Multi-critical unitary random matrix ensembles and

arXiv:math-ph/0508062v131Aug2005Multi-criticalunitaryrandommatrixensemblesandthegeneralPainlev´eIIequationT.Claeys,A.B.J.Kuijlaars,andM.VanlessenAbstractWestudyunitaryrandommatrixensemblesoftheformZ−1n,N|detM|2αe−NTrV(M)dM,whereα−1/2andVissuchthatthelimitingmeaneigenvaluedensityforn,N→∞andn/N→1vanishesquadraticallyattheorigin.Inordertocomputethedoublescalinglim-itsoftheeigenvaluecorrelationkernelneartheorigin,weusetheDeift/ZhousteepestdescentmethodappliedtotheRiemann-Hilbertproblemfororthogonalpolynomialsonthereallinewithrespecttotheweight|x|2αe−NV(x).Herethemainfocusisontheconstructionofalocalparametrixneartheoriginwithψ-functionsassociatedwithaspecialsolutionqαofthePainlev´eIIequationq′′=sq+2q3−α.Weshowthatqαhasnorealpolesforα−1/2,byprovingthesolvabilityofthecorrespondingRiemann-Hilbertproblem.Wealsoshowthattheasymptoticsoftherecurrencecoefficientsoftheorthogonalpolynomialscanbeexpressedintermsofqαinthedoublescalinglimit.1Introductionandstatementofresults1.1UnitaryrandommatrixensemblesForn∈N,N0,andα−1/2,weconsidertheunitaryrandommatrixensembleZ−1n,N|detM|2αe−NTrV(M)dM,(1.1)onthespaceofn×nHermitianmatricesM,whereV:R→Risarealanalyticfunctionsatisfyinglimx→±∞V(x)log(x2+1)=+∞.(1.2)Becauseof(1.2)andα−1/2,theintegralZn,N=Z|detM|2αe−NTrV(M)dM(1.3)convergesandthematrixensemble(1.1)iswell-defined.Itiswellknown,seeforexample[8,30],thattheeigenvaluesofMaredistributedaccordingtoadeterminantalpointprocesswithacorrelationkernelgivenbyKn,N(x,y)=|x|αe−N2V(x)|y|αe−N2V(y)n−1Xk=0pk,N(x)pk,N(y),(1.4)1wherepk,N=κk,Nxk+···,κk,N0,denotesthek-thdegreeorthonormalpolynomialwithrespecttotheweight|x|2αe−NV(x)onR.Scalinglimitsofthekernel(1.4)asn,N→∞,n/N→1,showaremarkableuniversalbehaviorwhichisdeterminedtoalargeextentbythelimitingmeandensityofeigenvaluesψV(x)=limn→∞1nKn,n(x,x).(1.5)Indeed,forthecaseα=0,BleherandIts[4](forquarticV)andDeiftetal.[13](forgeneralrealanalyticV)showedthatthesinekernelisuniversalinthebulkofthespectrum,i.e.,limn→∞1nψV(x0)Kn,nx0+unψV(x0),x0+vnψV(x0)=sinπ(u−v)π(u−v)wheneverψV(x0)0.Inaddition,theAirykernelappearsgenericallyatendpointsofthespectrum.Ifx0isarightendpointandψV(x)∼(x0−x)1/2asx→x0−,thenthereexistsaconstantc0suchthatlimn→∞1cn2/3Kn,nx0+ucn2/3,x0+vcn2/3=Ai(u)Ai′(v)−Ai′(u)Ai(v)u−v,whereAidenotestheAiryfunction,seealso[10].Theextrafactor|detM|2αin(1.1)introducessingularbehaviorat0ifα6=0.Thepointwiselimit(1.5)doesnotholdifψV(0)0,sinceKn,n(0,0)=0ifα0andKn,n(0,0)=+∞ifα0,duetothefactor|x|α|y|αin(1.4).However(1.5)continuestoholdforx6=0andalsointhesenseofweak∗convergenceofprobabilitymeasures1nKn,n(x,x)dx∗→ψV(x)dx,asn→∞.ThereforewecanstillcallψVthelimitingmeandensityofeigenvalues.ObservethatψVdoesnotdependonα.However,atamicroscopicleveltheintroductionofthefactor|detM|2αchangestheeigenvaluecorrelationsneartheorigin.Indeed,forthecaseofanon-criticalVforwhichψV(0)0,itwasshownin[29]thatlimn→∞1nψV(0)Kn,nunψV(0),vnψV(0)=π√u√vJα+12(πu)Jα−12(πv)−Jα−12(πu)Jα+12(πv)2(u−v),(1.6)whereJνdenotestheusualBesselfunctionoforderν.Wenoticethatuniversalityresultsfororthogonalandsymplecticensemblesofrandommatriceshavebeenobtainedonlyveryrecently,see[9,10,11].1.2Multi-criticalcaseItisthegoalofthispapertostudy(1.1)inacriticalcasewhereψVvanishesquadraticallyat0,i.e.,ψV(0)=ψ′V(0)=0,andψ′′V(0)0.(1.7)2Thebehavior(1.7)isamongthepossiblesingularbehaviorsthatwereclassifiedin[12].TheclassificationdependsonthecharacterizationofthemeasureψV(x)dxastheuniqueminimizerofthelogarithmicenergyIV(μ)=ZZlog1|x−y|dμ(x)dμ(y)+ZV(x)dμ(x)(1.8)amongallprobabilitymeasuresμonR.ThecorrespondingEuler-Lagrangevariationalconditionsgivethatforsomeconstantℓ∈R,2Zlog|x−y|ψV(y)dy−V(x)+ℓ=0,forx∈supp(ψV),(1.9)2Zlog|x−y|ψV(y)dy−V(x)+ℓ≤0,forx∈R.(1.10)InadditiononehasthatψVissupportedonafiniteunionofdisjointintervals,andψV(x)=1πqQ−V(x),(1.11)whereQVisarealanalyticfunction,andQ−Vdenotesitsnegativepart.NotethattheendpointsofthesupportcorrespondtozerosofQVwithoddmultiplicity.Thepossiblesingularbehaviorsareasfollows,see[12,26].SingularcaseIψVvanishesatanendpointtohigherorderthanasquareroot.ThiscorrespondstoazeroofthefunctionQVin(1.11)ofoddmultiplicity4k+1withk≥1.(Themultipicity4k+3cannotoccurinthesematrixmodels.)SingularcaseIIψVvanishesataninteriorpointofsupp(ψV),whichcorrespondstoazeroofQVintheinteriorofthesupport.Themultiplicityofsuchazeroisnecessarilyamultipleof4.SingularcaseIIIEqualityholdsinthevariationalinequality(1.10)forsomex∈R\supp(ψV).Ineachoftheabovecases,Viscalledsingular,otherwiseregular.Theaboveconditionscorrespondtoasingularendpoint,asingularinteriorpoint,andasingularexteriorpoint,respectively.Ineachofthesingularcasesoneexpectsafamilyofpossiblelimitingkernelsinadoublescalinglimitasn,N→∞andn/N→1atsomecriticalrate[3].Assaidbeforeweconsiderthecase(1.7)whichcorrespondstothesingularcaseIIwithk=1atthesingularpointx=0.Fortechnicalreasonsweassumethattherearenoothersingularpointsbesides0.Settingt=n/N,andlettingn,N→∞suchthatt→1,wehavethattheparametertdescribesthetransitionfromthecasewhereψV(0)0(fort1)throughthemulti-criticalcase(t=1)tothecasewhere0liesinagapbetwe