Asymptotic analysis of random matrices with extern

arXiv:math-ph/0610050v120Oct2006ASYMPTOTICANALYSISOFRANDOMMATRICESWITHEXTERNALSOURCEANDAFAMILYOFALGEBRAICCURVESK.T.-R.MCLAUGHLINAbstract.Wepresentasetofconditionswhich,ifsatisfied,provideforacompleteasymptoticanalysisofrandommatriceswithsourcetermcontainingtwodistincteigenvalues.Theseconditionsareshowntobeequivalenttotheexistenceofaparticularalgebraiccurve.Forthecaseofaquarticexternalfield,thecurveinquestionisproventoexist,yieldingpreciseasymptoticinformationaboutthelimitingmeandensityofeigenvalues,aswellasbulkandedgeuniversality.1.Introduction1.1.RandomMatriceswithSource.Considertheprobabilitymeasureμn(dM)=1Zne−nTr(V(M)−AM)dM,(1)definedonn×nHermitianmatrices,M.HeredMdenotesLebesguemeasureonthematrixentries,dM=Qnj=1dMjjQ1≤jk≤ndRe(Mjk)dIm(Mjk),thematrixAisafixedn×nmatrix,andtheparameterZnisanormalizationconstantchosensothat(1)isaprobabilitymeasure.Thisdefinesanensembleofrandommatrices,inwhichthematrixAplaystheroleofanexternalsource.ThefunctionVshouldberealandgrowsufficientlyrapidlythattheabovemeasurepossessesallfinitemoments;itistypicallyassumedtobeapolynomial.Thefamilyofmeasuresdescribedby(1)arenotinvariantunderunitarytransformations,andtheanal-ysisofeigenvaluestatisticsunderthesemeasuresisinitsinfancyincomparisontotheso-called“unitaryensembles”forwhichagreatdealisknown.FollowingtheworkofBr´ezin-Hikami[5],[6],andP.Zinn-Justin[13][14],ateamofresearchers(BleherandKuijlaars[2],[3],[4]andAptekarev,BleherandKuijlaars[1])haveconsideredthelargenbehaviorofeigenvaluestatisticsunder(1)fromthepointofviewofRiemann–Hilbertproblems.In[13],Zinn-Justinshowedthattheeigenvaluesofsuchrandommatricesareinfactadeterminantalpointprocess,andin[2],theauthorsshowedthatthisrepresentationmaybedescribedintermsofthesolutionofamatrixRiemann–Hilbertproblem,thesizeofwhichdependsonthenumberofdistincteigenvaluesofthematrixA.InthepresentpaperweshallassumethatthematrixApossessestwoeigenvalues,−aanda,withmultiplicitiesn1andn2,respectively(withn1+n2=n).Inthiscase,theRiemann–Hilbertproblemisasfollows:Riemann-HilbertProblem1.1.(a)AisanalyticonC\R.(b)TheboundaryvaluesofAsatisfy(2)A+(x)=A−(x)1w1w2010001,forz∈Rwherew1=e−nV1=e−n(V(x)+ax),w2=e−nV2=e−n(V(x)−ax).(3)Date:February7,2008.1(c)Asz→∞,wehave(4)A(z)=I+O1zzn000z−n1000z−n2.IntermsofthisRiemann–Hilbertproblem,theprobabilitymeasureoneigenvaluesinducedby(1)maybere-writtenasfollows:dμn(x1,...,xn)=det(Kn(xi,xj))1≤i,j≤ndnx,(5)[13],[2]wherenowKn(x,y)=e−n(V(x)+V(y))2πi(x−y)enayY(y)−1Y(x)21+e−nayY(y)−1Y(x)31oTheformula(5)isnotjustaconciserepresentationoftheprobabilitymeasure,itturnsoutthatallstatisticalpropertiesofeigenvaluescanberelatedtoKn.Forexample,Prob{noeigenvaluesin(a,b)}=det(1−Kn)L2[(a,b)]whereKnistheintegraloperatorwithkernelKn(x,y):Knf=ZbaKn(x,y)f(y)dy.TherepresentationofthecorrelationfunctionsintermsofthekernelKnisduetoZinn-Justin[13],andtherepresentationofthiskernelintermsofaRiemann–HilbertproblemisduetoBleherandKuijlaars[2].Remark:InthecasethatthematrixApossessespdistincteigenvalueswithmultiplicitiesn1,n2,...,np,theaboverepresentationsgeneralizedirectly,buttheassociatedRiemann–Hilbertproblemisp+1×p+1.In[5][6],theauthorsconsideredthecasethatVisaquadratic.Intheseriesofpapers[3],[1],and[4],theauthorsalsoconsideredtheGaussiancase,butfromthepointofviewofRiemann–Hilbertproblems.Thegoaloftheseworkswastostudythebehavioroftheeigenvaluestatisticsinthelargenlimit.In[14],Zinn-Justinalsostudiedthelargenlimit,forthecaseofgeneralV.Inthatwork,oneimportantissuewasthedescriptionofthelimitingdensityofstatesintermsofafunctionforwhichanexistencetheoremislacking.Heexplainsverycarefullytheanalyticitypropertiesandbranch-cutstructurerequiredofthisfunction,intheso-called“one-cut”case,undertheassumptionthatthedensityoftheeigenvaluesofAhasasmoothlimitwhenntendsto∞.HereweconsiderthecasethatApossessestwoeigenvalues,ofmultiplicitiesn1andn2,andourinterestisinthebehaviorwhenn,n1,n2→∞sothatnjn→xj.Themaingoalofourworkistoprovideanewandexplicitcharacterizationofthelimitingdensityofstatesintermsofanalgebraiccurve.Thefollowingisasummaryoftheresultsinthispaper:(1)InSection2,wepresentalistofconditions(referredtoasan“idealsituation”)which,iftrue,yieldatransformationoftheRiemann–Hilbertproblem(1.1)toa“normalform”,thatis,aRiemann–Hilbertproblemfromwhichsubsequentasymptoticanalysis(forn→∞)isstraightforward.(2)InSection3,weshowthatiftheseconditionsaresatisfied,thenthereexistsanalgebraiccurvewhoserootsyieldthedesiredtransformation.Thecurveisalwaysoftheformw3−V′(z)w2+C1(z)w−C0(z)=0,(6)whereC1andC2areanalyticfunctionsofz.(3)Forarbitrarypolynomialexternalfields,thealgebraiccurvemaybedetermineduptoafinitenumberoffreeparameters(seeSection4).(4)InSection6,weconsidertheGaussianandQuarticcases.FortheGaussiancase,thisalgebraiccurvehasnofreeparameters,andinthecasen1=n2=n/2,isequivalenttothealgebraiccurveusedbyBleherandKuijlaars[3,4]andbyAptekarev,BleherandKuijlaars[1].2(5)Forthequarticcase,V=x4/4,withn1=n2=n/2,weprovethatforasufficientlylargethereexistsachoiceofthefreeparameterssothatthealgebraiccurveyieldsthedesiredtransformation.Eachofthetwo