Universality of the Distribution Functions of Rand

arXiv:math-ph/9909001v131Aug1999UniversalityoftheDistributionFunctionsofRandomMatrixTheoryCraigA.TracyDepartmentofMathematicsInstituteofTheoreticalDynamicsUniversityofCaliforniaDavis,CA95616,USAHaroldWidomDepartmentofMathematicsUniversityofCaliforniaSantaCruz,CA95064,USAAugust31,19991RandomMatrixModelsInprobabilitytheoryandstatisticsacommonfirstapproximationtomanyrandomprocessesisasequenceX1,X2,X3,...ofindependentandidenticallydistributed(iid)randomvariables.LetFdenotetheircommondistribution.Tomotivatethematerialbelow,wetaketheserandomvariablesandconstructaparticularlysimpleN×Nrandommatrix,diag(X1(ω),X2(ω),···,XN(ω)).Theorderstatisticsaretheeigenvaluesorderedλ1≤λ2≤···≤λN,andthedistributionofthelargesteigenvalue,λmax(N)=λN,isProb(λmax(N)≤x)=Prob(X1≤x,...,XN≤x)=F(x)N.SincethedistributionFisarbitrary,weobservethatsotooisthedistributionofthelargesteigenvalueofaN×Nrandommatrix.However,oneisreallyinterestedinlimitinglawsasN→∞.Thatis,weaskifthereexistconstantsaNandbNsuchthatλmax(N)−aNbN(1.1)1convergesindistributiontoanontriviallimitingdistributionfunctionG.Inthepresentsitua-tionacompleteanswerisprovidedbyTheorem:If(1.1)convergesindistributiontosomenontrivialdistributionfunctionG,thenGbelongstooneofthefollowingforms:1.e−e−xwithsupportR.2.e−1/xαwithsupport[0,∞)andα0.3.e−(−x)αwithsupport(−∞,0]andα0.Thistheoremisamodelforthetypeofresultswewantfornondiagonalrandommatrices.Arandommatrixmodelisaprobabilityspace(Ω,P,F)whereΩisasetofmatrices.Herearesomeexamples•CircularUnitaryEnsemble(CUE,β=2)–Ω=U(N)=N×Nunitarymatrices.–P=Haarmeasure.•GaussianOrthogonalEnsemble(GOE,β=1)–Ω=N×Nrealsymmetricmatrices.–P=unique1measurethatisinvariantunderorthogonaltransformationsandthematrixelements(sayonandabovethediagonal)areiidrandomvariables.•GaussianUnitaryEnsemble(GUE,β=2)–Ω=N×Nhermitianmatrices.–P=uniquemeasurethatisinvariantunderunitarytransformationsandtherealandimaginarymatrixelements(sayonandabovethediagonal)areiidrandomvariables.•GaussianSymplecticEnsemble(GSE,β=4)–Ω=2N×2NHermitianself-dualmatrices.2–P=uniquemeasurethatisinvariantundersymplectictransformationsandtherealandimaginarymatrixelements(sayonandabovethediagonal)areiidrandomvariables.1Uniquenessisuptocenteringandanormalizationofthevariance.2Identifythe2N×2NmatrixwiththeN×Nmatrixwhoseentriesarequaternions.IfthequaternionmatrixelementssatisfyMji=Mijwherethebarisquaternionconjugation,thenthe2N×2NmatrixiscalledHermitianself-dual.EacheigenvalueofaHermitianself-dualmatrixhasmultiplicitytwo.2Expectedvaluesofrandomvariablesf:Ω→CarecomputedfromtheusualformulaEΩ(f)=ZΩf(M)dP(M).Iff(M)dependsonlyontheeigenvaluesofM∈Ω,thenonecanbemoreexplicit:•CUE(Weyl’sFormula)EU(N)(f)=1N!(2π)NZπ−π···Zπ−πf(θ1,...,θN)Y1≤μν≤NΔ(eiθ1,...,eiθN)2dθ1···dθN,•GaussianEnsembles(β=1,2,4):ENβ(f)=cNβZ∞−∞···Z∞−∞f(x1,...,xN)|Δ(x1,...,xN)|βe−β2Px2jdx1···dxN,wherecNβischosensothatENβ(1)=1andΔ(x1,...,xN)=Q1≤ij≤N(xi−xj).Thefactore−β2Px2jexplainsthechoiceoftheword“gaussian”inthenamesoftheseensembles.AcommonlystudiedgeneralizationofthesegaussianmeasuresistoreplacethesumofquadratictermsappearingintheexponentialwithPV(xi)whereVis,say,apolynomial(withtheobviousrestrictionstomakethemeasurewell-defined).Choosingf=Qi1−χJ(xi),χJthecharacteristicfunctionofasetJ⊂R,wegettheimportantquantity3ENβ(f)=ENβ(0;J):=probabilitynoeigenvalueslieinJ,andintheparticularcaseJ=(t,∞)FNβ(t):=Prob(λmax≤t)=ENβ(0,J).Thelevelspacingdistribution4isexpressibleintermsofthemixedsecondpartialderivativeofENβ(0;(a,b))withrespecttotheendpointsaandb.2FredholmDeterminantRepresentationsThoughENβ(0;J)areexplicitN-dimensionalintegrals,theseexpressionsarenotsousefulinestablishinglimitinglawsasN→∞.WhatturnedouttobeveryusefulareFredholm3Thisquantityhasanobviousextensiontootherrandommatrixmodels.4Lettheeigenvaluesbeordered.Theconditionalprobabilitythatgivenaneigenvalueata,thenextoneliesbetweensands+dsiscalledthelevelspacingdensity.3determinantrepresentationsforENβ(0;J).In1961M.Gaudinprovedforβ=2(usingthenewlydevelopedorthogonalpolynomialmethodofM.L.Mehta)thatEN2(0;J)=det(I−KN2)whereKN2isanintegraloperatoractingonJwhosekernelisoftheformϕ(x)ψ(y)−ψ(x)ϕ(y)x−y,(2.1)withϕ(x)=cNe−x2/2HN(x),ψ(x)=cNe−x2/2HN−1(x),andHj(x)aretheHermitepolyno-mials.5Forβ=1or4,generalizingF.J.Dyson’s1970analysisofthen-pointcorrelationsforthecircularensembles,itfollowsfromworkbyMehtathefollowingyearthatthesquareofENβ(0;J)againequalsaFredholmdeterminant,det(I−KNβ),butnowthekernelofKNβisa2×2matrix.63ScalingLimits(LimitingLaws)3.1BulkScalingLimitLetρN(x)denotethedensityofeigenvaluesatxandpickapointx0,independentofNwithρN(x0)0.Wescaledistancessothatresultingdensityisoneatx0,ξ:=ρN(x0)(x−x0),andwecallthelimitN→∞,x→x0,suchthatξisfixed,thebulkscalinglimit.Forβ=2,EN2(0;J)→E2(0;J)=det(I−K2)wheretheintegraloperatorK2(actingonL2(J))hasasitskernel(thesinekernel)1πsinπ(ξ−ξ′)ξ−ξ′.(WeusethesamesymbolJtodenotethescaledsetJ.)Furthermore,p2(s)=−d2ds2E2(0;(0,s))isthe(limiting)level-spacingdensityforGUE;knownastheGaudindistribution.7Weobservethatthelimitingkernelistranslationallyinvariantandindependentofx0.5Forth