transversal fluctuations for a first passage percolation model yuri bakhtin论文.pdf

该【transversal fluctuations for a first passage percolation model yuri bakhtin论文 】是由【宝钗文档】上传分享，文档一共【20】页，该文档可以免费在线阅读，需要了解更多关于【transversal fluctuations for a first passage percolation model yuri bakhtin论文 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..?rstpassagepercolationmodelinaPoissonianenvi-,?uctuationexponentforpoint-to-lineactionminimizersisatleast3/,weintroduceanew?rstpassagepercolation(FPP)?nesthemodeldependsnotonlyonthegeometryofthepathasaplanarsetbutalsoonthespeedoftravelor,equivalently,,forpoint-to-lineactionminimizers,weareabletoobtainalowerboundforthetransversal?uctuationexponentξ≥3/5,[HW65]asamodelofthe?uid??nedonthesquarelatticeZd,viaafamily(τe),,ordistanceT(x,y)betweentwoverticesx,yisthein?mumofe∈rτ(e),thenthereisalmostsurelyaPuniquepathr?connectingxtoysuchthatT(x,y)=τ(e).Suchoptimalpathse∈r?|x?y|→∞.Werefertoarecentsurvey[AHD15]?cultiesforthestandardFPPonZdariseduetolatticee?=limBr,whereB={x∈Zd:T(0,x)<r}.Theexistenceofthislimit,r→∞rrappropriatelyunderstood,isimpliedbytheshapetheorem([Ric73],seealso[Kes86]).ForanicedistributionF,itisnaturaltoconjecturethattheboundaryofBissmoothanduniformlycurved(asde?nedin[New95]),:[]19May2016Moreover,=χ(d)andξ=ξ(d).Roughlyspeaking,χandξarede?nedsothatthestandarddeviationofχ?T(x,y)isoforder|x?y|,andthe?uctuationofthegeodesicraboutthestraightlineξconnectingxtoyisoforder|x?y|.ThestandardFPPisbelievedtobelongtotheKPZuniversalityclass[KPZ86],thusind=2onebelievesthatχ=1/3andξ=2/,itwasprovedbyChatterjeethatχ=2ξ?1[Cha13].However,,thebestboundsforVarT(x,y)arethatithasalog|x?y|lowerbound(byNewmanandPiza[NP95]),andsublinearupperbound(byBenjamini,KalaiandSchramm[BKS03],seealso[DHS13]foranextension).Withcurvatureassumptions,itisknownthatχ≥1/8andξ≤3/4([NP95]).Forthelowerboundforξ,thebestknownresultisbyLicea,Newmanand1:..2YURIBAKHTINANDWEIWUPiza[LNP96],whode?nedanotherexponentξ′thatshouldbecloselyrelatedtoξandprovedthatξ′≥3/′afterthestatementofourmainresult,,duetolatticee?ects,itisdi?culttoadapttheargumentin[LNP96]forξitself,,,weintroduceananalogueofthepassagetimebetweenthemthat,besidestheendpointlocations,dependsalsoononeextraparameter,?mumofacertainactionfunctionalgivenbythedi?erenceicenergytermandthenumberofPoissonpointstouchedbythepath(see(1)and(2)belowforprecisede?nitions).Bakhtin,CatorandKhanin[BCK14]studiedasimilarlastpassagepercolationmodelin1+1dimension,?erentmodelsofFPPonRdhavebeenstudiedbyHowardandNewman([HN97],alsothesurvey[HN01]),andbyVahidi-AslandWierman[VAW90].Similarlytothemodelsstudiedin[HN97]and[VAW90],-to-lineactionminimizers,thetransversal?uctua-tionexponentξsatis?esξ≥3/?(suchastherelationχ=2ξ?1,χ≥1/8andξ≤3/4),=2,thereexistnodoublyin?nitegeodesicsinthestandardFPPonZd(adoublyin?nitegeodesicisadoublyin?nitepathsuchthatevery?nitesegmentofthepathisa?nitegeodesicbetweentheendpoints).IticIsingmodelonZ2hasonlytwogroundstates(namely,all+andall?,see[LN96]).Thisconjectureispartiallycon?rmedin[LN96],whereitisshown(underacurvatureassumptiononthelimitshape)thatforLebesguealmostevery?x,y?∈S1,?nitegeodesicthathasasymptoticdirections(?x,y?).Thisresultisstrengthenedbytherecentwork[DH15]thatrulesouttheexistenceofthedoublyin?nitegeodesicwithanyasymptoticdirections(assumingthatthelimitshapeboundaryisdi?erentiable).Theconjectureisstillopen,sincetheremayexistdoublyin?nitegeodesicwithin?[KS91]anexistaslongasξ>1/(ξ≥3/5)fortheFPPmodelthatwestudyseemstobesu?cient,itisstillaninterestingopenproblemtomaketheheuristicargumentsin[KS91][NP95]and[LNP96][Wut98]-,ifoneslightlyperturbsthelabelτ(e)ononeedge(retainingthevalueofalltheotherlabels),,,modifyingthepathinordertoincludethisextrapointmayleadtoasigni?cantchangeofthepathaction(sinceitisquadraticinthetotallengthofthepath).However,[BCK14]toobtainmomentboundsforaction.:..,wede?nepreciselythenewFPPmodelandstateourmainresult,?erenceoftwominimizers,,:-1007524andDMS--?nitepointcon?(i)givenaBorelsetB?R2,thenumberω(B)ofcon?gurationpointsinBisaPoissonrandomvariablewithmean|B|(theLebesguemeasureofB);(ii)fordisjointboundedBorelsetsA1,...,Am,therandomvariablesω(A1),...,ω(Am),weidentifylocally?nitepointcon?gurationsωwithinteger-valuedlocallyboundedBorelmeasureswithaunitatomateachpointofthecon??gurationωandanys>0wedenotebyC([0,s]:R2)thesetofωR2-valuedpiecewiselinearpathsde?nedon[0,s]∈C([0,s]:R2),hefollowingactionfunctionalωZss12A(γ)=|γ˙(u)|du?ωpp(γ),20whereωpp(γ)isthenumberofPoissonpointstouchedbyγand|·|?rstterm(“icenergy”)dependsonlyonthegeometryofthepath,whereasthesecondterm(“potentialenergy”)>0,wecande?neactionbetweenapointx∈R2,andasetS?R2byAs(x,S)=infAs(γ).γ∈Cω([0,s]:R2)γ(0)=x,γ(s)∈SWealsodenoteAs(S)=As(0,S).Onecanwritetheoptimizationproblemintwosteps:?rstminimizethevelocityconditionedonthepointcon?gurations,?nitionoftheaction(seeSection6foraproof):?2??PN??i=0|xi+1?xi|?As(x,S)=inf?N,(1)N≥0,(xi)N+1,xi6=xj?2s?i=0??x0=x,xN+1∈SNwherethein?mumistakenoverthenumberN∈N∪{0},locations(xi)i=1ofdistinctPoissonpoints,andtheterminalpointxN+1inthesetS.:..4YURIBAKHTINANDWEIWUThisresultshowsthatitissu?cienttoworkwithpathsunderstoodassequencesofpoints(x0,x1,...,xN,xN+1),wherex1,...,xNaredistinctPoissonpoints,assigningactionL2(x,x,...,x,x)s01NN+1A(x0,x1,...,xN,xN+1)=?(x0,x1,...,xN,xN+1)=i=0|xi+1?xi|.Wewillbemostlyinterestedinthepoint-to-∈S1,wede?neL={au:a∈R}.Also,forz=(x,y)∈R2(orequivalently,x+iy∈C),letΛuzdenotethelinepassingthroughzthatisperpendiculartoLz/|z|.Themainobjectinthispaperistheactionfrom0toΛt=Λt+i0=Λ(t,0):As(0,Λ)=infAs(γ).(2)t2γ∈Cω([0,s]:R)γ(0)=x,γ(s)∈ΛtBythenatureofthePoissonpointprocessandthecontinuityoftheactionwithrespecttoindividualparticlelocations,itfollowsthatthegeodesic,.,theminimizerin(2),.-(0,Λ),weareinterestedinthespace-timescalings=ct,foratconstantc>?∈(0,∞)withthefollowingproperties:ifc<c?,icenergydominates,andtheminimizerhasKPZ?uctuations;whenc>c?,theenvironmentcontributiondominates,andtheminimizerkeepswanderinginordertocollectmorePoissonpoints,whichmayleadtoalarger?,wewillfocusontheformercaseandprovidealowerboundforthetransversal?uctuationexponentofthepoint-to-linegeodesicsfortheactionde?nedby(2)(or(1)).Givenw>0andu∈S1,hecylinderC(w)symmetricaboutLandofuuwidthwas2 Cu(w)=z∈R:dist({z},Lu)≤w,whereforA,B?R2,dist(A,B):=infinf|x?y|.Givenx∈R2,S?R2,andx∈Ay∈Bt>0,wedenotebyM(x,S,t)∈C([0,t]:R2)thepathprovidingtheminimalactioninthede?nitionofAt(x,S).WedenoteM(S,t)=M(0,S,t)and,fors∈[0,t],useM(s)forM(S,t)(s),whenSandtareclearlyde?(x,y,t)=M(x,{y},t).hetransversal?uctuationexponentξforMbyξ=supγ:limsupsupP(M(Λ,ct)?C(tγ))<1.(3)tuut→∞u∈S1Equivalently,wecande?neξbyξ=supγ:limsupP(M(Λ,ct)?C(tγ))<1,(4)te1t→∞wheree1=(1,0).∈R2,S?R2,θ∈[0,2π),andletRbearotationde?nedbyR(S)= θθeiθy:y∈(x,Λ)=As(R(x),Λiθ)andM(x,S,t)=M(R(x),R(S),t)::..?∈(0,1]suchthatforallc<c?,wehaveξ≥3/?<1su?cesforTheorem3tohold,wedonottrytooptimizethevalueofc?.InthecaseofstandardFPPonZ2,Theorem3isestablishedin[LNP96],foranotherexponentξ′,ξ′de?nedin[LNP96]hastwoweaknesses:itonlyguaranteesM(Λ,ct)notbeingcon?nedinC(tγ),forsomesequences(u,t)tnunnunnnnsuchthatt→∞;also,thede?nitionofξ′,,wewritef(n)4g(n),orf(n)=O(g(n))ifthereexistsC<∞,suchthatforalln,f(n)≤Cg(n).Wewritef(n)?g(n)iff(n)4g(n)andg(n)4f(n).(n).GivenafunctionX:Z2→R,theweightofaPlatticeanimalA∈A(n),isde?nedbyNA=NA(X)=k∈?nedasNn=maxNA(X).(5)A∈A(n)Thefollowingtailboundfortheweightofgreedylatticeanimalsisaversionofageneralestimateestablishedin[CGGK93](seetheremarkafter()in[CGGK93])(Xj)j∈>>0suchthatify≥y=(e3λ)∨ρ,then0P(N>yn)≤e?yn,n∈=exp(λ(et?1))forallt>