsinf0,1g.Letsn=ån 1i=0u(i).Wedenethedelayedversionx(u)ofx,byx(u)(n)=x(sn).Thus,ifi=snthenx(u)(n)=x(i),andx(u)(n+1)=x(i)orx(i+1)dependingonwhetheru(n)=0or1.Ifu(n)=0thenwecansaythatx(u)isdelayedattimen+1.Fortwoinnitesequencesx,ywesaythattheydonotcollideifthereisadelaysequenceusuchthatforeachnwehavex(u)(n)6=y(1 u)(n).Here,1 uisthedelaysequencecomplementarytou:thus,y(1 u)isdelayedattimenifandonlyifx(u)isnot.Foragivenm�1,supposethatX=(X(0),X(1),...)isaninnitesequenceofindepen-dentrandomvariables,andY=(Y(0),Y(1),...)isanothersuchsequence,alsoindepen-dentofX,whereallvariablesareuniformlydistributedoverf0,...,m 1g.Thefollowingtheoremhasbeenconjecturedin[2].Proposition1.1(See[3]).Ifmissufcientlylargethenwithpositiveprobability,XdoesnotcollidewithY.ThesequencesX,YcanbeviewedastwoindependentrandomwalksonthecompletegraphKm,andthentheproblemiswhetheraclairvoyantdemon,i.e.abeingwhoknowsinadvancebothinnitesequencesXandY,canintroducedelaysintothesewalksinsuchawaythattheynevercollide.Keywordsandphrases.Dependentpercolation,scheduling,distributedcomputing.1 2PETERG´ACS1.2.Graphreformulation.WedeneagraphG=(V,E)asfollows.V=Z2�0isthesetofpoints(i,j)wherei,jarepositiveintegers.Letusdenethedistanceoftwopoints(i,j),(k,l)asjk ij+jl jj(L1distance).WhenrepresentingthesetVofpoints(i,j)graphically,therightdirectionistheoneofgrowingi,andtheupwarddirectionistheoneofgrowingj.ThesetEofedgesconsistsofallpairsoftheform((i,j),(i+1,j))and((i,j),(i,j+1)).GivenX,Yasinthetheorem,letussaythatapoint(i,j)hascolorkifX(i)=Y(j)=k.Otherwise,ithascolor 1,whichwewillcallwhite.ItiseasytoseethatXandYdonotcollideifandonlyifthereisaninnitedirectedpathinGstartingfrom(0,0)andproceedingonwhitepoints.Indeed,eachpathcorrespondstoadelaysequenceusuchthatu(n)=1ifandonlyiftheedgeishorizontal.Thus,thetwosequencesdonotcollideifandonlyifthegraphofwhitepointspercolates.Wewillsaythatthereispercolationiftheprobabilitythatthereisaninnitepathispositive.Ithasbeenshownindependentlyin[1]and[4]thatifthegraphisundirectedthenthereispercolationevenform=4.Intraditionalpercolationtheory,whenthereispercolationthentypically(unlesstheprobabilityofblockingisatacriticalpoint)theprobabilitythatthereispercolationtoadistancenbutnopercolationtoinnityisexponentiallysmallinn.Thisisthecasealsointhepaperscitedabove,butitisnottrueforthedirectedpercolationproblemwearefacing.Theorem1.IfthereispercolationfromtheorigintoinnitywithpositiveprobabilitythentheprobabilityofpercolatingfromtheorigintodistancenbutnottoinnityisatleastCn aforsomeconstantsC,a�0dependingonlyonm.2.THEPROOFLetbm=(0,1,2,...,m 1)becalledthebasiccolorsequenceoflengthm:itissimplythelistofalldifferentcolors.Letb0mbethereverseofbm,i.e.b0m(i)=bm(m i 1).LetEn,kbetheeventthatforalli2[0,k 1],j2[0,m 1]wehaveY(n+im+j 1)=b0m(j),i.e.startingwiththeindexn 1,thesequenceYhaskconsecutiverepetitionsofb0m.WesaythatiisanindexoftheoccurrenceofbminthesequenceXifX(i+j)=bm(j)forj2[0,m 1].Fori�0,lettibethei-thindexofoccurrenceofbminX.LetFn,kbetheeventthatforalli2[1,n]wehaveti+1 m tik 1,(1)andalsot1k 1.Lemma2.1.Ifforsomeintegern�0bothEn,kandFn,kholdthenthereisnowhiteinnitedirectedpath.Proof.Letusassumethatthereisawhiteinnitepath.SincetherearenconsecutivecopiesofbminX,thei-thonestartingatindexti,foreachp2[1,n]theremustbeaverticalstepinthepathwithanxprojectionin[tp,tp+m 1].Thereforethepathascendstothesegment[0,tn+m 1]fn 1g,beforeitsxprojectionreachestn.Foreach1pn,0qkthereisadiagonallydescendingsequenceofmcoloredpointsf(tp+j,n+(q+1)m j 2):j2[0,m 1]g.Foraxedpthereareksuchdiagonalbarriersstackedaboveeachother,forminganim-penetrablecolumnofheightkm.Thepathwouldhavetoascendbetweentwoofthese THECLAIRVOYANTDEMONHASAHARDTASK3columns,saybetweencolumniandi+1.Thedistanceoftwoconsecutivecolumnsfromeachotheristi+1 m tik 1.Sincetherearekconsecutivecopiesofb0minYstartingatindexn 1,foreachq2[0,k 1]theremustbeahorizontalstepinthepathwithaheightinn+qm 1+[0,m 1].Sincethedistanceofthetwocolumnsisatmostk 1,itisnotpossibleforthepathtopassbetweenthetwocolumns.(Thesameholdsforthespacebeforetherstcolumn.)Lemma2.2.Thereisaconstantasuchthatforalls�0,thereisakwithProb(En,k)m mn a(s+1),Prob(Fn,k)1 n s.Proof.Withp1=m m,wehaveProb(En,k)=pk1.LetusestimatetheprobabilityofFn,k.Theprobabilitythatt1�k 1isupperboundedbytheprobabilitythatacopyofbmdoesnotbeginatiinXforanyiinf0,m,...,b(k 1)/mcmg,whichis(1 p1)b(k 1)/mc+1(1 p1)k/me p1k/m.Thesameestimateholdsfortheprobabilityof(1)assumingthatthesimilarconditionsforsmallerihavealreadybeensatised.HenceProb(Fn,k)�1 ne p1k/m.Letuschoosek=d(s+1)(mlogn)/p1e(2)forsomes�0,thenProb(Fn,k)�1 n s,whileProb(En,k)p1nmlogp1p1(s+1).ProofofTheorem1.AssumethatthereisaninnitepathwithsomepositiveprobabilityP0.Choosessuchthatn s0.5P0andchoosekasafunctionofsasin(2).ThentheprobabilitythatFn,kholdsandthereisaninnitepathisatleast0.5P0.LetGnbetheeventthatthereisapathleavingtherectangle[0,tn][0,n 1].ThenProb(Fn,k^Gn)0.5P0.SinceFn,k^GnisindependentofEn,k,wehaveProb(En,k^Fn,k^Gn)0.5P0m mn a(s+1).TheauthoristhankfultoJ´anosKoml´osandEndreSzemer´edifortellinghimabouttheproblem.REFERENCES[1]P.N.Balister,B.Bollobas,andA.N.Stacey,Dependentpercolationintwodimensions,Tech.report,1999.1.2[2]D.Coppersmith,P.Tetali,andP.Winkler,Collisionsamongrandomwalksonagraph,SIAMJ.DiscreteMath6(1993),no.3,363374.1.1[3]PeterG´acs.Clairvoyantschedulingofrandomwalks.math.PR/0109152.1.1[4]P.Winkler,Dependentpercolationandcollidingrandomwalks,Tech.report,1999.1.2 4PETERG´ACSCOMPUTERSCIENCEDEPARTMENT,BOSTONUNIVERSITYE-mailaddress:gacs@bu.edu