Diffusion at the random matrix hard edge

arXiv:0803.2043v3[math.PR]30Jul2008DiffusionattherandommatrixhardedgeJos´eA.Ram´ırezBrianRiderMarch13,2008AbstractWeshowthatthelimitingminimaleigenvaluedistributionsforanaturalgeneral-izationofGaussiansample-covariancestructures(the“betaensembles”)aredescribedbythespectrumofarandomdiffusiongenerator.ByaRiccatitransformation,weob-tainaseconddiffusiondescriptionofthelimitingeigenvaluesintermsofhittinglaws.Thispicturepertainstotheso-calledhardedgeofrandommatrixtheoryandsitsincomplementtotherecentwork[15]oftheauthorsandB.Vir´agonthegeneralbetarandommatrixsoftedge.Infact,thediffusiondescriptionsfoundonbothsidesareusedheretoprovethereexistsatransitionbetweenthesoftandhardedgelawsatallvaluesofbeta.1IntroductionTheoriginsofrandommatrixtheorycanbetracedtotheintroductionofWishart’sen-sembles,matricesoftheformXX†withrectangularXcomprisedentirelyofindependentrealorcomplexGaussiansofmeanzeroandmean-squareone.Thespectrumoftheseob-jectsareoffundamentalimportanceinmathematicalstatistics(seethecomprehensivetext[14]),andcontinuetogeneratewideinterestduetotheirrelevancetosuchdisparateareasasinformationtheory[20],numericalanalysis[7],and,alongwiththeirquaternion-entriedcounterparts,theoreticalphysics[27].HereweconsiderscalinglimitsforWishart-typeeigenvaluesatthehardedge.Toexplain,letXben×m.Ifm≃nasn↑∞theminimaleigenvaluesofthe(non-negative)XX†willfeelthe“hard”constraintattheorigin,whileifm/nisstrictlylargerthanoneinthelargedimensionallimit,theminimaleigenvaluesseparateformzeroandonehas“soft”edgefluctuations(onwhichmorebelow).Infact,ifm=n+awithfixedaasn↑∞onediscoversaninterestingfamilyoflimitlawsindexedbyaforthebottomofthespectrum.TheknownresultsatthehardedgehavethusfarbeenbasedontheexplicitjointdensityfortheWisharteigenvalues0≤λ0,λ1,···,λn−1.Inparticular,whenXisn×(n+a)with1integera−1,thatdensityisPβ,a(λ1,...,λn)=1Zβ,aYjk|λj−λk|β×n−1Yk=0λβ2(a+1)−1keβ2λk,(1.1)withnormalizerZβ,a∞andβ=1,2,or4forreal,complex,orquaternionGaussianentries.Moreimportantly,withthesechoicesofβallfinitedimensionalcorrelationfunctionsoftheeigenvaluesarecomputableintermsofLaguerrepolynomials(thusthecommontag“Laguerreensembles”).Atβ=2andallvalida,[22]provesthelimitingdistributionoftheminimaleigenvalueisdescribedbytheFredholmdeterminantofakerneloperatorgivenintermsofBesselfunctions,andbasedonthisderivesaseconddescriptionofthelimitlawasafunctionalofthefifthPainlev´etranscendent.Otherworkatβ=1,2,4hardedgeinclude[4],[10],and[26].Theseagainrelyontheunderlyingorthogonalpolynomialstructure(thefirstandthirdreferenceuseRiemann-Hilbertmethodstoreplacetheexponentialweighte−(β/2)λin(1.1)withamoregenerale−V(λ)potential),anddescribetheeventuallimitlawthroughFredholmdeterminantsorFredholmpfaffians.Whilethedistributiononnpointsλ1,...,λn∈R+definedby(1.1)makessenseforallβ0anda−1,theorthogonalpolynomialapproachbreaksdownoutsidetheclassictripleinβ.Forsomespecialchoicesoftheparametersbeyondβ=1,2,4,[9]wasabletoexploitthenicetiesoftheexponentialweighttoobtainlimitlawsintermsofhypergeometricfuncions.Still,eventheexistenceofthegeneralβhardedgelimitlawremainedopenuntilnow.Ourapproach,similartothatin[15],restsontheexistenceoftridiagonalmatrixmodelsforallβ.Setforanya−1andβ0,Lβ,a=1√βχ(a+n)βχ(n−1)βχ(a+n−1)βχ(n−2)β......χ(a+2)βχβχ(a+1)β(1.2)inwhicheachχrthatappearsisanindependentχrandomvariableoftheindicatedindex.(Wesuppressherethedimensionparameternonthen×nrandommatrixLβ,a).Then,asdiscoveredbyDumitriuandEdelman[5],theeigenvaluesofLβ,aLTβ,ahave(1.1)astheirjointdensityfunction.Notethat,whenβ=1or2,thebidiagonal(1.2)maybearrivedatbyperformingHouseholdertransformationsonthecorresponding“full”Wishartensemble;thisfactwasusedpreviouslyinarandommatrixcontextbySilverstein[16].Viewingthen↑∞limitasgivingrisetoacontinuumapproximationtothediscreteoperatorsLβ,a,anentry-wiseexpansionintherandomχvariablesledEdelmanandSuttontothefollowingconjectureforthefullβ0hardedge.2Conjecture(Edelman-Sutton[6])Letskdenotethek-thsmallestsingularvalueofthebidiag-onaloperatorLβ,a.Then,asn↑∞thefamily{√nsk}convergesinlawtothecorrespondingsingularvaluesofLβ,a=−√xddx+a2√x+1√βb′(x)inwhichx7→b(x)isaBrownianmotion.Here,Lβ,aisunderstoodtoactonfunctionsf∈L2[0,1]subjecttof(1)=0and(Lβ,af)(0)=0.Ourmainresultestablishesthisconjecture,thoughweprefertophrasemattersinadifferentway,backintermsofeigenvaluesofthesymmetricensemblesLβ,aLTβ,a,henceforthreferredtoasthe(β,a)-Laguerreensembles.Towardthis,introducetherandomoperatorofsecondorder,Gβ,a=−exp[(a+1)x+2√βb(x)]ddxnexp[−ax−2√βb(x)]ddxo,(1.3)whereagainb(x)isaBrownianmotionanda−1,β0.FormalmanipulationswilltakeyoufromLβ,aLTβ,atoGβ,a,butthelatterisbetterunderstooduponrecognizing,inthespiritofthetitle,that−Gβ,ageneratesthediffusionwith(random)speedandscalemeasuresm(dx)=e−(a+1)x−2√βb(x)dxands(dx)=eax+2√βb(x)dx.Thismotionmaybeconstructedpath-wiseintheclassicalmode(seeforexample[12]),placing(1.3)onfirmground.ThelimitingspectralproblemwillrequireconsiderationofGβ,aactingonthepositivehalf-linewithDirichletconditionsattheorigin,andthiscarriesoverintokillingtheunderlyingprocesswhenreachingt