The Level Densities of Random Matrix Unitary Ensem

arXiv:math-ph/0503022v110Mar2005THELEVELDENSITIESOFRANDOMMATRIXUNITARYENSEMBLESANDTHEIRPERTURBATIONINVARIABILITYWANGZHENGDONG1ANDYANKUIHUA1,2Abstract.Usingoperatormethods,wegenerallypresentthelevelden-sitiesforkindsofrandommatrixunitaryensemblesinweaksense.Asacorollary,thelimitspectraldistributionsofrandommatricesfromGaussian,LaguerreandJacobiunitaryensemblesarerecovered.Atthesametime,westudytheperturbationinvariabilityoftheleveldensitiesofrandommatrixunitaryensembles.Aftertheweightfunctionasso-ciatedwiththe1-levelcorrelationfunctionisappendedapolynomialmultiplicativefactor,theleveldensityisinvariantintheweaksense.1.IntroductionInclassicalquantummechanics,thestatisticalpropertiesofenergylev-elscanbedescribedbythek-levelcorrelationfunctionsdefinedas(seeMehta[13])Rknβ(x1,x2,···,xk)=n!(n−k)!Z···ZPnβ(x1,x2,···,xn)dxk+1···dxn,wherePnβ(x1,x2,···,xn)=cnβ·exp(−βH)isthejointprobabilitydensityfunctionofneigenvaluesx1,x2,···,xnofan×nrandommatrix,cnβisthenormalizedconstant.HistheHamiltonianofthelogarithmicalinteractingnparticlessystemonastraightline,whichisgivenbyconstrainingone-bodypotentialandlogarithmicrepulsivetwo-bodypotential,i.e.H=nXi=1V(xi)−X1≤ij≤nlog|xi−xj|,xi∈R.Ingeneral,V(x)iscalledpotentialfunctionandβDyson’sindex.β=1,2and4arecorrespondingtotheorthogonal,unitaryandsymplecticensemblesrespectively.Whenk=1,R1nβ(x)canbeexplainedasthedistributiondensityofenergylevelswhichcanbefoundnearbyx.Theleveldensitydenotedbyσβ(x),whichisaglobalquantity,isdefinedbythelimitofthe1-levelcorrelationfunctionR1nβ(x).Thenhowtodeterminetheleveldensity?ItcanbetracedbacktoWigner’spioneeringwork[20,21].Theresultsofearlyworkarereviewedin[13,16].Recently,therearemany1authorstoconcentrateonthisproblem(SeeSpohn[17],BaiandYin[2],NagaoandWadati[14],HaagerupandThorbjørnsen[9],Girko[8],KiesslingandSpohn[10],Due˜nez[4],Ledoux[11],etc.).Noticethatgivendifferentorspecialone-bodypotentialV(x),itwillexhibitedkindsofimagesforus.Asamatteroffact,inthecaseofclassicalGaussianensembles,V(x)=x22,x∈R.Theleveldensityisthefamous“semicirclelaw”firstderivedbyWigner[20,21],i.e.σβ(x)=1π√2n−x2x2≤2n0x2≥2n.Inaddition,inthecaseofLaguerreensembles,theleveldensitycanbeevaluatedbyaphysicalargument(SeeBronk[3]),i.e.σβ(x)=1π√x√2n−x0x≤2n0x≥2n.InthecaseofJacobiensembles,theleveldensitycanalsobeevaluatedbyaphysicalargument(SeeLeff[12]),i.e.σβ(x)=nπ√1−x2−1x10otherwise.Inthecaseofunitaryensembles,R1n2(x)canbeexpressedtoaconciseformula(see[13],[14])whichiscloselycorrelatedtoclassicalorthogonalpolynomials,i.e.(1)R1n2(x)=n−1Xm=0p2m(x)̟(x),where̟(x)=¯cexp(−2V(x)),¯cisthenormalizedconstantandpm(x)bethem-ordernormalizedorthogonalpolynomialsassociatedwiththenormalizedweightfunction̟(x),i.e.Zpm(x)pn(x)·̟(x)dx=δmn.In[9],HaagerupandThorbjørnsenonlystudiedtheGaussianunitaryensemble(GUE)whichwasdenotedbySGRM(n,σ2)there.Usingthefol-lowingpropertyofHermitepolynomialsHk(x+a)=kXj=0Cjk(2a)k−jHj(x),a∈R,2theauthorsdirectlyobtainedanequalityforthecomplexLaplacetransformof1nR1n2(x),i.e.ZRexp(sx)1nR1n2(x)dx=ZRexp(sx)1n√πn−1Xk=0ˆH2k(x)e−x2dx=exp(s24)Φ(1−n,2;−s22),s∈C,whereˆHk(x)isthek-ordernormalizedHermitepolynomial,Φ(a,c;x)istheconfluenthyper-geometricfunction(CHGF)withparametersaandc.ThentheygaveashortproofofWigner’ssemicirclelaw.Asweknow,theCHGFsarecomplicatedseriesexpansions.In[11],LedouxpushedforwardtheinvestigationbyHaagerupandThorbjørnsenandonlyconcentratedonthedifferentialaspectsofCHGFs.Theauthorcon-structedanabstractframeworkofMarkovdiffusiongenerators,andinwhichderivedthebasicdifferentialequationsonLaplacetransformsofp2m(x)̟(x).UsingtherecurrenceformulaofHermitepolynomialsandtheuniformintegrabilityofrandomvariablesequence,bytheobtaineddifferentialequa-tion,theauthorshowedthatlimm→∞Zfx2√m1mm−1Xk=0p2k(x)̟(x)dx=Ef(√XY),forallf∈Cb(R)whichdeterminestheleveldensityofGUEintheweaksense,whereXandYaretwoindependentrandomvariableswiththeuniformdistributionon[0,1]andthearcsinedistributionon(−1,+1)respectively.Bytheanalogoustechnique,theauthoralsoobtainedtheleveldensitiesofLaguerreandJacobiunitaryensemblesrespectively.Inthispaper,wewillgenerallydealwiththeproblemforkindsofunitaryensembles(i.e.β=2)byoperatormethod.Itisnoconfusiontoomitthesubscript2inthebelowtext.Moreover,aswewillsee,thismethodcaneffectivelybeusedtostudytheperturbationinvariabilityofleveldensity.Itiswellknownthatthenormalizedorthogonalpolynomialspn(x)satisfythefollowingrecursionformula(seesection2fordetails)xpn(x)=αnpn+1(x)+βnpn(x)+γnpn−1(x),n=1,2,3,···.Inthepaper,weassumethatαnandβnsatisfythefollowingexponentialgrowthconditions:(2)αn=ξnt(1+ξn),βn=ζnt(1+ζn)+ηn,whereξ6=0,ζ≥0,0≤t≤1areconstantsandlimn→∞ξn=limn→∞ζn=limn→∞ηn=0.3NotethattheclassicalHermite,LaguerreandJacobipolynomialsallsat-isfyexponentialgrowthconditions(seesection2).Wedefinethe“ascending”,“equilibrating”and“descending”operatorsasfollow,A+pn(x)=αnpn+1(x),A0pn(x)=βnpn(x),A−pn(x)=γnpn−1(x).Insection3,weusetheseoperatorstoobtainthemomentinequalityoftheprobabilitydensityσn(x)=√DnnR1n(xpDn)andtoshowthatthelimitofk-thmomentM(k)nofσn(x)exists(seetheorem3.2),whereDnisthe2ndmomen