A LAW OF LARGE NUMBERS FOR FINITE-RANGE DEPENDENT

ALAWOFLARGENUMBERSFORFINITE-RANGEDEPENDENTRANDOMMATRICESGREGANDERSONANDOFERZEITOUNIAbstract.Weconsiderrandomhermitianmatricesinwhichdistantabove-diagonalentriesareindependentbutnearbyentriesmaybecorrelated.Wefindthelimitoftheempiricaldistributionofeigenvaluesbycombinatorialmethods.WealsoprovethatthelimithasalgebraicStieltjestransformbyanargumentbasedondimensiontheoryofnoetherianlocalrings.1.IntroductionStudyoftheempiricaldistributionofeigenvaluesofrandomhermitian(orrealsymmetric)matriceshasalonghistory,startingwiththeseminalworkofWigner[Wig55]andWishart[Wis28].Exceptincaseswherethejointdistributionofeigen-valuesisexplicitlyknown,mostresultsavailableareasymptoticinnatureandbasedononeofthefollowingapproaches:(i)themomentmethod,i.e.,evaluationofex-pectationsoftracesofpowersofthematrix;(ii)appropriaterecursionsfortheresolvant,asintroducedin[PM67];or(iii)thefreeprobabilitymethod(especiallythenotionofasymptoticfreeness)originatingwithVoiculescu[Voi91].Agoodre-viewofthefirsttwoapproachescanbefoundin[Ba99].Forthethird,see[Voi00],andforasomewhatmorecombinatorialperspective,[Sp98].Theseapproacheshavebeenextendedtosituationsinwhichthematrixanalyzedneitherpossessesi.i.d.entriesabovethediagonal(asintheWignercase)norisittheproductofmatriceswithi.i.d.entries(asintheWishartcase).Wementioninparticularthepapers[MPK92],[KKP96],[Sh96]and[Gu02]forresultspertainingtothemodelof“randombandmatrices”,allwithindependentabove-diagonalentries.Inourrecentwork[AZ06]westudiedconvergenceoftheempiricaldistributionofeigenvaluesofrandombandmatrices,anddevelopedacombinatorialapproach,basedonthemomentmethod,toidentifythelimit(andalsotoprovidecentrallimittheoremsforlinearstatistics).Herewedevelopthemethodfurthertohandleaclassofmatriceswithlocaldependenceamongentries(wepostponetheprecisedefinitionoftheclasstoSection2).Toeachrandommatrixoftheclassweassociatearandombandmatrixwiththesamelimitofempiricaldistributionofeigenvaluesbyaprocessof“Fouriertransformation”,thusmakingitpossibletodescribethelimitintermsofourpreviouswork(seeTheorem2.5).WealsoprovethattheStieltjestransformofthelimitisalgebraic(seeTheorem2.6),doingsobyageneral“soft”(i.e.,nonconstructive)methodbasedonthetheoryofnoetherianlocalrings(seeTheorem6.2)whichoughttobeapplicabletomanymorerandommatrixproblems.Date:September10,2006.O.Z.waspartiallysupportedbyNSFgrantnumberDMS-0503775.12GREGANDERSONANDOFERZEITOUNITogettheflavorofourresults,thereadershouldimagineaWignermatrix(i.e.,anN-by-Nrealsymmetricrandommatrixwithi.i.d.above-diagonalentries,eachofmean0andvariance1/N),onwhichalocal“filtering”operationisperformed:eachentrynotnearthediagonaloranedgeisreplacedbyhalfthesumofitsfourneighborstonortheast,southeast,southwestandnorthwest.AttheendofSection3(Theorem2.5takenforgranted)weanalyzethe“(NE+SE+SW+NW)-filteredWignermatrix”describedabove.Wefindthatthelimitmeasureisthefreemultiplicativeconvolutionofthesemicirclelaw(density∝1|x|2√4−x2)andthearcsinelaw(density∝10x2/px(2−x)).Theappearanceinthisexampleofafreemultiplicativeconvolutionhasasimpleexplanation(seeProposition3.6).WealsowritedownthequarticequationsatisfiedbytheStieltjestransformofthelimitmeasure.Recentlyotherauthorshaveconsideredtheempiricaldistributionofeigenvaluesformatriceswithdependententries,see[GoT05],[Ch05],[SSB05].Theirclassofmodelsdoesnotoverlapsignificantlywithours.Inparticular,inalltheseworksandunlikeinourmodel,thelimitoftheempiricalmeasureisalwaysthesameasthatofasemicirclelawmultipliedbyarandomordeterministicconstant.Closesttoourworkistherecentpaperby[HLN05],thatbuildsuponearlierworkby[BDM96]and[Gi01].TheyconsiderGrammatricesoftheformXNX∗NwhereXNisasequenceof(non-symmetric)Gaussianmatricesobtainedbyapplyingafilteringoperationtoamatrixwith(complex)Gaussianindependententries.Theyalsoconsiderthecase(XN+AN)(XN+AN)∗withANdeterministicandToeplitz.TheGaussianassumptionallowsthemtodirectlyapproximatethematrixXNbyaunitarytransformationofaGaussianmatrixwithindependent(butnotidenticallydistributed)entries,towhichtheresultsof[Gi01]and[AZ06]apply.Inparticular,theydonotneedanassumptiononthesupportofthefilter.Unlikethepresentwork,theapproachin[HLN05]and[BDM96]isbasedonstudyingresolvants,ratherthanmoments.Wementionnowmotivationfromelectricalengineering.Theanalysisofthelimitingempiricaldistributionofeigenvaluesofrandommatriceshasrecentlyplayedanimportantroleintheanalysisofcommunicationsystems,see[TV04]foranextensivereview.Inparticular,whenstudyingmulti-antennasystems,oneoftenmakesthe(unrealistic)assumptionthatgainsbetweendifferentpairsofantennasareuncorrelated.Themodelsstudiedinthisworkwouldallowcorrelationbetweenneighboringantennapairs.Wedonotdevelopthisapplicationfurtherhere.Thestructureofthepaperisasfollows.Inthenextsectionwedescribetheclassofmatriceswetreat,andstateourmainresults,namelyTheorem2.5(assert-ingalawoflargenumbers)andTheorem2.6(assertingalgebraicityofaStieltjestransform).InSection3wediscussthelimitmeasureindetail,andinparticularwritedownalgebro-integralequationsforitsStieltjestransform,whichwecallcolorequations.Section4providesacomputationo