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1、ACENTRALLIMITTHEOREMFORCONVOLUTIONEQUATIONSANDWEAKLYSELF-AVOIDINGWALKSERWINBOLTHAUSENANDCHRISTINERITZMANNAbstract.Themainresultofthispaperisageneralcentrallimittheoremfordistributionsdefinedbycertainrenewaltypeequations.Weapplythistoweaklyself-avoidingrandomwalks.Wegivegooderrores
2、timatesandGaussiantailestimateswhichhavenotbeenobtainedbyothermethods.Weusethe‘laceexpansion’andatthesametimedevelopanewperspectiveonthismethod:WeworkwithafixedpointargumentdirectlyinZdwithoutusingLaplaceorFouriertransformation.Contents1.IntroductionandResults12.DeterminingtheMass
3、Constants63.LocalEstimatesinHighDimensions104.ApplicationtotheWeaklySelf-AvoidingWalk23AppendixA.ALCLTandDiscretizationEstimates25AppendixB.TheLaceExpansion30References351.IntroductionandResults1.1.Introduction.ThestandardsimplerandomwalkonthehypercubiclatticeZdisgivenbytheunifor
4、mdistributiononthesetofnearest-neighbourpathsstart-ingin0,andoflengthn.Thelawoftheself-avoidingrandomwalkissimplytheuniformdistributiononthesetofwalkshavingnoself-intersections.Aninterpo-lationbetweenthestrictlyself-avoidingwalkandthestandardrandomwalkistheso-calledweaklyself-avo
5、idingwalk(alsoknownas‘Domb-Joycemodel’).Here,self-arXiv:math/0103218v2[math.PR]11Jan2002intersectionsarenotcompletelyforbidden,butpenalizedbyafactor1−λforeveryself-intersection,whereλ∈(0,1)isaparameter.Notmuchisknownrigorouslyfortheseself-avoidingrandomwalksindimensionsd=2,3and4.
6、AquantityofbasicinterestisthesocalledconnectivityC(x),x∈Zd,n∈N,nwhichisobtainedbysummingtheweightsofallpathsfrom0toxoflengthn:Intherandomwalkcase,thepathsallgetweight1,inthestrictlyself-avoidingcase,onlythepathswithoutself-intersectionsgetweight1,theothers0,andintheweaklyself-avo
7、idingcase,theweightisgivenintermsofthenumberofself-intersections(andtheparameterλ)asindicatedabove.Afternormalization,thisdefinesthedistributionoftheend-pointofthewalk.Thefirstresultsinthecased≥5wereobtainedbyBrydgesandSpencer[2]inthemiddleoftheeighties.Theyintroducedaperturbativee
8、xpansiontechnique,basedonthesocalledlace
