A Central Limit Theorem for the spectrum of the mo

arXiv:math-ph/0406015v314Nov2004ACENTRALLIMITTHEOREMFORTHESPECTRUMOFTHEMODULARDOMAINZE´EVRUDNICKThestatisticsofthehigh-lyingeigenvaluesoftheLaplacianonaRie-mannianmanifoldhavebeenintensivelystudiedinthepastfewyearsbyphysicistsworkingon“quantumchaos”.Anumberoffundamentalinsightshaveemergedfromthesestudies,thoughtodatethesehaveyettobesetonrigorousfooting.Inthecasethemanifoldathandisofarithmeticorigin,thesestudiesarerelatedtosomeprofoundnumbertheoreticalproblemsandassuchmaybemoreamenabletoinvestiga-tion.InthisnoteImakeuseofthearithmeticstructureofthemodulardomaintoestablishGaussianfluctuationsinitsspectrumforcertainsmoothcountingfunctions.Contents1.Background21.1.Numbervariance21.2.Fluctuations31.3.Themodulardomain42.Formulationofresults52.1.Definitionofthesmoothcountingfunctions52.2.Theresults63.Themodulargroup73.1.Conjugacyclasses73.2.Theamplitudeβ(n)84.Thelengthspectrum95.AnexpansionforNf,L115.1.TheSelbergTraceFormula115.2.TransformingNf,L126.ThemeanandvarianceofSf,L13Date:November10,2004.AversionofthispaperwaspresentedintheIAS/ParkCityMathematicsInsti-tutesummersessiononAutomorphicFormsandApplicationsinJuly2002aspartoftheauthor’smini-courseonArithmeticQuantumChaos.SupportedbyagrantfromtheIsraelScienceFoundation,foundedbytheIsraelAcademyofSciencesandHumanitiesandaLeverhulmeTrustLinkedFellowshipatBristolUniversity.12ZE´EVRUDNICK6.1.Theaveragingprocedure136.2.TheexpectedvalueofSf,L146.3.ThevarianceofSf,L147.Highermoments167.1.Reductiontothepre-diagonal167.2.Off-diagonalterms177.3.Thediagonalcontribution198.Conclusion19References211.BackgroundTosetthestage,Istartwithdescribingsomeofwhatiscurrentlybelievedtoholdforthestatisticsoftheeigenvalues.GivenacompactRiemanniansurfaceM,Weyl’slawfortheeigenvaluesEjoftheLapla-ciansaysthatthenumberofeigenvaluesbelowxgrowslinearlywithx:#{Ej≤x}∼area(M)4πx,asx→∞.Letn(E,L)bethenumberoflevelsinawindowaroundEforwhichtheleadingterminWeyl’slawpredictsLlevels:n(E,L)=#{E−2πarea(M)·LEjE+2πarea(M)·L}andmoregenerallyforatestfunctionfdefine(1.1)nf(E,L):=Xjf(area(M)4π·(Ej−E)L)whichcountsthelevelslyingina“soft”windowoflength4πarea(M)LaboutE.IntheaboveL=L(E)dependsonthelocationE.Inwhatfollowswewillusuallywriten(L)forn(E,L),thedependenceonEimplicitlyunderstood.Tostudythestatisticalbehaviourofn(L)weneedtoconsiderEasrandom,drawnfromacertaindistributionontheline.Wedenotebyh·ithiskindofenergyaveraging,e.g.hFi=1ER2EEF(E′)dE′.Weyl’slawleadsustoexpectthatthemeanvalueofn(L)isLandlikewisethatofnf(L)isL·Rf(x)dx.1.1.Numbervariance.Thevarianceofnf(E,L)fromitsexpectedvalueis:Σ2f(E,L)=|nf(L)−hnf(L)i|2CLTFORTHESPECTRUMOFTHEMODULARDOMAIN3ItiscustomarytoexpressthenumbervariancebymeansofanintegralkernelKE(τ),calledthe“formfactor”,sothatasE→∞Σ2f(E,L)∼L·Z∞−∞bf(u)2KE(uL)du=Z∞−∞(Lbf(Lτ))2KE(τ)dτ.wherebf(y)=R∞−∞f(x)e−2πixydxistheFouriertransformoff.For“generic”surfaces,Berry[3,4]arguedthatasE→∞,thebehaviourofΣ2f(E,L)forLintherange1(1.2)1≺≺L≺≺Lmax=√Eisuniversal,dependingonlyonthecoarsedynamicalnatureofthegeodesicflowonthesurface,andfollowsthatofoneofasmallnumberofrandommatrixensembles:Iftheflowisintegrable(asinthecaseofaflattorus)thenΣ2f(E,L)∼L·R∞−∞f(x)2dxforL→∞,asinthePoissonmodelofuncorrelatedlevels.Iftheflowischaotic(asinthecaseofnegativecurvature)thenthebehaviourisasintheGaussianOrthogonalEnsemble(GOE):Forthesharpwindow(f=1[−1/2,1/2]),thisisgivenbyΣ2(E,L)∼2π2logLforL→∞.Forsufficientlysmoothf,intheGOEwehaveΣ2f(E,L)∼2R∞−∞bf(u)2|u|du,thatisthevarianceofsufficientlysmoothstatisticstendstoafinitevalueasL→∞.TheformfactorsfortherandommodelsareKpois(τ)≡1,andKGOE(τ)=(2|τ|−|τ|log(1+2|τ|),|τ|≤12−|τ|log1+2|τ|2|τ|−1,|τ|1.Itistobeemphasizedthattheabovebehaviourisonlyvalidintheuniversalregime1≺≺L≺≺√E;forL≻≻√Etheintegrablecaseisfairlywellunderstood(atarigorouslevel),seethesurvey[5]:Thevariancegrowsas√E(aclassicalresult[12]inthecaseofthestandardflattorus).Inthechaoticcaseitisbelieved[3,4]thatgenerically,thenumbervariancecontinuestobesmallasintheuniversalregime.1.2.Fluctuations.Ourmaininteresthereisinthevaluedistributionofthenormalizedlinearstatisticnf(E,L)−hnf(L)iqΣ2f(E,L)asEvaries.Inallthestatisticalmodels(PoissonandGOE/GUE),itisastandardGaussian[22,14,13].1Thesymbolf(x)≺≺g(x)meansthatf(x)/g(x)→04ZE´EVRUDNICKIntheintegrablecase,whenL≻≻√E,thedistributionisknown([17],[5]),andisdefinitelynotGaussian.Insidetheuniversalregime(1.2),thedistributionisbelievedtobeGaussianinboththeintegrable[6]andchaotic[1,9,25]cases.Inthespecialcaseofthestandardflattorus,thishasbeenprovedinasmallpartoftheuniversalregimenear√E[18].1.3.Themodulardomain.Westartwiththeupperhalf-planeH={x+√−1y:y0}equippedwiththehyperbolicmetricds2=y−2(dx2+dy2),whichhasconstantcurvatureequal−1.TheLaplace-BeltramioperatorforthismetricisgivenbyΔ=y2(∂2∂x2+∂2∂y2).Theorientation-preservingisometriesofthemetricds2arethelinearfrac-tionaltransformationsPSL(2,R)=SL(2,R)/{±I}.ThemodulardomainistheRiemannsurfaceobtainedbyidentifyingpointsintheupperhalfplanewhichdifferbyalinearfractionaltrans-formationwithintegercoefficients,thatisbyelementsofthemodulargroupΓ:=PSL(2,Z)=SL(2,Z)/{±I}.Theresultingsurface