2P.BourgadeHarvardCollegeMathematicsReviewnaturalquestion.Choosenindep - PDF document

HarvardCollegeMathematicsReviewTeaTimeinPrincetonHesaid:\That'stheformfactorforthepaircorrelationofeigenvaluesofrandomHermitianmatrices!".Thisnoteisaboutwho\He"is,what\That"is,andwhyyoushouldnevermissteatime.SincetheseminalworkofRiemann,itiswell-knownthatthedistributionofprimenumbersiscloselyrelatedtothebehaviorofthefunction.Mostimportantly,itwasconjecturedin[13]thatallits(non-trivial)zerosarealigned1,andHilbertandPolyaputforwardtheideaofaspectraloriginforthisphenomenon.IspenttwoyearsinGottingenendingaroundthebeginof1914.ItriedtolearnanalyticnumbertheoryfromLandau.Heaskedmeoneday:\Youknowsomephysics.DoyouknowaphysicalreasonthattheRiemannhypothesisshouldbetrue?"Thiswouldbethecase,Ianswered,ifthenontrivialzerosofthe-functionweresoconnectedwiththephysicalproblemthattheRiemannhy-pothesiswouldbeequivalenttothefactthatalltheeigenvaluesofthephysicalproblemarereal.GeorgePolya,correspondencewithAndrewOdlyzko,1982.Despitethelackofprogressconcerningthehorizontaldistributionofthezeros(i.e.alltheirrealpartsbeingsupposedlyequal),somesupportfortheHilbert-Polyaideacamefromtheverticaldistribution,i.e.thedistributionofthegapsbetweentheimaginarypartsofthenon-trivialzeros.Indeed,in1972,thenumbertheoristHughMontgomeryevaluatedthepaircorrelationofthesezeros,andthemathematicalphysicistFreemanDysonrealizedthattheyexhibitthesamerepulsionastheeigen-valuesoftypicallargerandomHermitianmatrices.Inthisexpositorynote,weaimatexplainingMontgomery'sresult,placingemphasisonthecommonpointswithrandommatrices.ThesestatisticalconnectionshavesincebeenextendedtomanyotherL-functions(e.g.overfunctionelds,cf.[12]);forthesakeofbrevityweonlyconsidertheRiemannzetafunction,andreferforexampleto[8]formanyotherconnectionsbetweenanalyticnumbertheoryandrandommatrices.1IndependentrandompointsAsarststeptowardstherepulsionbetweensomeparticles,eigenvaluesorzerosofthezetafunction,wewishtounderstandwhathappenswhenthereisnorepul-sion,inparticularforindependentrandompoints.Forthis,considerthefollowing 1.ForadenitionoftheRiemannzetafunctionandtheRiemannhypothesis,seethebeginningofSection2. 2P.BourgadeHarvardCollegeMathematicsReviewnaturalquestion.Choosenindependentanduniformpointsontheinterval[0;1].Whatisthetypicalspacingbetweentwosuccessivesuchpoints?Agoodwaytomakethisquestionmorepreciseistoassumethatamongstthesepointsx1;:::;xn,welabelone,sayx1,andweconsidertheprobabilitythatithasnoright-neighboruptodistance.Denoting(I)thenumberofxi'sinanintervalI,theprobabilityofsuchaneventisZ10P(((y;y+])=0jx1=y)dy;becausex1isuniformlydistributed.Now,asallthexi'sareindependent,theinte-grandisalso(wheny+1)P(\ni=2fxi62(y;y+]g)=nYi=2P(xi62(y;y+])=(1�)n�1: Figure1{Histogramof105nearest-neighborspacings(i.e.n
).Dashed:there-scalede�ucurve.Choosing=u nandconsideringthelimitn!1,wegetthattheprobabilitythatthegapbetweenx1anditsrightneighborisgreaterthanu nconvergestoe�u.Morege-nerally,denotingby
thegapbetweenx1anditsright-neighbor,weobtainthat,forany0ab,P(n
2[a;b])�!n!1Zbae�udu:(1)Anotherwaytoquantifythemicrosco-picstructureoftheseindependentpointsconsistsinlookingatthefollowingstatis-tics,r(f;n)=1 nP1j;kn;j6=kf(n(xj�xk));foragenerictestfunctionf.Thereaderwilleasilyprovethefollowingasymptotics:E(r(f;n))�!n!1ZRf(y)dy:(2)Thislimitingexponentialdistribution(1)andthepaircorrelation(2)appearuniversally,i.e.whenthesampledpointsaresucientlyclosetoindependence,nomatterwhichdistributiontheyhave2.Itisanaturalquestionwhetherthisremainsvalidforotherrandompoints,andwewillexplainwhathappenswhenconsideringthezeroswithlargeimaginarypartortheeigenvaluesofrandommatrices.Thegapsstatisticswillbeverydierent,bothfortheformer(Section2)andthelatter(Section3),forwhichacommontypeofcorrelationsappearsinthelimit.Thefollowingsectionsarewidelyindependent. 2.Forexample,thereadercouldconsiderindependentpointswithstrictlypositivedensitywithrespecttotheuniformmeasureon[0;1],andhewouldobtainanexponentiallawinthelimitaswell. Vol.00,2012TeaTimeinPrinceton32Thepaircorrelationofthezeros.Inthissection,westatesomeelementarypropertiesoftheRiemannzetafunc-tion,mentioningalongthewayaformalanalogybetweenthezerosandtheeigenva-luesoftheLaplacianonsomesymmetricspaces.WethencometomorequantitativeestimatesthroughMontgomery'sresultontherepulsionbetweenthezeros.For=(s)&#x]TJ/;༖ ;.9;Ւ ;&#xTf 8;&#x.634;&#x 0 T; [0;1,theRiemannzetafunctioncanbedenedasaDirichletseriesoranEulerproduct:(s)=1Xn=11 ns=Yp2P1 1�1 ps;wherePisthesetofallprimenumbers.Thesecondequalityisaconsequenceoftheexpansion(1�p�s)�1=Pk0p�ksanduniquenessoffactorizationofintegersintoprimenumbers.Remarkably,asprovedinRiemann'soriginalpaper,canbemeromorphicallyextendedtoC�f1g,andthisextensionsatisesafunctionalequation(seee.g.[15]foraproof):writing(s)=�s=2�(s=2)(s),wehave(s)=(1�s):Consequently,thezetafunctionadmitstrivialzerosats=�2;�4;�6;:::corres-pondingtothepolesof�(s=2).Alltheotherzerosareconnedinthecriticalstrip01,andtheyaresymmetricallypositionedabouttherealaxisandthecriticalline=1=2.TheRiemannHypothesisstatesthatallofthisnon-trivialzerosareexactlyontheline=1=2.Traceformulas.Therstsimilaritybetweenthezetazerosandspectralproper-tiesofoperatorsoccurswhenlookingatlinearstatistics.Namely,westatetheWeilexplicitformulaconcerningthezerosandSelberg'straceformulafortheLaplacianonsurfaceswithconstantnegativecurvature.FirstconsidertheRiemannzetafunction.Forafunctionf:(0;1)!C,deneitsMellintransformF(s)=R10f(s)xs�1dx.Thentheinversionformula(whereischoseninthefundamentalstrip,i.e.wheretheimagefunctionFconverges)f(x)=1 2iZ+i1�i1F(s)x�sdsholdsundersuitablesmoothnessassumptions,inasimilarwayastheinverseFouriertransform.Hence,forexample,1Xn=2(n)f(n)=1Xn=2(n)1 2iZ2+i12�i1F(s)n�sds=1 2iZ2+i12�i1�0 (s)F(s)ds;whereisVanMangoldt'sfunction3.Toderivetheaboveformula,weusethat�0 (s)=Pn2(n) ns,whichisobtainedbyderivingtheformula�log(s)=PPlog(1�p�s).Now,changingthelineofintegrationfrom(s)=2to(s)=�1,alltrivialandnon-trivialpoles(aswellass=1)arecrossed,leadingtothefollowingformula,XF()+Xn0F(�2n)=F(1)+Xp2P;m2N(logp)f(pm); 3.(n)=logpifn=pkforsomeprimep,0otherwise. 4P.BourgadeHarvardCollegeMathematicsReviewwheretherstsumisovernon-trivialzeroscountedwithmultiplicities.Whenrepla-cingtheMellintransformbytheFouriertransform,theaboveformulalinkinglinearstatisticsofzerosandprimestakesthefollowingform,knownastheWeilexplicitformula.Theorem.Lethbeeven,analyticonj=(z)j1=2+,bounded,anddecreasingash(z)=O(jzj�2�)forsome&#x]TJ/;༖ ;.9;Ւ ;&#xTf 1;.89; 0 ;&#xTd [;0.Here,thesumisoverall n'ssuchthat1=2+i nisanon-trivialzero,and^h(x)=1 2R1�1h(y)e�ixydy:X nh( n)�2hi 2=1 2ZRh(r)�0 �1 4+i 2r�logdr�2Xp2Pm2Nlogp pm=2^h(mlogp):(3)Inaverydistinctcontextholdsasimilarrelation,theSelberg'straceformula.Inoneofitssimplestmanifestations,itcanbestatedasfollows.Let�nHbeaquo-tientofthePoincarehalf-plane,where�isasubgroupofPSL2(R),theorientation-preservingisometriesofH=fx+iy;y�0gendowedwiththemetric(ds)2=(dx)2+(dy)2 y2:(4)TheLaplace-Beltramioperator
=�y2(@xx+@yy)isself-adjointwithrespecttotheinvariantmeasureassociatedto(4),d=dxdy y2,i.e.Rv(
u)d=R(
v)ud,soalleigenvaluesof
arerealandpositive.If�nHiscompact,thespectrumof
restrictedtoafundamentaldomainDofrepresentativesoftheconjugationclassesisdiscrete,noted001:::TostateSelberg'straceformula,weneed,aspreviously,afunctionhanalyticonj=(z)j1=2+,even,bounded,anddecreasingash(z)=O(jzj�2�),forsome&#x]TJ/;༖ ;.9;Ւ ;&#xTf 1;.42; 0 ;&#xTd [;0.Theorem.Undertheabovehypotheses,settingk=sk(1�sk),sk=1=2+irk,then1Xk=0h(rk)=(D) 2Z1�1rh(r)tanh(r)dr+Xp2P;m2N`(p) 2sinhm`(p) 2^h(m`(p));(5)where^histheFouriertransformofh(^h(x)=1 2R1�1h(y)e�ixydy),Pisnowthesetofallprimitive4periodicorbits5and`isthegeodesicdistancecorrespondingto(4).Thesimilaritybetween(3)and(5)maymakeyouwishthatprimenumberswouldcorrespondtoprimitiveorbits,withlengthslogp,p2P.Noresultinthisdirectionisknownhowever,anditseemssafernottothinkaboutthisanalogyasaconjecture,butratherjustasatoolguidingintuition(asdonee.g.in[3]tounderstandthepaircorrelationsbetweenthezerosof).Nevertheless,thereadercouldprovethat,asaconsequenceofSelberg'straceformula,thenumberofprimitiveorbitswithlengthlessthanxisjf`(p)xgjx!1ex x: 4.i.e.nottherepetitionofshorterperiodicorbits5.ofthegeodesic owon�nH Vol.00,2012TeaTimeinPrinceton5Similarly,bytheprimenumbertheorem,jflog(p)xgjx!1ex x:Montgomery'stheorem.Amorequantitativeconnectionofanalyticnumbertheorywithaspectralproblemsappearedintheearly70'sthankstoaconversation,duringteatime,inPrinceton,aboutsomeresearchonthespacingsbetweenthezeros.Hereisahowtheauthorofthiswork,HughMontgomery,relatesthis\serendipity"moment[6].ItookafternoonteathatdayinFuldHallwithChowla.FreemanDysonwasstan-dingacrosstheroom.IhadspentthepreviousyearattheInstituteandIknewhimperfectlywellbysight,butIhadneverspokentohim.Chowlasaid:\HaveyoumetDyson?"Isaidno,Ihadn't.Hesaid:\I'llintroduceyou."Isaidno,Ididn'tfeelIhadtomeetDyson.Chowlainsisted,andsoIwasdraggedreluctantlyacrosstheroomtomeetDyson.Hewasverypolite,andaskedmewhatIwasworkingon.ItoldhimIwasworkingonthedierencesbetweenthenon-trivialzerosofRiemann'szetafunction,andthatIhaddevelopedaconjecturethatthedistributionfunctionforthosedierenceshadintegrand1��sinu u
2.Hegotveryexcited.Hesaid:\That'stheformfactorforthepaircorrelationofeigenvaluesofrandomHermitianmatrices!"I'dneverheardtheterm\paircorrelation."Itreallymadetheconnection.ThenextdayAtle(Selberg)hadanoteDysonhadwrittentomegivingreferencestoMehta'sbook,placesIshouldlook,andsoon.TothisdayI'vehadoneconversationwithDysonandoneletterfromhim.Itwasveryfruitful.Isupposebythistimetheconnectionwouldhavebeenmade,butitwascertainlyfortuitousthattheconnectioncamesoquickly,becausethenwhenIwrotethepaperfortheproceedingsoftheconference,Iwasabletousetheappropriateterminologyandgivethereferencesandgivetheinterpretation.Iwasamusedwhen,afewyearslater,Dysonpublishedapapercalled\MissedOpportunities."I'msuretherearelotsofmissedopportunities,butthiswasacounterexample.ItwasrealserendipitythatIwasabletoencounterhimatthiscrucialjuncture.SowhatwasitexactlythatMontgomeryproved?Tostatehisresult,weneedtorstintroducesomenotation.Firstbychoosingforhanappropriateapproximationofanindicatorfunction,fromtheexplicitformula(3)onecanprovethefollowing:thenumberofzeroscountedwithmultiplicitiesin0=()tisasymptoticallyN(t)t!1t 2logt:(6)Inparticular,themeanspacingbetweenzerosatheighttis2=logt.Now,wewriteaspreviously1=2i nforthezetazeroscountedwithmultiplicity,assumingtheRiemannhypothesisandtheordering 1 2:::Let!n= n 2log n 2.From(6)weknowthatn=!n+1�!nhasameanvalue1asn!1.Amorepreciseunderstandingofthezetazerosinteractionsreliesonthestudyofthespacingsdistributionfunctionbelowfort!1,1 N(t)jf(n;m)2J1;N(t)K2:!n�!m;n6=mgj; 6P.BourgadeHarvardCollegeMathematicsReviewandmoregenerallyontheoperator~r(f;t)=1 N(t)X1j;kN(t);j6=kf(!j�!k):Aswesawin(2),ifthe!k'sbehavedasindependentrandomvariables(uptotheordering),~r(f;t)wouldconvergetoRRf(y)dyast!1.ThefollowingresultbyMontgomery[10]provesthatthezerosareactuallynotasymptoticallyindependent,butpresentsomestatisticalrepulsioninstead.Weincludeanoutlineofaproofdirectlyfollowingthestatementfortheinterestedreader. Figure2{Thefunction~r(y)andthehisto-gramofthenormalizedspacingbetweennon-necessarilyconsecutivezeros,atheight1013(anumberof2109zeroshavebeenusedtocom-putetheempiricaldensity,representedassmallcircles).Source:XavierGourdon[7]Theorem.AssumetheRiemannhypo-thesis.Supposefisatestfunctionwiththefollowingproperty:itsFou-riertransform6isC1andsupportedin(�1;1).Then~r(f;t)�!t!1ZRf(y)~r(y)dy;where~r(y)=1�sin(y) y2.InfactanimportantconjectureduetoMontgomeryassertsthattheaboveresultholdswithnoconditiononthesupportoftheFouriertransform.Ho-wever,weakeningtherestrictioneventosupp^f(�1�";1+")forsome"�0outofreachwithknowntechniques.TheMontgomeryconjecturewouldhaveim-portantconsequencesforexampleintermsofthestatisticsofgapsbetweentheprimenumbersp1p2::::forexample,itwouldimplythatpn+1�pnp pnlogpn.SketchofproofofMontgomery'sTheorem.ConsiderthefunctionF(;t)=1 t 2logtX0 ; 0ti( � 0)4 4+( � 0)2;wherethe 'saretheimaginarypartsofthezeros.ThisistheFouriertransformofthenormalizedspacings,uptothefactor4=(4+( � 0)2),presentherejustfortechnicalconvergencereasons.ThisfunctionnaturallyappearswhencountingthesecondordermomentsZt0jG(s;t)j2ds=F(;t)tlogt+O(log3t);G(s;x)=2X xi 1+(s� )2:(7) 6.ContrarytotheWeilandSelbergformulas(3)and(5),thechosennormalizationhereis^f(x)=R1�1f(y)e�i2xydy Vol.00,2012TeaTimeinPrinceton7AsGisalinearfunctionalofthezeros,itcanbewrittenasasumoverprimesbyanappropriateexplicitformulalike(3):MontgomeryprovedthatG(s;x)=�p x Xnx(n)x n�1 2+is+X�nx(n)x n3 2+is!+"(s;x);where"(s;x)isanerrortermwhich,undertheRiemannhypothesis,canbeboundedecientlyandmakesnocontributioninthefollowingasymptotics.Themoment(7)canthereforebeexpandedasasumoverprimes,andtheMontgomery-Vaughaninequality(cf.theexercisehereafter)leadstoZt0jG(s;t)j2ds=(t�2logt++o(1))tlogt:(8)TheseasymptoticscanbeprovedbytheMontgomeryVaughaninequality,butonlyintherange2(0;1),whichexplainsthesupportrestrictioninthehypotheses.GatheringbothasymptoticexpressionsforthesecondmomentofGyieldsF(;t)=t�2logt++o(1).Finally,bytheFourierinversionformula,1 t 2logtX0 ; 0tf( � 0)logt 24 4+( � 0)2=ZRF(;t)^f()d:Ifsupp^f(�1;1),thisisapproximatelyZR^f()(t�2jj+jj)d=ZRe�2jj^f(=logt)d+ZRjj^f()d=^f(0)+f(0)�ZR(1�jj)^f()d+o(1)=f(0)+ZRf(x) 1�sinx x2!dx+o(1);bythePlancherelformula. (Dicult)Exercise.Let(ar)becomplexnumbers,(r)distinctrealnumbersandr=mins6=rjr�sj.ThentheMontgomery-Vaughaninequalityassertsthat1 tZt0jXrareirsj2ds=Xrjarj21+3 trforsomejj1.Inparticular,Zt01Xn=1an nis2ds=t1Xn=1janj2+O 1Xn=1njanj2!:Provethattheaboveresultimplies(8). 8P.BourgadeHarvardCollegeMathematicsReview Figure3{Thedistributionfunctionofasymptoticgapsbetweeneigenvaluesofran-dommatricescomparedwiththehistogramofgapsbetweensuccessivenormalizedze-ros,basedonabillionzerosnear#1:3
1016.TonumericallytestMontgomery'sconjec-ture,Odlyzko[11]computedthenormalizedgaps,!i+1�!i,andproducedthejointhis-togram.Inparticular,notethatthelimitingdensityvanishesat0,contrastingwithFi-gure1,andthatthistypeofrepulsioncoin-cidesremarkablywiththeshapeofgapsforrandommatrices.Moreover,Montgomery'sresulthasbeenextendedintheworkbyRudnickandSarnak[14],whoprovedthatforsomesta-tisticsdependingonmorethanjustonegap,thezerosalsopresentthesamelimitdis-tributionaspredictedbyRandomMatrixTheory.Thisurgesustoexplaininmorede-tailswhatwemeanbyrandommatrices.3EigenvaluesrepulsionforrandommatricesLetbeapointprocess,i.e.arandomsetofpointsfx1;x2;:::g,inametricspace,identiedwiththerandompunctualmeasurePixi.Thekthcorrelationfunctionforthispointprocess,k,isdenedastheasymptotic(normalized)proba-bilityofhavingexactlyoneparticleinrespectiveneighborhoodsofkxedpoints.Moreprecisely,iftheui'saredistinctin,k(u1;:::;uk)=lim"!0P((Bui;")=1;1ik) Qkj=1(Bui;");providedthatthelimitexists(hereBui;"denotestheballwithradius"andcenteru,andthemeasurewillbespeciedlater).Ifconsistsalmostsurelyofnpoints,thecorrelationfunctionssatisfytheintegrationproperty(n�k)k(u1;:::;uk)=Zk+1(u1;:::;uk+1)d(uk+1):(9)Interestingly,manypropertiesaboutapointprocessarewell-understoodwhenthecorrelationfunctionsarealsodeterminants.Moreprecisely,assumenowthat=C.IfthereexistsafunctionK:CC!Csuchthatforallk1and(z1;:::;zk)2Ckk(z1;:::;zk)=det�K(zi;zj)ki;j=1
;thenissaidtobeadeterminantalpointprocesswithrespecttotheunderlyingmeasureandwithcorrelationkernelK.Thedeterminantalconditionforallcorrelationfunctionsisquiterestrictive.Ne-vertheless,asstatedinthefollowingtheorem,anybidimensionalsystemofparticleswithquadraticinteractionisdeterminantal(see[1]foraproof).Theorem.Letdbeany7nitemeasureonC(eventuallyconcentratedonaline). 7.WejustneedadecreasingofthemassatinnityoftypeRjzj�td(z)t�kforanyk�0. Vol.00,2012TeaTimeinPrinceton9Considertheprobabilitydistributionwithdensityc(n)Y1knjzl�zkj2withrespecttoQnj=1d(zj),wherec(n)isthenormalizationconstant.Forthisjointdistribution,fz1;:::;zngisadeterminantalpointprocesswiththefollowingexplicitkernel,K(x;y)=n�1Xk=0Pk(x) Pk(y)wherePk(0kn�1)isapolynomialwithdegreekandthePj'sareorthonormalfortheHermitianproductf;g7!Rf gd:Weapplytheaboveresulttothefollowingexamples,whichareamongthemoststudiedrandommatrices.First,considertheso-calledGaussianunitaryensemble(GUE).Thisistheensemble(orset)ofrandomnnHermitianmatriceswithindependent(uptosymmetry)Gaussianentries:M(n)ij= M(n)ji=1 p n(Xij+iYij);1ijn;wheretheXij'sandYij'sareindependentcenteredrealGaussiansentrieswithmean0andvariance1=2andM(n)ii=Xii=p nwithXiirealcenteredGaussianswithvariance1,stillindependent.Theserandommatricesarenaturalinthesensethattheyareuniquelycharacterizedbytheindependence(uptosymmetry)oftheirentries,andinvariancebyunitaryconjugacy.Asimilarnaturalsetofmatrices,whentheentriesarenowrealGaussian,calledGOE(Gaussianorthogonalensemble)willappearinthenextsection.FortheGUE,thedistributionoftheeigenvalueshasanexplicitdensity,1 Zne�nPni=12i=2Y1inji�jj2(10)withrespecttoLebesguemeasure(seee.g.[1]foraderivationofthisresult).Wedenoteby(hn)theHermitepolynomials,morepreciselythesuccessivemonicpoly-nomialsorthogonalwithrespecttotheGaussianweighte�x2=2dx,andconsidertheassociatednormalizedfunctions k(x)=e�x2=4 p p 2k!hk(x):ThenfromthepreviousTheorem,onecanprovethatthesetofpointf1;:::;ngwithlaw(10)isadeterminantalpointprocesswhosekernel(withrespecttotheLebesguemeasureonR)isgivenbyKGUE(n)(x;y)=n n(xp n) n�1(yp n)� n�1(xp n) n(yp n) x�y;extendedbycontinuitywhenx=y.Hereweusedasimplication:thesumoverallorthogonalpolynomialscansimplifyasasumoverjusttwoofthem,thisistheChristoel-Darbouxformula.ThePlancherel-RotachasymptoticsfortheHermitepolynomialsimpliesthat,asn!1,KGUE(n)(x;x)=nhasanon-triviallimit. 10P.BourgadeHarvardCollegeMathematicsReview Figure4{HistogramoftheeigenvaluesfromtheGaussianUnitaryEnsembleindi-mension104.Dashed:therescaledsemicirclelaw.Moreprecisely,theempiricalspectraldistri-bution1 nPiconvergesinprobabilitytothesemicirclelawwithdensitysc(x)=1 2p (4�x2)+withrespecttoLebesguemeasure.Thisistheasymptoticbehaviorofthespectruminthemacroscopicregime.Themicrosco-picinteractionsbetweeneigenvaluesalsocanbeevaluatedthankstoasymptoticsoftheHermiteorthogonalpolynomials:foranyx2(�2;2),u2R,1 nsc(x)KGUE(n)x;x+u nsc(x)�!n!1K(u)=sin(u) u:Thisleadstoarepulsivecorrelationstructurefortheeigenvaluesatthescaleoftheaveragegap:forexamplethetwo-pointcorrelationfunctionasymptoticsare1 nsc(x)2GUE(n)2x;x+u nsc(x)�!n!1~r(u)=1�sin(u) u2;thestrictanaloguetoMontogmery'sresult,ananalogyidentiedbyDysonasmen-tionedinSection2. Figure5{Upperline:asampleofindependentpointsdistributedaccordingtothesemicirclelawafterzoominginthebulk.Middleline:asampleeigenvaluesoftheGUEafterzoominginthebulkofthespectrum.Lowerline:asequenceofimaginarypartsofthezeros,aboutheight105.Aremarkablefactabouttheabovelimitingsinekernelisthatitappearsuni-versallyinthelimitingcorrelationfunctionsofrandomHermitianmatriceswithindependent(uptosymmetry)entries(notnecessarilyGaussian);thesedeepuni-versalityresultswereachieved,stillfortheHermitiansymmetryclass,inrecentworksbyErd}os,Yauetal,orbyTao,Vu.Inthecaseofothersymmetryclasses8,theuniversalityofthelocaleigenvaluesstatisticshasalsobeenprovedbyErd}os,Yauetal.Finallywewanttomentionthefollowingstructuralreasonfortherepulsionoftheeigenvaluesoftypicalmatrices:asanexercise,thereadercouldprovethatthespaceofHermitianmatriceswithatleastonerepeatedeigenvaluehascodimension3inthespaceofallHermitianmatrices.Repeatedeigenvaluesthereforeoccurwithverysmallprobabilitycomparedtoindependentpoints(onaproductspace,thecodimensionofthesubspacewheretwopointscoincideis1).LaszloErd}osaskedmeaboutastructural,heuristic,argumentfortherepulsionofthezeros.Unabletoanswerit,Itransmitthequestiontothereaders. 8.i.e.forrandomsymmetricmatricesorrandomsymplecticmatrices Vol.00,2012TeaTimeinPrinceton114EigenvaluesrepulsionforquantumbilliardsToconcludethisexpositorynote,wewishtomentionsomeconjecturesabouttheasymptoticdistributionofeigenvalues,fortheLaplacianoncompactspaces.Theexamplesweconsideraretwo-dimensionalquantumbilliards9.Forsomebilliards,theclassicaltrajectoriesareintegrable10andforotherstheyarechaotic. Figure6{Anintegrablebilliard(ellipse)andachaoticone(stadium)Onthequantumside,weconsidertheHelmholtzequationinsidethebilliard,describingthestandingwaves:�
n=n n;wherethespectrumisdiscreteasthedomainiscompact,withorderedeigenva-lues012:::,andappropriateDirichletorNeumannboundaryconditions. Figure7{Somechaoticbilliards,fromlefttoright,uptodown:thestadium,Sinai'sbillard,thecardioid,andabilliardwithnoname.Thequestionsaboutquantumbilliardsweareinterestedhereisabouttheasymptoticbehaviorofthen's,i.e.whethertheywillpresentasympto-ticindependenceoraRandomMatrixTheorytypeofrepulsion.Thesitua-tionisstillsomehowmysterious:thereisaconjecturaldichotomybetweenthechaoticandintegrablecases.First,in1977,BerryandTabor[4]putforwardtheconjecturethatformostintegrablesystems,thelargeeigenvalueshavethestatisticsofaPoissonpointprocess,i.e.rescaledgapsbeingasymptoticallyexponentialrandomvariables,likeinSection1.Moreprecisely,byWeyl'slaw,weknowthatthenumberofsucheigen-valuesuptoisjfi:igj!1area(D) 4:(11)Toanalyzethecorrelationsbetweeneigenvalues,considerthepointprocess(n)=1 nXin4 area(D)(i+1�i):Itsexpectationconvergesto1(asn!1)from(11). 9.Abilliardisacompactconnectedsetwithnonemptyinterior,withagenerallypiecewiseregularboundary,sothattheclassicaltrajectoriesarestraightlinesre ectingwithequalanglesofincidenceandre ection10.Roughlyspeakingthismeansthattherearemanyconservedquantitiesalongthetrajectory,andthatexplicitsolutionscanbegivenforthespeedandpositionoftheballatanytime 12P.BourgadeHarvardCollegeMathematicsReview Figure8{EnergylevelsforthecircularbilliardcomparedtothoseoftheGaussianensemblesandPoissonianstatistics(dataandpicturefrom[2]).BytheconjecturedlimitingPoisso-nianbehavior,thespacingdistributionconvergestoanexponentiallaw:foranyIR+(n)(I)�!n!1ZIe�xdx:(12)Inthechaoticcase,thesituationdiersradically:theeigenvaluesaresupposedtorepeleachother,withgapsstatisticsconjecturallysimilartothoseofarandommatrix,fromanensembledependingonthesymmetrypropertiesofthesystem(e.g.time-reversibilityforourquantumbilliardscorrespondtotheGaussianOrthogonalEnsemble).ThisisknownastheBohigas-Giannoni-SchmidtConjecture[5]. Figure9{EnergylevelsforthecardioidbilliardcomparedtothoseoftheGaussianensemblesandPoissonianstatistics(dataandpicturefrom[2]).Numericalexperimentswereper-formedin[5]givingacorrespondencebetweentheeigenvaluespacingsstatis-ticsforSinai'sbilliardandthoseoftheGaussianOrthogonalEnsemble.Thejointgraphs,byA.Backer,presentsimi-larexperimentsforanintegrablebilliard(Figure8)andachaoticone(Figure9).ThesestatisticsareperfectlycoherentwithboththeBerry-TaborandtheBohigas-Giannoni-Schmidtconjectures.ThisdeepenstheinterestintheseRan-domMatrixTheorydistributions,whichappearincreasinglyinmanyelds,in-cludinganalyticnumbertheory.References[1]G.W.Anderson,A.Guionnet,O.Zeitouni,AnIntroductiontoRandomMa-trices,CambridgeUniversityPress,2009.[2]A.Backer,Ph.D.thesis,UniversitatUlm,Germany,1998.[3]M.V.Berry,J.P.Keating,TheRiemannzerosandeigenvalueasymptotics,SIAMReview41(1999),236{266.[4]M.V.Berry,M.Tabor,Levelclusteringintheregularspectrum,Proc.Roy.Soc.Lond.A356(1977),375{394.[5]O.Bohigas,M.-J.Giannoni,C.Schmidt,Characterizationofchaoicquantumspectraanduniversalityoflevel uctuationlaws,Phys.Rev.Lett.52(1984),1{4. Vol.00,2012TeaTimeinPrinceton13[6]J.Derbyshire,PrimeObsession:BernhardRiemannandtheGreatestUnsol-vedProbleminMathematics(PlumeBooks,2003)[7]X.Gourdon,The1013rstzerosoftheRiemannZetafunction,andzeroscomputationatverylargeheight.[8]J.P.Keating,N.C.Snaith,Randommatrixtheoryandnumbertheory,inTheHandbookonRandomMatrixTheory,491{509,editedbyG.Akemann,J.Baik&P.DiFrancesco,OxforduniversityPress,2011.[9]M.L.Mehta,Randommatrices,Thirdedition,PureandAppliedMathematicsSeries142,Elsevier,London,2004.[10]H.L.Montgomery,Thepaircorrelationofzerosofthezetafunction,Analy-ticnumbertheory(ProceedingsofSymposiuminPureMathemathics24(St.LouisUniv.,St.Louis,Mo.,1972),AmericanMathematicalSociety(Provi-dence,R.I.,1973),pp.181{193.[11]A.M.Odlyzko,Onthedistributionofspacingsbetweenthezerosofthezetafunction,Math.Comp.48(1987),273{308.[12]N.M.Katz,P.Sarnak,RandomMatrices,FrobeniusEigenvaluesandmono-dromy,AmericanMathematicalSocietyColloquiumPublications,45.Ameri-canMathematicalSociety,Providence,Rhodeisland,1999.[13]B.Riemann,UberdieAnzahlderPrimzahlenuntereinergegebenenGrosse,MonatsberichtederBerlinerAkademie,GesammelteWerke,Teubner,Leipzig,1892.[14]Z.Rudnick,P.Sarnak,ZeroesofprincipalL-functionsandrandommatrixtheory,DukeMath.J.81(1996),no.2,269{322.AcelebrationofJohnF.Nash.[15]E.C.Titchmarsh,TheTheoryoftheRiemannZetaFunction,London,OxfordUniversityPress,1951.