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arXiv:math/0608400v3[math.PR]5Feb2008OrderofcurrentvarianceanddiffusivityintheasymmetricsimpleexclusionprocessM´artonBal´azs∗,TimoSepp¨al¨ainen†February5,2008AbstractWeprovethatthevarianceofthecurrentacrossacharacteristicisofordert2/3inastationaryasymmetricsimpleexclusionprocess,andthatthediffusivityhasordert1/3.Theproofproceedsviacouplingstoshowthecorrespondingmomentboundsforasecondclassparticle.Keywords:Asymmetricsimpleexclusionprocess,diffusivity,currentfluctua-tions,secondclassparticle2000MathematicsSubjectClassification:60K35,82C221IntroductionTheasymmetricsimpleexclusionprocess(ASEP)isaMarkovprocessthatde-scribesthemotionofparticlesontheone-dimensionalintegerlatticeZ,subjecttotheexclusioninteractionthatallowsatmostoneparticleateachsite.Particlesintheprocessjumponesteptotherightwithratepandonesteptotheleftwithrateq=1−p,andweassume0≤qp≤1.Particlesattemptjumpsindependentlyofeachother,butanyattempttojumpontoanalreadyoccupiedsiteissuppressed.InSection2belowwegivearigorousconstructionofASEPintermsofPoissonclocksthatgovernthejumpattempts.ThisprocessisamongtheinteractingparticlesystemsintroducedinSpitzer’sseminalpaper[25].WereferthereadertoLiggett’smonographs[16,17]forcoverageofmostoftheworkonASEPuptothelate1990’s.In1994FerrariandFontes[10]provedacentrallimittheoremforthenetparticlecurrentseenbyanobservermovingatafixedspeedv.Thisquantity∗MTA-BMEStochasticsResearchGroup†UniversityofWisconsin-MadisonM.Bal´azswaspartiallysupportedbytheHungarianScientificResearchFund(OTKA)grantsT037685,K60708,TS49835,andF67729,theBolyaiScholarshipoftheHungarianAcademyofSciences,andNationalScienceFoundationgrantDMS-0503650.PartofthisworkwascompletedwhileBal´azswasavisitingassistantprofessoratUniversityofWisconsin-Madison.T.Sepp¨al¨ainenwaspartiallysupportedbyNationalScienceFoundationgrantsDMS-0402231andDMS-0701091.1thatwedenotebyJ(v)(t)isthenumberofparticlesthatpasstheobserverfromlefttorightminusthenumberthatpassfromrighttoleftduringtimeinterval(0,t].TheparticleprocessisassumedtobestationarywithBernoulli-distributedoccupationvariablesatsomeaveragedensity̺∈(0,1).Theresultistheweaklimitofthediffusivelyrescaledandcenteredcurrent:limt→∞J(v)(t)−E[J(v)(t)]t1/2=χv.ThelimitχvisacenteredGaussianrandomvariablewithvarianceσ2=̺(1−̺)|(p−q)(1−2̺)−v|.Theinterestingphenomenonisthevanishingofthevarianceatthecharacter-isticspeedv=V̺≡(p−q)(1−2̺).Asweexplainbelow,V̺isthespeedatwhichperturbationstravelinthesystem,bothatthemicroscopicparticlelevel(intheexpectedsenseasgivenin(2.7)below)andatthemacroscopicp.d.e.level.Physicalreasoning[26]impliedthatthecorrectorderofthefluctuationsofthecurrentacrossthecharacteristicshouldbet1/3.Thiswouldexplainthedegeneratelimitundert1/2normalization.These“t1/3fluctuations”remainedelusivethroughoutthe1990’s.TheseminalpapersofBaik,DeiftandJohansson[2]andJohansson[13]gavethefirstrigorousproofsofsuchfluctuations.Thecorrectorderwasverifiedtobet1/3,andthelimitingfluctuationswerefoundtoobeyTracy-Widomdistributionsfromrandommatrixtheory.Thefirstpaperdealtwiththelast-passageversionofHammersley’sprocess,andthesecondwiththelast-passageversionofthetotallyasymmetricsimpleexclusionprocess(TASEP).Totalasymmetrymeansherethatparticlesareallowedtojumponlyinonedirectionataconstantrate,sothisisthecasep=1,q=0.Thesepapersdidnotstudystationaryparticleprocesses,butinsteadprocessesstartedfromspecialjam-typedeterministicinitialconditions.ForTASEPthismeansthatinitiallyallsitestotheleftoftheoriginareoccupiedandallsitestotherightempty.Withsuchinitialconditionstheprocessescouldberepresentedbylast-passagepercolationmodels,apointthathadbeenexploitedalreadyinthepast(amongtheseminaloneswere[1],[22]and[23]).Theactualanalysiswasthenperformedentirelyoncombinatorialdescriptionsofthelast-passagemodel.Lateralast-passagerepresentationwasalsofoundforastationaryTASEP[18],andthentheTracy-Widomlimitprovedforthecurrentacrossthecharacteristicinthatsetting[12].TheearlyproofsoffluctuationsreliedonacountingargumentthatutilizestheRobinson-Schensted-KnuthcorrespondenceforYoungtableaux,andGessel’sformulathatconvertscertainSchurfunctionsumsintoToeplitzdeterminants.Laterthisstephasbeenreplacedbyamoredirectconnectionbetweenthelast-passagemodelandadeterminantalpointprocess.Thefluctuationlimitsarethenderivedbyanalyzingtheasymptoticsofthedeterminantintheappropriatescalingregime.Consequently,whileagenuinebreakthroughhasbeenachieved,thedelicatestepsoftheproofrestrictthereachoftheresultsinseveralways.2Inparticular,theparticlesofthesystemsarepermittedtomoveinonlyonedirectionandadmitonlythesimplesttypeofjumps.Inthepresentpaperwegivethefirstproofoftheaccurateorderofthefluc-tuationsinsystemsthatareonlypartiallyasymmetric.Namely,weshowintheoriginalsettingofFerrari-Fontes[10]thatthevarianceofthecurrentacrossthecharacteristicin(p,q)ASEPisofordert2/3.Ourargumentsareentirelyprob-abilisticandutilizecouplingsofseveralprocessesandboundsonsecondclassparticles.Informallyspeaking,secondclassparticlesareperturbationsinthesystemthatdonotdisturbthemotionoftheregularparticlesbutareinfluencedbytheambientsystem.Precisede
本文标题:Order of current variance and diffusivity in the a
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