Universality of the double scaling limit in random

arXiv:math-ph/0501074v131Jan2005UniversalityofthedoublescalinglimitinrandommatrixmodelsTomClaeysandArnoB.J.KuijlaarsFebruary7,2008AbstractWestudyunitaryrandommatrixensemblesinthecriticalcasewherethelimitingmeaneigenvaluedensityvanishesquadraticallyataninteriorpointofthesupport.Weestablishuniversalityofthelimitsoftheeigenvaluecorrelationkernelatsuchacriticalpointinadoublescalinglimit.ThelimitingkernelsareconstructedoutoffunctionsassociatedwiththesecondPainlev´eequation.ThisextendsaresultofBleherandItsforthespecialcaseofacriticalquarticpotential.ThetwomaintoolsweuseareequilibriummeasuresandRiemann-Hilbertproblems.Inourtreatmentofequilibriummeasuresweallowanegativedensitynearthecriticalpoint,whichenablesustotreatallcasessimultaneously.TheasymptoticanalysisoftheRiemann-HilbertproblemisdonewiththeDeift/Zhousteepestdescentanalysis.FortheconstructionofalocalparametrixatthecriticalpointweintroduceamodificationoftheapproachofBaik,Deift,andJohanssonsothatweareabletosatisfytherequiredjumppropertiesexactly.1IntroductionWeconsidertheunitaryrandommatrixmodelZ−1n,Nexp(−NTrV(M))dM(1.1)definedonHermitiann×nmatricesMinacriticalregimewherethelimitingmeandensityofeigenvaluesvanishesataninteriorpoint.Itisabasicfact1ofrandommatrixtheory[17,34]thattheeigenvaluesoftherandommatrixensemble(1.1)followadeterminantalpointprocesswithcorrelationkernelKn,N(x,y)=e−N2V(x)e−N2V(y)n−1Xk=0pk,N(x)pk,N(y),(1.2)wherepk,Ndenotesthekthdegreeorthonormalpolynomialwithrespecttotheweighte−NV(x)onR.WeassumeinthispaperthattheconfiningpotentialV:R→Rin(1.1)isrealanalyticandthatitsatisfiesthegrowthconditionV(x)log(x2+1)→+∞as|x|→+∞.(1.3)Theseassumptionsensurethatthemeaneigenvaluedensity1nKn,N(x,x)hasalimitasn,N→∞,n/N→1,seee.g.[17],whichwedenotebyψV(x).ItisknownthatψVisthedensityofthemeasureμVwhichminimizestheweightedenergyIV(μ)=ZZlog1|s−t|dμ(s)dμ(t)+ZV(t)dμ(t)(1.4)amongallprobabilitymeasureonR.ThemeasureμViscalledtheequilib-riummeasureintheexternalfieldV.ThefactthatVisrealanalyticensuresthatthesupportSV=supp(μV)consistsofafiniteunionofintervals[18].Itisaremarkablefactthatlocalscalinglimitsofthekernel(1.2)dependonlyonthenatureofthedensityψV.ThishasbeenprovedrigorouslyinthebulkofthespectrumforaquarticVin[9]andforgeneralrealanalyticVin[20].Indeed,ifψV(x∗)0,thenlimn→∞1nψV(x∗)Kn,nx∗+unψV(x∗),x∗+vnψV(x∗)=Kbulk(u,v)(1.5)exists,andKbulk(u,v)=sinπ(u−v)π(u−v).(1.6)Thescalinglimitsaredifferentatspecialpointsofthespectrum.AtedgepointsofthespectrumthedensityψVtypicallyvanisheslikeasquareroot,andthenitisknownthatforsomeconstantc0,limn→∞1(cn)2/3Kn,nx∗+u(cn)2/3,x∗+v(cn)2/3=Kedge(u,v)(1.7)2whereKedge(u,v)=Ai(u)Ai′(v)−Ai′(u)Ai(v)u−v(1.8)andAiistheAiryfunction.TheAirykernelisrelatedtotheTracy-Widomdistribution[39].In(1.7)wehaveassumedthatx∗isarightedgepoint.Foraleftedgepointwechangeu7→−u,v7→−vintheleft-handsideof(1.7).Otherspecialpointsinthespectruminclude•Edgepointswherethedensityvanishestoahigherorder.Thepossibleedgepointbehaviors(atarightendpointx∗)areψV(x)=c(x∗−x)2k+12(1+o(1))asx→x∗+(1.9)wherec0andkisanon-negativeinteger.•Interiorpointswherethedensityvanishes.ThenψV(x)=c(x−x∗)2k(1+o(1))asx→x∗(1.10)wherec0andkisapositiveinteger.Inthesecriticalcasesitisbelievedthatthelocalscalinglimitatx∗ofthekernelonlydependsontheorderofvanishingofthedensityatx∗[8].ThecasewhereψVvanishesquadraticallyataninteriorpointofSV,thatis,thecasek=1in(1.10),wasconsideredbyBleherandIts[10]forthecaseofacriticalquarticpotentialV(x)=g4x4+t2x2,withg0andt=tc=−2√g.ThenψV(x)=12πgx2s4√g−x2,forx∈[−2g−1/4,2g−1/4],sothatψVvanishesquadraticallyattheorigin.BleherandItsconsiderthedoublescalinglimitwheretchangeswithnandtendstotcasn→∞insuchawaythatn2/3(t−tc)remainsconstant.Forthequarticpotentialthisisequivalenttoconsidering(1.1)wheren,N→∞,n/N→1,suchthatlimn,N→∞n2/3nN−1(1.11)3exists.BleherandItsgaveaone-paremeterfamilyKcrit(u,v;s)oflimitingkernels,dependingons∈R,sothatforsomec0,limn,N→∞1(cn)1/3Kn,Nu(cn)1/3,v(cn)1/3=Kcrit(u,v;s)(1.12)wheresisproportionaltothevalueofthelimit(1.11).Thecriticalkernelsareexpressedintermsofso-calledψ-functionsasso-ciatedwiththeHastings-McLeodsolutionofthePainlev´eIIequation[28].Considerasin[10]thelineardifferentialequationsfora2-vector(or2×2matrix)Ψ=Ψ(ζ;s),ddζΨ=AΨ,∂∂sΨ=BΨ(1.13)whereA=A(ζ;s)=4ζq4ζ2+s+2q2+2r−4ζ2−s−2q2+2r−4ζq,(1.14)andB=B(ζ;s)=qζ−ζ−q.(1.15)Thecompatibilityconditionfor(1.13)isthatq=q(s)satisfiesthePainlev´eIIequationq′′=sq+2q3andthatr=r(s)=q′(s).Weassumethatq(s)istheHastings-McLeodsolutionofPainlev´eII,whichischaracterizedbytheasymptoticconditionq(s)=Ai(s)(1+o(1))ass→+∞.Thecriticalkernelsop[10]aregivenbyKcrit(u,v;s)=Φ1(u;s)Φ2(v;s)−Φ2(u;s)Φ1(v;s)π(u−v)(1.16)whereΦ1(ζ;s)Φ2(ζ;s)isthespecialsolutionto(1.13)whichisrealforrealζ,satisfiesΦ1(−ζ;s)=Φ1(ζ;s),Φ2(−ζ;s)=−Φ2(ζ;s)4andhasasymptoticsonthereallineΦ1(ζ;s)=cos43ζ3+sζ+O(ζ−1),Φ2(ζ;s)=−sin43ζ3+sζ+O(ζ−1)asζ→±∞.IfweputΦ1=Φ1+iΦ2,Φ2=Φ1−iΦ2(1.17)thenKcrit(u,v;s)=−Φ1(u;s)Φ2(v;s)+Φ2(u;s)Φ1(v;s)2πi(u−v)(1.18)andΦ1Φ2isaspecialsolutionofthedifferentialequationsddζΨ(ζ;s)=−4iζ2−i(s+2q2)4ζq+2ir4ζq−2ir4iζ2+i(s+2q)Ψ(ζ;s)(1.19)and∂∂sΨ