Limit Correlation Functions for Fixed Trace Random

arXiv:math/0610149v2[math.PR]12Dec2007LIMITCORRELATIONFUNCTIONSFORFIXEDTRACERANDOMMATRIXENSEMBLESFRIEDRICHG¨OTZE1ANDMIKHAILGORDIN1,2Abstract.Universallimitsfortheeigenvaluecorrelationfunc-tionsinthebulkofthespectrumareshownforaclassofnon-determinantalrandommatricesknownasthefixedtraceortheHilbert-Schmidtensemble.TheseuniversallimitshavebeenprovedbeforefordeterminantalHermitianmatrixensemblesandforsomespecialclassesoftheWignerrandommatrices.1.IntroductionandthestatementoftheresultLetHNbethesetofallN×N(complex)Hermitianmatrices,andlettrA=PNi=1aiidenotethetraceofasquarematrixA=(aij)Ni,j=1.HNisarealHilbertspaceofdimensionN2withrespecttothesymmetricbilinearform(A,B)7→trAB.LetlNdenotetheuniqueLebesguemeasureonHNwhichsatisfiestherelationlN(Q)=1foreverycubeQ⊂HNwithedgesoflength1.AGaussianprobabilitymeasureonHNinvariantwithrespecttoallorthogonallineartransformationsofHNisuniquelydefineduptoascalingtransformation.Suchmeasuresformaone-parameterfamily(μsN)s0,wherethemeasureμsNisspecifiedbyitsdensity(1.1)gsN(A)=1(√s2π)N2exp−12strA2withrespecttolN.Thus,forarandommatrixXdistributedaccordingtoμsNwehave(1.2)EμsNtrX2=sN2.ThesetHNendowedwiththemeasureμsNiscalledtheGaussianUni-taryEnsemble(GUE).LetXbearandomN×NHermitianmatrixKeywordsandphrases.Randommatrices,fixedtraceensemble.1ResearchsupportedbySonderforschungsbereich701”SpektraleStrukturenundTopologischeMethodeninderMathematik”.2PartiallysupportedbygrantsRFBR-05-01-00911,DFG-RFBR-04-01-04000,andNS-2258.2003.1.1(thatisarandomvariabletakingvaluesinHN).Weconsidertheeigen-valuesλ1,λ2,...,λNoftherandommatrixXasafinitesequenceofex-changeablerandomvariables.Bydefinition,thismeansthattheirjointdistributionPXNdoesnotchangeunderanypermutationofthesevari-ables.Letforeachn,1≤n≤N,PXn,NdenotethejointdistributionofsomenoftheseNvariables.Obviously,PXn,NisapermutationinvariantprobabilitymeasureinRn.Inparticular,themeasurePX1,Ndescribesthedistributionofasingleeigenvalue.Bydefinition,then-pointcorre-lationmeasureρXn,NofarandommatrixXisanon-normalizedmeasuredefinedbytherelation(1.3)ρXn,N=N!(N−n)!PXn,N.ForameasurablesetA⊂RnthequantityρXn,N(A)canbeinterpretedastheaveragenumberofn−tuplesofeigenvaluesinthesetA.IfthemeasureρXn,NisabsolutelycontinuouswithrespecttotheLebesguemeasureonRn,itsRadon-NikodymderivativeRXn,Niscalledthen-pointcorrelationfunctionoftherandommatrixX.Inparticular,themeasureρX1,NhastotalmassN.ForameasurablesetE⊂R1,thequantityρX1,N(E)expressestheexpectednumberoftheeigenvaluesbe-longingtoE.ThecorrespondingdensitywithrespecttotheLebesguemeasureinR1,ifitexists,iscalledtheeigenvaluedensityorthedensityofstates(caution:underthesamenamesthenormalizedversionsofthesamemeasuresareconsideredintheliteratureaswell).LetXNbearandommatrixwiththedistributionμsN.Forn=1,...,NwesetPGUE,sn,N=PXNn,NandρGUE,sn,N=ρXNn,N.AclassicalresultfortheGUEsaysthatwehave(1.4)PGUE,1/N1,N→N→∞W,wherethemeasuresconvergeintheweaksense,andWisthestandardWignermeasureon[−2,2]definedbythedensity(1.5)w(x)=(2π)−1p(4−x2)+,x∈R.Intermsofthecorrelationmeasuresthesamerelationreads(1.6)1NρGUE,1/N1,N→N→∞W.Forthen−pointcorrelationmeasureswehaveasimilarrelation(1.7)1NnρGUE,1/Nn,N→N→∞W×···×Wntimes,whichmeansthattheeigenvaluesbecomeindependentinthelimit.However,forn≥2,thestudyofafinerasymptoticsnearapointfrom2theprincipaldiagonalinthecube(−2,2)nshows[10,11]:foreveryu∈(−2,2)andt1,...,tn∈R1limN→∞1(Nw(u))nRGUE,1Nn,Nu+(t1/Nw(u)),...,u+(tn/Nw(u))
(1.8)=detsinπ(ti−tj)π(ti−tj)ni.j=1.Thislimitrelationpresentsapatternformanyotherresults,inpartic-ular,forthatofthepresentpaper.Therighthandsideofthisrelationrepresentsanexampleofthecorrelationfunctionofaso-calleddetermi-nantal(orfermionic)randompointprocess[15].Ingeneralthen−pointcorrelationfunctionRnofsuchaprocessisgivenbytheformulaR(u1,...,un)=detK(ui,uj)|ni,j=1,whereKisthekernelofanintegraloperatorontheline,whichistrace-classhavingbeenrestrictedtoanyfiniteintervalinR,andsubjecttosomefurtherconditions(see[15]foradetailedexposition).Moreover,intheasymptoticHermitianrandommatrixtheory(KN)N≥0arethereproducingkernelsofthesubspacesofpolynomialsofdegree≤N−1withrespecttosomeweightontheline.Inthiscasewecallthecorre-spondingmatrixensembledeterminantal.TheGUEgivesanexampleofsuchanensemble.Toalargeextenttheasymptoticstudyofdeter-minantalensemblesreducestothatoftherespectivekernels[3,5].Inparticular,thesamelocallimitasinthecaseofGUEisestablishedin[12,3]fortwobroadclassesofdeterminantalmatrixensembles.Itisexpectedthatthesin-kernellimit(firstdiscoveredbyF.Dyson)israthercommon.Thisisknownastheuniversalityconjecture.OutsidetheclassofdeterminantalHermitianrandommatricesonlyveryfewresultsontheasymptoticsofthecorrelationfunctionsareknown(see,forinstance,[10],whereamixtureofdeterminantalmeasuresiscon-sidered).Inthepresentpaperweinvestigatethefollowingnon-determinantalensembleofHermitianrandommatrices.Let(1.9)SrN={A∈HN:trA2=r2}bethesphereinHNoftheradiusr0centeredattheorigin.Setr=√sN.ThesphereS√sNNcarriesauniqueprobabilitymeasureνsNinvariantwithrespecttoallorthogonallineartransformationsinthespaceHN.Wecallthismeasurethefixed