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1、RandomMatrices:Invertibility,Structure,andApplicationsRomanVershyninUniversityofMichigan2011CanadianMathematicalSocietySummerMeetingJune3,UniversityofAlberta,EdmontonRomanVershynin(UniversityofMichigan)RandomMatrices2011CMSSummerMeeting1/37ChaosandOrderManycomplexsyste
2、msthatoccurinnatureandsocietyexhibitchaosonthemicroscopiclevelandorderonthemacroscopiclevel.RomanVershynin(UniversityofMichigan)RandomMatrices2011CMSSummerMeeting2/37ChaosandOrderGasmolecules:Statisticalmechanics:randomnessatthemicroscopiclevelaveragesoutatthemacroscop
3、iclevel.RomanVershynin(UniversityofMichigan)RandomMatrices2011CMSSummerMeeting3/37ProbabilityTheoryMicroscopic:independentrandomvariablesX1;X2;:::Macroscopic:functionf(X1;:::;Xn)wherenislarge.Example:Bernoullir.v'sX=1withprobabilities1.i2Ateachgame,gain$1orlose$1indep
4、endently.Macroscopicquantity:averagegainX1++Xnf(X1;:::;Xn)=:nRomanVershynin(UniversityofMichigan)RandomMatrices2011CMSSummerMeeting4/37ProbabilityTheoryLimittheoremsdescribethemacroscopicpictureasn!1.LawofLargeNumbers:X1++Xn!0almostsurelynRomanVershynin(Universit
5、yofMichigan)RandomMatrices2011CMSSummerMeeting5/37ProbabilityTheoryCentralLimitTheorem:pX1++XnN(0;n)indistributionRomanVershynin(UniversityofMichigan)RandomMatrices2011CMSSummerMeeting6/37ProbabilityTheoryMicroscopic:independentrandomvariablesX1;X2;:::Macroscopic:f
6、unctionf(X1;:::;Xn).FunctionsmaybemorecomplexthanthesumX1++Xn.Example:randommatrixtheory.RomanVershynin(UniversityofMichigan)RandomMatrices2011CMSSummerMeeting7/37RandomMatrixTheoryMicroscopic:independentrandomvariablesXij,arrangedinamatrix01X11X12X1nBBX21X22X
7、2nCCH=@:::::::::::::::::::AXn1Xn2XnnMacroscopic:theeigenvaluesofH1(H);:::;n(H):RomanVershynin(UniversityofMichigan)RandomMatrices2011CMSSummerMeeting8/37RandomMatrixTheoryOnecanmakeHsymmetricbyplacingindependentrv'sabovethediagonalandre
ecting:Xij=XjiThisisaWigner
8、randommatrix:01X11X12X1nBBX12X22X2nCCH=@:::::::::::::::::::AX1nX2nXnnRomanVershynin(UniversityofMichigan)Ran
