A central limit theorem for weighted averages of s

arXiv:math/0405358v1[math.PR]18May2004AcentrallimittheoremforweightedaveragesofspinsinthehightemperatureregionoftheSherrington-Kirkpatrickmodel.DmitryPanchenko∗†77MassachusettsAvenue,Room2-181Cambridge,MA,02451email:[e-mail]:617-253-2665fax:[phone]February1,2008AbstractInthispaperweprovethatinthehightemperatureregionoftheSherrington-KirkpatrickmodelforatypicalrealizationofthedisordertheweightedaverageofspinsPi≤NtiσiwillbeapproximatelyGaussianprovidedthatmaxi≤N|ti|/Pi≤Nt2iissmall.Keywords:spinglasses,Sherrington-Kirkpatrickmodel,centrallimittheorem.1Introduction.ConsideraspaceofconfigurationsΣN={−1,+1}N.Aconfigurationσ∈ΣNisavector(σ1,...,σN)ofspinsσieachofwhichcantakethevalues±1.Consideranarray(gij)i,j≤Nofi.i.d.standardnormalrandomvariablesthatiscalledthedisorder.Givenparametersβ0andh≥0,letusdefineaHamiltonianonΣN−HN(σ)=β√NX1≤ij≤Ngijσiσj+hXi≤Nσi,σ=(σ1,...,σN)∈ΣN∗DepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts.†Suggestedrunninghead:Weightedaveragesofspins.1anddefineaGibbs’measureGonΣNbyG({σ})=exp(−HN(σ))/ZN,whereZN=Xσ∈ΣNexp(−HN(σ)).ThenormalizingfactorZNiscalledthepartitionfunction.Gibbs’measureGisarandommeasureonΣNsinceitdependsonthedisorder(gij).Theparameterβphysicallyrepresentstheinverseofthetemperatureandinthispaperwewillconsideronlythe(very)hightemperatureregionoftheSherrington-Kirkpatrickmodelwhichcorrespondstoββ0(1.1)forsomesmallabsoluteconstantβ00.Theactualvalueβ0isnotspecifiedherebut,inprincipal,itcanbedeterminedthroughcarefulanalysisofallargumentsofthispaperandreferencestootherpapers.Foranyn≥1andafunctionfontheproductspace(ΣnN,G⊗n),hfiwilldenoteitsexpectationwithrespecttoG⊗nhfi=XΣnNf(σ1,...,σn)G⊗n({(σ1,...,σn)}).TheSherrington-Kirkpatrickmodelhasbeenstudiedextensivelyoverthepastthirtyyears(see,forexample,[1]-[8],[11]-[18]).Inthispaperwewillprovethefollowingresultconcerningthehightemperatureregion(1.1).Givenavector(t1,...,tN)suchthatt21+...+t2N=1(1.2)letusconsiderarandomvariableon(ΣN,G)definedasX=t1σ1+...+tNσN.(1.3)Themaingoalofthispaperistoshowthatinthehightemperatureregion(1.1)thefollowingholds.Ifmaxi≤N|ti|issmallthenforatypicalrealizationofthedisorder(gij)therandomvariableXisapproximatelyGaussianr.v.withmeanhXiandvariancehX2i−hXi2.Bythe“typicalrealization”weunderstandthatthestatementholdsonthesetofmeasurecloseto1.Thisresultistheanalogueofaveryclassicalresultforindependentrandomvariables.Namely,givenasequenceofindependentrandomvariablesξ1,...,ξNsatisfyingsomeinte-grabilityconditionstherandomvariableξ1+...+ξNwillbeapproximatelyGaussianifmaxi≤NVar(ξi)/Pi≤NVar(ξi)issmall(see,forexample,[9]).Inparticular,ifσ1,...,σNin(1.3)werei.i.d.BernoullirandomvariablesthenXwouldbeapproximatelyGaussianprovidedthatmaxi≤N|ti|issmall.Itisimportanttonoteatthispointthatthemainclaimofthispaperinsomesenseisawellexpectedresultsinceitiswellknownthatinthehightemperatureregionthespinsbecome“decoupled”inthelimitN→∞.Forexample,Theorem2.4.10in[15]states2thatforafixedn≥1,foratypicalrealizationofthedisorder(gij)thedistributionG⊗nbecomesaproductmeasurewhenN→∞.Thus,intheveryessencetheclaimthatXin(1.3)isapproximatelyGaussianisacentrallimittheoremforweaklydependentrandomvariables.However,theentiresequence(σ1,...,σN)isamuchmorecomplicatedobjectthanafixedfinitesubset(σ1,...,σn),andsomeunexpectedcomplicationsarisethatwewilltrytodescribeafterwestateourmainresult-Theorem1below.InsteadofdealingwiththerandomvariableXwewilllookatitssymmetrizedversionY=X−X′,whereX′isanindependentcopyofX.IfwecanshowthatYisapproximatelyGaussianthen,obviously,XwillalsobeapproximatelyGaussian.Themainreasontocon-siderasymmetrizedversionofXisverysimple-itmakesitmucheasiertokeeptrackofnumerousindicesinalltheargumentsbelow,eventhoughitwouldbepossibletocarryoutsimilarargumentsforacenteredversionX−hXi.Inordertoshowthatforatypicalrealization(gij)andasmallmaxi≤N|ti|,Yisap-proximatelyGaussianwithmean0andvariancehY2iwewillproceedbyshowingthatitsmomentsbehavelikemomentsofaGaussianrandomvariable,i.e.hYli≈a(l)hY2il/2,(1.4)wherea(l)=Egl,forastandardnormalrandomvariableg.Sincethemomentsofthestandardnormalrandomvariablearealsocharacterizedbytherecursiveformulasa(0)=0,a(1)=1anda(l)=(l−1)a(l−2),(1.4)isequivalenttohYli≈(l−1)hY2ihYl−2i.Letusdefinetwosequences(σ1(l))l≥0and(σ2(l))l≥0ofjointlyindependentrandomvari-ableswithGibbs’distributionG.Wewillassumethatallindices1(l)and2(l)aredifferentandonecanthinkofσ1(l)andσ2(l)asdifferentcoordinatesoftheinfiniteproductspace(Σ∞N,G⊗∞).LetusdefineasequenceSlbySl=NXi=1ti¯σli,where¯σl=σ1(l)−σ2(l).(1.5)Inotherwords,SlareindependentcopiesofY.ThefollowingTheoremisthemainresultofthepaper.Theorem1Thereexistsβ00suchthatforββ0thefollowingholds.Foranynaturalnumbersn≥1andk1,...,kn≥0andk=k1+...+kn,wehaveEhnYl=1(Sl)kli−nYl=1a(kl)EhS21ik/2=Ø(maxi≤N|ti|),(1.6)whereØ(·)dependsonβ0,n,kbutnotonN.3Remark.Theorem1answersthequestionraisedintheResearchproblem2.4.11in[15].Theorem1easilyimpliesthatEhnYl=1(Sl)kli−nYl=1a(kl)hS21ik/22=Ø(maxi≤N|ti|).(1.7)Indeed,EhnYl=1(Sl)kli−nYl=1a(kl)hS21ik/22=EhnYl=1(Sl)kli2−2nYl=1