Limit Theorems for Products of Large Random Matrices Alexander - PowerPoint PPT Presentation

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples The Notation ◮ Denote by M p , q the space of p × q matrices. ◮ Let M = M n o , n 1 ⊗ M n 1 , n 2 ⊗ · · · ⊗ M n m − 1 , n m . ◮ Let F denote a matrix-value map F : M → M n , p (3) with some p = p ( n ) ≥ n . ◮ We define matrix F X = ( f jk ) = F ( X ( 1 ) , . . . , X ( m ) ) . ◮ We shall interesting for spectra of matrices W X ( α ) = ( F X − α I )( F X − α I ) ∗ (4) for any α = x + iy ∈ C . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples The Notation ◮ Let Y ( q ) be independent Gaussian random variables with jk covariance cov ( Re Y jk , Im Y ( q ) jk ) = cov ( Re X ( q ) jk , Im X ( q ) jk ) . ◮ We shall assume that Y ( q ) and X ( q ) jk , for q = 1 , . . . , m , are jk defined on the same probability space and mutually independent. ◮ We shall consider matrices Y ( q ) = √ n q − 1 ( Y ( q ) 1 jk ) , for q = 1 , . . . , m , and 1 ≤ j ≤ n q − 1 , 1 ≤ k ≤ n q . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples The Notation ◮ Denote by F Y := F ( Y ( 1 ) , . . . , Y ( m ) ) and W Y ( α ) = ( F Y − α I )( F Y − α I ) ∗ .Here and in what follows I denotes the unit matrix of corresponding dimension. ◮ To compare asymptotic behaviour of empirical spectral distributions of matrices W X ( α ) and W Y ( α ) we introduce the matrices     O F X O F Y   ,   . V X = V Y = F ∗ F ∗ O O X Y A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples The Notation ◮ We define the matrix   O α I   , J ( α ) = α = x − iy , α I O and consider shifted matrices V X ( α ) := V X − J ( α ) and V X ( α ) := V X − J ( α ) . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples ◮ Furthermore, we denote by s 2 1 ( X , α ) ≥ . . . ≥ s 2 n ( X , α ) the eigenvalues of matrix W Y ( α ) and by s 2 1 ( Y , α ) ≥ . . . ≥ s 2 n ( Y , α ) the eigenvalues of matrix W Y ( α ) correspondingly. A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples ◮ In these notation the eigenvalues of matrices V X ( α ) and V Y ( α ) are ± s 2 1 ( X , α ) , . . . ± s 2 ± s 2 1 ( Y , α ) , . . . ± s 2 n ( X , α ) and n ( Y α ) ◮ Define the empirical spectral distribution of matrices W X ( α ) ( W Y ( α ) resp.) and V X ( α ) ( V Y ( α ) resp.) n � G n ( x , X , α ) := 1 I { s 2 j ( X , α ) ≤ x } , n j = 1 n n � � G n ( x , X , α ) := 1 I { s j ( X , α ) ≤ x } + 1 � I {− s j ( X , α ) ≤ x } . 2 n 2 n j = 1 j = 1 A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples ◮ Here I { B } denotes indicator of event B . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples ◮ The distributions G and � G are connected by formula G ( x ) = 1 + sign ( x ) G ( x 2 ) � . 2 ◮ We introduce now the resolvent matrices R X ( α, z ) = ( V X ( α ) − z I ) − 1 , R Y ( α, z ) = ( V Y ( α ) − z I ) − 1 . ◮ We define the following matrices Z ( q ) = X ( q ) cos ϕ + Y ( q ) sin ϕ, for any ϕ ∈ [ 0 , π 2 ] and any q = 1 , . . . , m . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples ◮ Let F ( ϕ ) = F ( Z ( 1 ) ( ϕ ) , . . . , Z ( m ) ( ϕ )) , V ( α, ϕ ) = V Z ( α ) . ◮ We have F ( π V ( π F ( 0 ) = F X , 2 ) = F Y , V ( α, 0 ) = V X ( α ) , 2 ) = V Y ( α ) . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples ◮ Furthermore, we define the corresponding resolvent matrices R := R ( z , α, ϕ ) = ( V ( α, ϕ ) − z I ) − 1 . ◮ Stieltjes transform of singular values distribution of matrix V ( α, ϕ ) , m n ( z , α, ϕ ) := 1 2 n Tr R ( z , α, ϕ ) . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Lindeberg condition We shall assume that r.v.’s X ( q ) satisfy the Lindeberg condition, jk i.e. n q − 1 n q � � √ 1 E | X ( q ) jk | 2 I {| X ( q ) L n ( τ ) = max jk | > τ n } → 0 as n → ∞ , n 2 1 ≤ q ≤ m j = 1 k = 1 for any τ > 0 . (5) A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples To formulate the conditions on the function F we need some additional notations.In what follows we shall omit argument ϕ in the notation. ◮ Define the function ∂ V g ( q ) := g ( q ) jk ( Z ( 1 ) , . . . , Z ( m ) ) := Tr R 2 . jk ∂ Z ( q ) jk ◮ Let θ be random variables distributed in [ 0 , 1 ] and independent on all Z ( q ) jk . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples ◮ We shall assume that there exist constants A 1 > 0 and A 2 > 0 and τ 0 > 0 such that � ∂ g ( q ) � � �� jk � Z ( q ) sup � E ( θ ) � ≤ A 1 a.s. , (6) jk ∂ Z ( q ) q , n , j , k ,ϕ jk ◮ and, for any τ ≤ τ 0 , � ∂ 2 g ( q ) � � �� √ � � � jk I {| Z ( q ) � Z ( q ) sup jk | ≤ τ n } � E 2 ( θ ) � ≤ A 2 a.s. (7) jk ∂ Z ( q ) q , n , j , k jk A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Universality of singular values distribution Theorem 2.1 Let X ( q ) jk ’s and Y ( q ) jk ’s be random variables as described above and assume that X ( q ) satisfy the Lindeberg condition (5) . jk Assume that function F is such that the conditions (6) and (7) hold. Then | E m n ( z , α, π 2 ) − E m n ( z , α, 0 ) | → 0 as n → ∞ . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Remark 2.2 Under conditions of Theorem 2.1 the expected distribution function of singular value of matrix F X ( α ) has the same limit as distribution function of singular values of matrix F Y ( α ) . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Example ◮ For m = 1 and F ( X ) = X    2 [ R 2 ] jk , for j � = k g jk =   [ R 2 ] jj , otherwise . ◮ It is straightforward to check that | ≤ Cv − 3 , | ∂ 2 g jk | ∂ g jk | ≤ Cv − 4 , ∂ Z 2 ∂ Z jk jk for z = u + iv . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Let µ a probability measure on the complex plane. Define the logarithmic potential of measure µ as � U µ ( α ) = − log | α − ζ | d ζ. C Let µ X (resp. µ Y ) denote the empirical spectral measure of the matrix F X (resp. F Y ), i.e. µ X (resp. µ Y ) is the uniform distribution on the eigenvalues { λ 1 ( X ) , . . . λ n ( X ) } (resp. { λ 1 ( Y ) , . . . λ n ( Y ) } of the matrix F X (resp. F Y ).Then � n � log | α − ζ | d µ X ( ζ ) = − 1 U X ( α ) = − log | λ j ( X ) − α | , n C j = 1 � n � U Y ( α ) = − 1 log | α − ζ | d µ Y ( ζ ) = − log | λ j ( Y ) − α | . n C j = 1 A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Let G X ( x , α ) = E G X ( x , α ) . We may represent � ∞ U X ( α ) = log | x | dG X ( x , α ) . −∞ The function log | x | is uniformly integrated with respect to distribution functions G X ( x , α ) if � ∞ � � t →∞ lim sup lim Pr | log | x | dG X ( x , α ) | > t = 0 . (8) n →∞ −∞ A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Definition 1 Let random matrices X ( 1 ) , . . . , X ( m ) be independent random matrices of order n 0 × n 1 , . . . n m − 1 × n m respectively. Assume that random variables X ( q ) are mutually independent , for jk q = 1 , . . . , m and j = 1 , . . . , n q − 1 , k = 1 . . . , n q . Let E X ( q ) = 0, jk jk | 2 = 1 and random variables X ( q ) E | X ( q ) have uniformly jk integrated second moment, i.e. E | X ( q ) jk | 2 I {| X ( q ) sup jk | > M } → 0 as n → ∞ . q , j , k , n Then we say that matrices X ( 1 ) , . . . , X ( m ) satisfy the conditions ( C 0 ) . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Definition 2 Let matrix-valued functions F X = F ( X ( 1 ) , . . . , X ( m ) ) is such that the function log | x | is uniformly integrated with respect to singular values distribution of matrices G X ( x , α ) . Then we say that matrices F X satisfy the condition ( C 1 ) . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Theorem 3.1 Let random matrices X ( 1 ) , . . . , X ( m ) and Y ( 1 ) , . . . , Y ( m ) satisfy the conditions ( C 0 ) . Let matrices F X = F ( X ( 1 ) , . . . , X ( m ) and F Y = F ( Y ( 1 ) , . . . , Y ( m ) satisfy the condition ( C 1 ) .Assume the functions F satisfy the conditions (6) and (7) of Theorem 2.1. Then the matrices F X and F Y have the same limit distribution of eigenvalues. A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples This Proposition is bounded on the following Lemma from Bordenave and Chafai "Around circular law", Probability surveys, vol. 9(2012). Lemma 3.1 Let ( X n ) be a sequence of random matrices. Let ν n ( · , z ) be the empirical distribution function of singular values of matrix X n − z I . Suppose a.a. z ∈ C there exists a probability measure ν ( · , z ) on [ 0 , ∞ ) such 1) ν n ( · , z ) → ν ( · , z ) weak as n → ∞ in probability; 2) the function log x is uniformly integrated in probability with respect to measures ν n ( · , z ) . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Then there exists a probability measure µ on the complex plane C such that empirical spectral measures µ n of matrices X n weakly convergence to the measure µ in probability. Moreover � � ∞ U µ ( z ) = − log | ζ − z | d µ ( ζ ) = − log xd ν n ( x , z ) . (9) 0 C A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples We recall the definition of Voiculescu asymptotic freeness. Two sequences of matrices ( A n ) n ∈ N and ( B n ) n ∈ N are asymptotic free if for all k ≥ 1 and all p 1 , m 1 , . . . , p k , m k the following relations ◮ there exist measures µ A and µ B such that � 1 n E Tr A p 1 x p 1 d µ A , n = M p 1 ( A ) := lim n →∞ � 1 n E Tr B p 1 x p 1 d µ B ; lim n = M p 1 ( B ) := (10) n →∞ A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples � n →∞ E 1 ( A p 1 n − M p 1 ( A ) I )( B m 1 lim n Tr − M m 1 ( B ) I ) · · · n � × ( A p k n − M p k ( A ) I )( B m k − M m k ( B ) I ) = 0 . n (11) Consider sequences of n × n random matrices X n , and define matrices   O F n   . A n = F n ∗ O A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples For any z = u + iv , introduce matrices   O − α I   . B n = J ( α ) = − α I O We apply the definition of asymptotic freeness to matrices ( A n ) n ∈ N ) and ( B n ) n ∈ N defined in such way. Note that   | α | 2 p I 2 p ,  if m = 2 p B m n = . (12)  | α | 2 p J ( α ) ,  if m = 2 p + 1 From here it follows immediately that n ( α ) − ( lim 1 J 2 p 2 m Tr J 2 p m ( α )) I 2 p = O . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples This implies relation (11) holds if at least one of the m 1 , m 2 , . . . , m k is even. We may rewrite relation (11) for our case as follows � n →∞ E 1 1 ( A n 1 m E Tr A n 1 lim n − ( lim m ) I ) J ( α ) · · · n Tr m →∞ � 1 ( A n k m E Tr A n k n − ( lim m ) I ) J ( α ) = 0 . (13) m →∞ A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples S-transform The Voiculescu S -transform was defined for non-negative distribution. By several authors it was extend to symmetric distributions. We define Voiculescu S-transform of distribution as follows. Let M ( z ) denote the generic moment function of random variable X with distribution function F X ( x ) , � ∞ M ( z ) = � ∞ k = 1 ϕ ( X k ) z k , where ϕ ( X k ) :== −∞ x k dF X ( x ) . Let M − 1 ( z ) denote inverse function of M ( z ) w.r.t. composition of functions. A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Define S-transform of distribution F ( x ) with ϕ ( X ) � = 0, by equality S X ( z ) := z + 1 M − 1 ( z ) . z It is well-known that for free random variables ξ and η with ϕ ( ξ ) � = 0 and ϕ ( η ) � = 0 S ηξ ( z ) = S η S ξ . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Consider now the case distribution with vanishing mean. Definition 3 Let X be random variable with ϕ ( X ) = 0 and ϕ ( X 2 ) � = 0. Then its two S -transform S X and � S X are defined as follows. Let χ and χ denote two inverses under composition of the series � ∞ � ϕ ( X n ) z n = ϕ ( X 2 ) z 2 + ϕ ( X 3 ) z 3 + · · · , ψ ( z ) := (14) n = 1 then S X ( z ) := χ ( z ) 1 + z χ ( z ) 1 + z � and S X ( z ) := � and (15) z z A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Theorem 4.1 Let X and Y be free random variables such that ϕ ( X ) = 0 , ϕ ( X 2 ) � = 0 and ϕ ( Y ) � = 0 .Then S XY ( z ) = � � S XY ( z ) = S X ( z ) S Y ( z ) and S X ( z ) S Y ( z ) . (16) A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples We may interpret this equality for random matrices as follows. Let X and Y be two asymptotic free random square matrices of order n × n .Denote by µ n and ν n the empirical spectral measures of matrices XX ∗ and YY ∗ respectively. Assume that the measures µ n and ν n weakly convergence to some measures µ and ν , µ n → µ and ν n → ν .Then the spectral measure of matrix XYY ∗ X ∗ convergence to some measure µ ⊠ ν and S µ ⊠ ν ( z ) = S µ ( z ) S ν ( z ) A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples R-transform of matrix J ( α ) Introduce the following 2 n × 2 n block-matrix    O − α I  , J ( α ) = (17) − α I O where O is n × n matrix withe zero entries, and I denotes n × n unit matrix. This matrix has a spectral distribution V ( · ) = 1 2 δ | α | + 1 2 δ −| α | , and δ a denote the unit atom in the point a . We calculate now the R -transform of distribution V ( x ) . A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples R-transform of matrix J ( α ) It is straightforward to check that generic moments function M ( z ) of distribution V ( x ) defined by equality | α | 2 z 2 M ( z ) = 1 − | α | 2 z 2 . From here it follows that � M − 1 ( z ) = 1 z 1 + z . | α | and � S ( z ) = 1 1 + z . | α | z A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Here and in the what follows we denote by f − 1 inverse function with respect to composition. Using relation between S - and R - transforms, we get � z ( 1 + z ) R − 1 ( z ) = zS ( z ) = . | α | From here it follows, R 2 ( z ) + R ( z ) − | α | 2 z 2 = 0 . Solving this equation, we obtain � 1 + 4 | α | 2 z 2 R ( z ) = − 1 + . 2 A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Equations for the Stieltjes transform of limit spectral of shifted matrices Theorem 4.2 Assume that spectral measure of matrix V has a limit µ V and corresponding R-transform R V ( z ) . Assume also that matrices V and J ( α ) are asymptotically free. Then Stieltjes transform s ( z , α ) of expected spectral distribution of matrix satisfies the following system of equations A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples w = z + R α ( − s ( z , α )) (18) s ( z , α ) s ( z , α ) = ( 1 + ws ( z , α )) S V ( − ( 1 + ws ( z , α )) . (19) A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Intorduction Universality of singular value distribution Universality of eigenvalue distribution Asymptotic freeness and S -transform Examples Density of probability distribution of eigenvalues We compute the density of the limit measure of empirical spectral √ √ distribution of matrix V F .Let κ ( x , α ) = − − 1 s ( − 1 x , α ) , where x > 0 . We shall assume that distribution function G F ( x , α ) has the density with respect to Lebesgue measure, g ( x , α ) = dG F ( x ,α ) . Shall dx assume as well that � ∞ � � 1 + u 2 ∂ lim log g ( u , α ) du = 0 (20) C 2 ∂ u C →∞ −∞ A. Tikhomirov, Syktyvkar, Russia Product of random matrices

Limit Theorems for Products of Large Random Matrices Alexander - PowerPoint PPT Presentation

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