Extreme value problems in Random Matrix Theory and

arXiv:cond-mat/0702244v1[cond-mat.stat-mech]9Feb2007ExtremevalueproblemsinRandomMatrixTheoryandotherdisorderedsystemsGiulioBiroli1,3,Jean-PhilippeBouchaud2,3,MarcPotters3February3,20081ServicedePhysiqueTh´eorique,OrmedesMerisiers–CEASaclay,91191GifsurYvetteCedex,France.2ServicedePhysiquedel’´EtatCondens´e,OrmedesMerisiers–CEASaclay,91191GifsurYvetteCedex,France.3Science&Finance,CapitalFundManagement,6BdHaussmann,75009Paris,France.AbstractWereviewsomeapplicationsofcentrallimittheoremsandextremevaluesstatisticsinthecontextofdisorderedsystems.Wediscussseveralproblems,inparticularconcerningRandomMatrixTheoryandthegen-eralisationoftheTracy-Widomdistributionwhenthedisorderhas“fattails”.Weunderlinetherelevanceofpower-lawtailsforDirectedPoly-mersandmean-fieldSpinGlasses,andwepointoutvariousopenproblemsandconjecturesonthesematters.WefindthatinmanyinstancestheassumptionofGaussiandisordercannotbetakenforgranted.1IntroductionMoststatisticalmodelsofdisorderstartbyassumethatrandomnesshasGaussianstatistics–fromtheclassicBrownianmotiontotheEdwards-Anderson(orDerrida)modelsofspin-glasses,Kraichnanmodelsofturbulentflows,KPZmodelsofsurfacegrowth,Black-Scholesmodelsoffinancialmarkets,etc.ThankstotheoutstandingmathematicalpropertiesofGaussianrandomvariables,thisassumptionisoftentech-nicallyveryconvenientandallowsonetousepowerfulanalyticaltechniques:stochasticcalculusandIto’slemma,fieldtheoryandreplicas,etc..Thereal(andoftenimplicit)justificationishowevertheexistenceofaCentralLimitTheorem.Thisshouldensurethatonelargeenoughlengthsscalesortimescales,thephysicalresultsareuniversal,independentofthedetailsofthemicroscopicrandomness–whichcantherefore,forsimplicityandcongeniality,bechosenasGaussian.TheparadigmofsuchamechanismistheBrownianmotion;inthiscase,providedelementaryhopsaresufficientlydecor-relatedfromoneanother,itiswellknownthatthesumofaverylargenumberofthese1smalldisplacementsleadstoaGaussiandiffusionprofile,quiteindependentlyofthedistributionofelementaryhops–wheneveritssecondmomentisfinite.Ifthesecondmomentdiverges,thewalkbecomesaL´evyflight,withanomalousdiffusiondescribedbythegeneralizedCentralLimitTheoremofL´evyandGnedenkowhichagainensuresacertaindegreeofuniversality[1]:onlytheextremetailsofthemicroscopicdistri-butionmatterinthemacroscopiclimit.Althoughthisdichotomybetweenfiniteandinfinitevarianceisasymptoticallyrigorous,finitetimeorsizeeffectscanbestrongandleadtoeffectiveviolationsoftheseCentralLimitTheorems.Animportantexampleiswhenthedistributionofelementaryhopshasafinitevariancebutpower-lawtails.Inthiscase,fattaileffectsarepersistentandconvergencetowardsGaussiandiffusionisveryslow.Thisisparticularlyrelevantinfinance,wheresignificantdeviationsfromGaussianstatisticsareobservedevenforlongtimelags[2].SumsofNiidrandomvariablesthereforeprovideabeautifulillustrationofuniver-salityanduniversalityclasses,aconceptthatextendsfarbeyondthissimple,exactlysolubleexample.TheexistenceofgeneralizedCentralLimitTheoremsformorecom-plicated(nonlinear)problemsinvolvingrandomvariablesshouldbegeneric,againleadingtosomeuniversality–universalityclassesshouldhoweverbedeterminedonacasebycasebasisandmightbedifferentfromtheL´evy-Gnedenkoclassification.Awellknownexampleisthestatisticsofextremevalues,saythelargestofNindependentrandomvariablesxi.Inthiscaseagain,thelimitingdistributionbecomestosomedegreeuniversal;onehastodistinguishthreedifferentcases,dependingonthe‘micro-scopic’distributionp(x):Weibull(fordistributionsp(x)whichstrictlyvanishbeyondafinitevaluex∗),Gumbel-Fisher-Tippett(fordistributionsdecayingfasterthananypower-law)andFr´echet(forpower-lawdistributions)[3].Interestingly,itispossibletoformulateaproblemwhichinterpolatesbetweensumsofrandomvariablesandextremesofrandomvariables,byconsideringthefollowingquantity:Sq=NXi=1xqi#,(1)whereoneassumesforsimplicitythatxi’sareallpositive.Clearly,q=1correspondstoasimplesum,whereaswhenq→∞atfixedN,S1/qqconvergestothelargestelementxmax.Definingxi≡exp−εi,itisclearthatSqplaystheroleofapartitionfunctionandqistheinversetemperatureforageneralizedRandomEnergyModel(REM),wheretheenergiesεi’sarenotnecessarilyGaussian.Thisproblemwasconsideredin[4,5]andhas,beyondtheREMinterpretation,manydifferentapplications.Forexample,supposeεiisagrowthrateofspecieiinthepopulation,orthereturnofassetiinaportfolio,andqisthetime.ThenSqisthetotalpopulationaftertimeqorthetotalvalueoftheportfolioaftertimeq;thedetailedstatisticsoftheseobjectsisthereforequiteinteresting.1TheresultdependsontherelativevalueofqandNwhenbothdivergetoinfinity.Moreprecisely,takingforsimplicityεitobeGaussianwithvarianceσ2,therelevantparameterisμ=√2lnN/qσ.ThestatisticsofSq(N)onlydependsonμand,quiteinterestingly,closelyfollowstheaboveGauss/L´evydichotomy:forμ2,Sq1Morecomplexsituations,forexamplediffusionofspeciesinarandomenvironment,canbeanalyzedalongsimilarlines[6].2isGaussian;forμ2itbecomesL´evydistributedandmoreandmoredominatedbyextremevalues.Infact,assoonasμ1,thewholesumSqiswellapproximatedbyafinitenumberofterms,whereaswhenμ→0,onlythelargestsurvives.Thetransitionatμ=1correspondsexactlytheglasstr