Vacant Set of Random Interlacements and Percolation

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the

VACANTSETOFRANDOMINTERLACEMENTSANDPERCOLATION

Alain-SolSznitman

PreliminaryDraft

arXiv:0704.2560v2 [math.PR] 21 May 2007AbstractWeintroduceamodelofrandominterlacementsmadeofacountablecollectionofdoublyin nitepathsonZd,d≥3.Anon-negativeparameterumeasureshowmanytrajectoriesenterthepicture.ThismodeldescribesinthelargeNlimitthemicroscopicstructureinthebulk,whichariseswhenconsideringthedisconnectiontimeofadiscretecylinder(Z/NZ)d 1×Zbysimplerandomwalk,orthesetofpointsvisitedbysimplerandomwalkonthediscretetorus(Z/NZ)dattimesoforderuNd.Inparticularwestudythepercolativepropertiesofthevacantsetleftbytheinterlacementatlevelu,whichisanin niteconnectedtranslationinvariantrandomsubsetofZd.Weintroduceacriticalvalueu suchthatthevacantsetpercolatesforu<u anddoesnotpercolateforu>u .Ourmainresultsshowthatu is nitewhend≥3andstrictlypositivewhend≥7.

DepartementMathematik

ETHZ¨urich

CH-8092Z¨urich

SwitzerlandMay2007

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the

0Introduction

Thisarticleintroducesamodelofrandominterlacementsconsistingofacountablecol-lectionofdoublyin nitetrajectoriesonZd,d≥3.Acertainnon-negativeparameterugovernstheamountoftrajectorieswhichenterthepicture.Theunionofthesupportsofthesetrajectoriesde nestheinterlacementatlevelu.Itisanin niteconnectedtransla-tioninvariantrandomsubsetofZd.Ourmainpurposeistostudywhetherthisrandom“fabric”is“rainproof”ornot,i.e.whetheritscomplement,thevacantsetatlevelu,doesnot,ordoescontainanin niteconnectedcomponent.Thisissueisrelatedtothebroadquestion“howcanrandomwalkpathscreateinterfacesinhighdimension?”.Themodelweconstructhasaspecialinterestbecauseinaheuristicsenseito ersamicroscopicdescriptionofthe“textureinthebulk”fortwoproblemsrelatedtothisbroadquestion.OneproblempertainstothepercolativepropertiesinthelargeNlimitofthevacantsetleftonthediscretetorus(Z/NZ)d,d≥3,bythetrajectoryofsimplerandomwalkwithuniformlydistributedstartingpoint,uptotimesthatareproportionaltothenumberofsitesofthediscretetorus,cf.[2].TheotherproblempertainstothelargeNbehaviorofthedisconnectiontimeofadiscretecylinder(Z/NZ)d 1×Z,d≥3,bysimplerandomwalk,cf.[6],[7],andalso[18].Inthisworkweestablishaphasetransition:foru<u ,thevacantsetatleveludoespercolate,whereasforu>u ,itdoesnot.Thecriticalvalueu isshowntobenon-degenerate(i.e.positiveand nite),whend≥7,and niteforalld≥3.Weexpectthattheresultspresentedherewillalsoleadtoprogressonthetwoproblemsalludedtoabove.

Wenowdescribethemodel.WeconsiderthespacesW+andWofin nite,respectivelydoublyin nite,nearestneighborpathsonZd,d≥3,thatspend nitetimeinboundedsubsetsofZd.WedenotewithPx,x∈Zd,thelawonW+ofsimplerandomwalkstartingatx.Thisismeaningfulsincethewalkistransientinviewoftheassumptiond≥3.WewriteXn,n≥0,orXn,n∈Z,forthecanonicalcoordinatesonW+,oronW.Wealsoconsiderthesetofdoubly-in nitetrajectoriesmodulotime-shifts

(0.1)W =W/～,wherew～w′,ifw(·)=w(·+k),forsomek∈Z.Wedenotewithπ :W→W ,thecanonicalprojection. ,u)TherandominterlacementsaregovernedbyaPoissonpointprocessω=i≥0δ(wii onW×R+,withintensitymeasureν(dw)du,whereνisacertainσ- nitemeasureonW ,whichwenowdescribe.Forany nitesubsetKofZd,wedenotewitheKthe

0equilibriummeasureofK,cf.(1.6),withWKthesubsetofWoftrajectoriesenteringK

attime0:

(0.2)0WK={w∈W;w(0)∈Kandw(n)∈/K,foralln<0},

0andwithWK=π (WK)thesubsetofW ofequivalenceclassesoftrajectoriesenteringK.WeshowinTheorem1.1thatthereisauniqueσ- nitemeasureνonW suchthat(0.3) d ν=π QK,forany nitesubsetKofZ,1WK

0whereQKisthe nitemeasuresupportedonWKsuchthat

i)QK(X0∈·)=eK(·),

(0.4)ii)wheneK(x)>0,conditionallyonX0=x,(Xn)n≥0,and(X n)n≥0are

independentwithrespectivedistributionsPxandPxconditioned

on{Xn∈/K,foralln≥1}.

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the

ThemotivationforsucharequirementstemsfromTheorem3.1and(3.13)of[2],wherethelargeNlimitofcertainsuitablyde nedexcursionstoaboxofsizeL<<N,bysimplerandomwalkon(Z/NZ)dwasinvestigated,andfromthealternativecharacterizationofQKgivenin(1.26),seealsoRemarks1.22)and1.63).Similarmeasuresappearin[20]and

[15],p.61,followinganoutlinein[10].Theconstructionwegiveherebypassesprojectivelimitarguments.Wedenotewith thecanonicalspacewhereωvaries,cf.(1.16),andwithPthelawturningωintoaPoissonpointprocesswithintensityν(dw )du.ThelawPenjoysanumberofremarkableproperties.Itisinvariantundertranslationoftrajectoriesbyaconstantvector,andundertime-reversaloftrajectories,cf.Proposition1.3.AlsowhenKisa nitesubsetofZd,weintroducetherandompointprocessonW+×R+:

,u),(0.5)µK(ω)=δ(wi,ui)1{wi∈WK},ifω=δ(wiii≥0i≥0

whereforwi∈WK,widenotestheuniquetrajectoryinW+startingattime0,wherewi entersK,andfollowingfromthenonwistepbystep,cf.(1.18)fortheprecisede nition.

WeshowinProposition1.3that

(0.6)µKisaPoissonpointprocesswithintensityPeK(dw)du,

wherePeK=xeK(x)Px.Furtherthepointproces …… 此处隐藏：26838字，全部文档内容请下载后查看。喜欢就下载吧 ……