A Sharp Small Deviation Inequality for the Largest Eigenvalue of a Random Matrix

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A Sharp Small Deviation Inequality for the Largest Eigenvalue of a Random Matrix Guillaume Aubrun Universite de Paris 6, Institut de Mathematiques, Equipe d'Analyse Fonctionnelle, Boite 186, 4 place Jussieu, 75005 PARIS Summary. We prove that the convergence of the largest eigenvalue ?1 of a n ? n random matrix from the Gaussian Unitary Ensemble to its Tracy-Widom limit holds in a strong sense, specifically with respect to an appropriate Wasserstein-like distance. This unifying approach allows us both to recover the limiting behaviour and to derive the inequality P(?1 > 2+t) 6 C exp(?cnt3/2), valid uniformly for all n and t. This inequality is sharp for “small deviations” and complements the usual “large deviation” inequality obtained from the Gaussian concentration principle. Following the approach by Tracy and Widom, the proof analyses several integral operators, which converge in the appropriate sense to an operator whose determinant can be estimated. Key words: Random matrices, largest eigenvalue, GUE, small deviations. Introduction Let Hn be the set of n-dimensional (complex) Hermitian matrices. The general element of Hn is denoted A(n), and its entries are denoted (a(n)ij ). We exclusively focus on the Gaussian Unitary Ensemble GUE, which can be defined by the data of a probability measure Pn on Hn which fulfills the following conditions 1.

  • find universal positive

  • widom distribution

  • gaussian law

  • random matrix

  • mn can

  • gue random

  • painleve function

  • ?1

  • universal constant

  • there exists


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ASharpSmallDeviationInequalityfortheLargestEigenvalueofaRandomMatrixGuillaumeAubrunUniversitedeParis6,InstitutdeMathematiques,EquipedAnalyseFonctionnelle,Boite186,4placeJussieu,75005PARISSummary.Weprovethattheconvergenceofthelargesteigenvalue1ofannrandommatrixfromtheGaussianUnitaryEnsembletoitsTracy-Widomlimitholdsinastrongsense,speci callywithrespecttoanappropriateWasserstein-likedistance.ThisunifyingapproachallowsusbothtorecoverthelimitingbehaviourandtoderivetheinequalityP(1>2+t)6Cexp(cnt3/2),validuniformlyforallnandt.Thisinequalityissharpfor“smalldeviations”andcomplementstheusual“largedeviation”inequalityobtainedfromtheGaussianconcentrationprinciple.FollowingtheapproachbyTracyandWidom,theproofanalysesseveralintegraloperators,whichconvergeintheappropriatesensetoanoperatorwhosedeterminantcanbeestimated.Keywords:Randommatrices,largesteigenvalue,GUE,smalldeviations.IntroductionLetHnbethesetofn-dimensional(complex)Hermitianmatrices.ThegeneralelementofHnisdenotedA(n),anditsentriesaredenoted(ai(jn)).WeexclusivelyfocusontheGaussianUnitaryEnsembleGUE,whichcanbede nedbythedataofaprobabilitymeasurePnonHnwhichful llsthefollowingconditions1.Then2randomvariables(ai(in)),(<ai(jn))i<j,(=ai(jn))i<jareindependent,)n(2.i,aiifollowstheGaussianlawN(0,1/n),3.i<j,<ai(jn)and=ai(jn)followtheGaussianlawN(0,1/2n).ThemeasurePnisuniquelydeterminedbythesethreeconditionsbecauseoftheextrasymmetryconstraintaij=aji;itcanalsobemadeexplicit.Hnisavectorspaceonwhichthescalarproducthu,vi:=tr(uv)inducesanEuclideanstructure,henceaLebesguemeasure.TheprobabilitymeasurePnhasadensitywithrespecttothisLebesguemeasure,whichcanbeshowntolauqe