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arXiv:math-ph/0008032v123Aug2000GapProbabilitiesforEdgeIntervalsinFiniteGaussianandJacobiUnitaryMatrixEnsemblesN.S.Witte,DepartmentofMathematicsandStatistics&SchoolofPhysics,UniversityofMelbourneParkville,Victoria3052,AUSTRALIAP.J.Forrester,DepartmentofMathematicsandStatisticsUniversityofMelbourneParkville,Victoria3052,AUSTRALIAandChristopherM.CosgroveSchoolofMathematicsandStatisticsUniversityofSydneySydney,NSW2006,AUSTRALIA.February7,2008AbstractTheprobabilitiesforgapsintheeigenvaluespectrumofthefinitedimensionN×NrandommatrixHermiteandJacobiunitaryensemblesonsomesingleanddisconnecteddoubleintervalsarefound.Thesearecaseswhereareflectionsymmetryexistsandtheprobabilityfactorsintotwootherrelatedprobabilities,definedonsingleintervals.OurinvestigationusesthesystemofpartialdifferentialequationsarisingfromtheFredholmdeterminantexpressionforthegapprobabilityandthedifferential-recurrenceequationssatisfiedbyHermiteandJacobiorthogonalpolynomials.InourstudywefindsecondandthirdordernonlinearordinarydifferentialequationsdefiningtheprobabilitiesinthegeneralNcase.ForN=1andN=2theprobabilitiesandthusthesolutionoftheequationsaregivenexplicitly.AnasymptoticexpansionforlargegapsizeisobtainedfromtheequationintheHermitecase,andalsostudiedisthescalingattheedgeoftheHermitespectrumasN→∞,andtheJacobitoHermitelimit;theselasttwostudiesmakecorrespondencetoothercasesreportedhereorknownpreviously.Moreover,thedifferentialequationarisingintheHermiteensembleissolvedintermsofanexplicitra-tionalfunctionofaPainlev´e-Vtranscendentanditsderivative,andananalogoussolutionisprovidedinthetwoJacobicasesbutthistimeinvolvingaPainlev´e-VItranscendent.AMSclassification-primary:15A52,secondary:34A34,34A05,33C4511IntroductionItisacelebrateddiscoveryofJimbo,Miwa,MˆoriandSato[14]thattheprobabilityofaneigenvaluefreeregioninthebulkoftheinfiniteGUE(GaussianunitaryensembleofrandomHermitianmatrices)canbeexpressedexactlyintermsofaPainlev´e-Vtranscendent.Thishasinitiatedanumberofstudies[18,1,11]which,inadditiontoclarifyingthegeneralsettingoftheexactresult,giveformalismsthatallowanalogousresultstobeobtainedinothercases.Forexample,theprobabilityofaneigenvaluefreeregionattheedgeoftheinfiniteGUE(appropriatelyscaled)hasbeenexpressedintermsofthePainlev´e-IItranscendent[17],whileaPainlev´e-IIItranscendenthasbeenshowntodeterminethedistributionofthesmallesteigenvalueintheinfiniteLUE(Laguerreunitaryensembleofnon-negativematricesoftheformA†AwithAcomplex[18]).Moreover,forthefiniteclassicalensembleswithunitarysymmetry,theprobabilityofasingleeigenvaluefreeregionwhichincludesanendpointofthesupportoftheweighthasbeenexpressedintermsofsolutionsofcertainnon-linearequations[18].Itistheobjectiveofthisworktoprovidethreenewevaluationsofgapprobabilitiesforparticularfiniteclassicalrandommatrixensembleswithunitarysymmetry.TheensemblesconsideredarethefiniteGUEandthesymmetricJacobiunitaryensemble(JUE).Werecalltheeigenvaluep.d.f.foranensemblewithunitarysymmetryisoftheformNYl=1w2(λl)Y1≤jk≤N|λk−λj|2,(1.1)wheretheweightfunctionw2(λ)determinesthespecificunitaryensemble:w2(λ)=e−λ2,GUE(1−λ)α(1+λ)β,JUE(1.2)(thesymmetricJacobiensemblereferstothecaseα=βoftheJUE).TheseensemblescanberealisedintermsofmatriceswithindependentGaussianelements.FortheGUE,eachmatrixXsaymustbeHermitianandhaveitsdiagonalelementsxjj(whichmustbereal)anduppertriangularelementsxjk=ujk+ivjkchosenwithp.d.f.N(0,1/√2)andN(0,1/2)+iN(0,1/2).FortheJUEonefirstconstructs[15]auxiliaryrectangularM1×NandM2×N(M1,M2≥N)matricesaandbrespectively,withcomplexelementsindependentlydistributedaccordingtoN(0,1/√2)+iN(0,1/√2).ThenwithA=a†a,B=b†b,itcanbeshownthatthedistributionoftheeigenvaluesofA(A+B)−1isanexampleof(1.1)withw2(λ)=λM1−N(1−λ)M2−N,0λ1.(1.3)Thechangeofvariablesλ7→(1−λ)/2showsthatthisisanexampleoftheJUE.WithE2(0;I;w2(λ);N)denotingtheprobabilitythattherearenoeigenvaluesintheintervalIofanensemblewitheigenvaluep.d.f.(1.1),theprobabilitiestobecalculatedareE2(0;(−∞,−s)∪(s,∞);e−λ2;N)(1.4)andE2(0;(−1,−s)∪(s,1);(1−λ2)α;N),E2(0;(−s,s);(1−λ2)α;N).(1.5)Thequantity(1.4)givestheprobabilitythattherearenoeigenvaluesintheGUEwithmodulusgreaterthans,asdoesthefirstquantityin(1.5)fortheJUE.Thefinalquantityin2(1.5)givestheprobabilitythattherearenoeigenvaluesintheJUEwithmoduluslessthans(theanalogousprobabilityfortheGUEhaspreviouslybeenevaluated[18]).Theseprobabilitiesarespecialbecause,likethesituationinwhichIisasingleintervalwhichincludesanendpointofthesupportoftheweightfunctionnotedabove,wewillshowthattheycanbeevaluatedexactlyintermsofthesolutionofcertainnon-linearequations.Anothermotivatingfactorforourstudyisarecentlydiscovered[9]generalidentitysatisfiedbytheprobabilityE2(0;I;w2(λ);N)applicablewheneverw2(λ)isevenandIissymmetricalabouttheorigin.TheidentitystatesE2(0;I;w2(λ);N)=E2(0;I+;y−1/2w2(y1/2);[(N+1)/2])×E2(0;I+;y+1/2w2(y1/2);[N/2])(1.6)whereI+denotestheportionofIonR+inthevariabley=λ2andw2(y1/2):=0fory0.Theprobabilities(1.4)and(1.5)arealloftheformrequiredbytheLHSofthisidentity,andaretherebyrelatedtoE2(0;(s2,∞);y∓1/2e−y;m)(1.7)andE2(0;(s2,1);y∓1/2(1−y)α;m),E2(0;(0,s2);y∓1/2(1−y
本文标题:Gap Probabilities for Edge Intervals in Finite Gau
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