Freeness and the Transpose ( Matrices Just Wanna be Free ) Jamie - PowerPoint PPT Presentation

Freeness and the Transpose ( Matrices Just Wanna be Free ) Jamie Mingo (Queen’s University) with Mihai Popa Recent Developments in Quantum Groups, Operator Algebras and Applications, February 6, 2015 1 / 20

random variables and their distributions ◮ ( A , ϕ ) unital algebra with state; ◮ C � x 1 , . . . , x s � is the unital algebra generated by the non-commuting variables x 1 , . . . , x s ◮ the distribution of a 1 , . . . , a s ∈ ( A , ϕ ) is the state µ : C � x 1 , . . . , x s � → C given by µ ( p ) = ϕ ( p ( a 1 , . . . , a s )) ◮ convergence in distribution of { a ( N ) , . . . , a ( N ) } ⊂ ( A N , ϕ N ) s 1 to { a 1 , . . . , a s } ⊂ ( A , ϕ ) means pointwise convergence of distributions: µ N ( p ) → µ ( p ) for p ∈ C � x 1 , . . . , x s � . freeness ◮ A 1 , A 2 ⊆ A unital subalgebras are free if given ◮ a 1 , · · · , a n ∈ A with ϕ ( a i ) = 0 for all i , ◮ a i ∈ A j 1 with j 1 � · · · � j n we have ϕ ( a 1 · · · a n ) = 0, ◮ a 1 and a 2 are free if alg ( 1, a 1 ) and alg ( 1, a 2 ) are free, 2 / 20

freeness and asymptotic freeness ◮ { a ( N ) , . . . , a ( N ) } ⊂ ( A N , ϕ N ) are asymptotically free if µ n → µ s 1 and x 1 , . . . , x s are free with respect to µ ◮ if a and b are free with respect to ϕ then ϕ ( abab ) = ϕ ( a 2 ) ϕ ( b ) 2 + ϕ ( a ) 2 ϕ ( b 2 ) − ϕ ( a ) 2 ϕ ( b ) 2 ◮ in general if a 1 , . . . , a s are free then all mixed moments ϕ ( x i 1 · · · x i n ) can be written as a polynomial in the moments of individual moments { ϕ ( a k i ) } i , k . ◮ a ( N ) , . . . , a ( N ) ∈ ( A n , ϕ N ) are asymptotically free if s 1 whenever we have b ( N ) ∈ alg ( 1, a ( N ) ) is such that i j i ϕ N ( b ( N ) ) = 0 and j 1 � j 2 � · · · � j m we have i ϕ N ( b ( N ) · · · b ( N ) m ) → 0 1 3 / 20

simple distributions: Wigner and Marchenko-Pastur 2 π e − t 2 / 2 be the density of the Gauss law 1 ◮ let f ( t ) = √ f ( is )) = s 2 ∞ s n � ◮ then log ( ˆ 2 = k n n ! with k 2 = 1 and k n = 0 for n = 1 n � 2, so the Gauss law is characterized by having all cumulants except k 2 equal to 0 ◮ µ a probability measure on R , z ∈ C + , � ( z − t ) − 1 d µ ( t ) is the Cauchy transform of µ and G ( z ) = z = κ 1 + κ 2 z + κ 3 z 2 + · · · is the R ( z ) = G � − 1 � ( z ) − 1 R -transform of µ √ 4 − t 2 dt is the semi-circle law we have κ n = 0 1 ◮ if d µ ( t ) = 2 π except for κ 2 = 1 ◮ if 0 < c and a = ( 1 − √ c ) 2 and b = ( 1 + √ c ) 2 we let √ ( b − t )( t − a ) d µ = dt for c � 1 and √ 2 π t ( b − t )( t − a ) d µ = ( 1 − c ) δ 0 + dt for 0 < c < 1, the 2 π t Marchenko-Pastur distribution: κ n = c for all n 4 / 20

GUE random matrices and asymptotic freeness 1 ◮ X N = X ∗ N = √ ( x ij ) ij a N × N self-adjoint random matrix N with x ij independent complex Gaussians with E ( x ij ) = 0 and E ( | x ij | 2 ) = 1 ( modulo self-adjointness) ◮ λ 1 � λ 2 � · · · � λ N eigenvalues of X N , µ N = 1 N ( δ λ 1 + · · · + δ λ N ) is the spectral measure of X N , � t k d µ N ( t ) = tr ( X k N ) 0.3 X N is the N × N GUE with limiting 0.2 eigenvalue distribution given by ◮ 0.1 Wigner’s semi-circle law - - 2 1 0 1 2 ◮ Y N another GUE with entries independent from those of X N ◮ for large N mixed moments of X N and Y N are close to those of freely independent semi-circular operators (thus asymptotically free ) 5 / 20

Wishart Random Matrices ◮ Suppose G 1 , . . . , G d 1 are d 2 × p random matrices where G i = ( g ( i ) jk ) jk and g ( i ) jk are complex Gaussian random variables with mean 0 and (complex) variance 1, i.e. E ( | g ( i ) jk | 2 ) = 1. Moreover suppose that the random variables { g ( i ) jk } i , j , k are independent. ◮   G 1 1 1 . � � G ∗ G ∗ ( G i G ∗ . W = · · · = j ) ij   . 1 d 1 d 1 d 2 d 1 d 2   G d 1 is a d 1 d 2 × d 1 d 2 Wishart matrix. We write W = d − 1 1 ( W ( i , j )) ij as d 1 × d 1 block matrix with each entry the d 2 × d 2 matrix d − 1 2 G i G ∗ j . 6 / 20

Partial Transposes on M d 1 ( C ) ⊗ M d 2 ( C ) · G i a d 2 × p matrix · W ( i , j ) = 1 d 2 G i G ∗ j , a d 2 × d 2 matrix, · W = 1 d 1 ( W ( i , j )) ij is a d 1 × d 1 block matrix with entries W ( i , j ) · W T = 1 d 1 ( W ( j , i ) T ) ij is the “full” transpose Γ = 1 · W d 1 ( W ( j , i )) ij is the “left” partial transpose · W Γ = 1 d 1 ( W ( i , j ) T ) ij is the “right” partial transpose p · we assume that → c , 0 < c < ∞ d 1 d 2 · eigenvalue distributions of W and W T converge to Marchenko-Pastur with parameter c and W Γ converge to a shifted Γ · eigenvalues of W semi-circular with mean c and variance c (Aubrun, 2012) · W and W T are asymptotically free (M. and Popa, 2014) · what about W Γ and W Γ ? 7 / 20

Semi-circle and Marchenko-Pastur Distributions p Suppose → c . d 1 d 2 ◮ limit eigenvalue distribution of W (Marchenko-Pastur) � b � ( b − t )( t − a ) � lim E ( tr ( W n )) = t n c # ( σ ) dt = 2 π t a σ ∈ NC ( n ) # ( σ ) is the number of blocks of σ , a = ( 1 − √ c ) 2 , and b = ( 1 + √ c ) 2 ◮ limit eigenvalue distribution of W Γ (semi-circle) � � c # ( σ ) = lim E ( tr (( W Γ ) n )) = κ π σ ∈ NC 1,2 ( n ) π ∈ NC ( n ) NC 1,2 ( n ) is the set of non-crossing partitions with only blocks of size 1 and 2. ( c.f. Fukuda and ´ Sniady (2013) and Banica and Nechita (2013)) 8 / 20

main theorem Γ , W Γ , W T } form an asymptotically ◮ thm : The matrices { W , W free family · let ( ǫ , η ) ∈ { − 1, 1 } 2 = Z 2 2 .  W if ( ǫ , η ) = ( 1, 1 )    Γ W if ( ǫ , η ) = (− 1, 1 ) · let W ( ǫ , η ) = W Γ if ( ǫ , η ) = ( 1, − 1 )    W T if ( ǫ , η ) = (− 1, − 1 ) · let ( ǫ 1 , η 1 ) , . . . , ( ǫ n , η n ) ∈ Z n 2 E ( Tr ( W ( ǫ 1 , η 1 ) · · · W ( ǫ n , η n ) )) � p � # ( σ ) � d f ǫ ( σ )+ # ( σ )− n d f η ( σ )+ # ( σ )− n = 1 2 d 1 d 2 σ ∈ S n where f ǫ ( σ ) = # ( ǫδγ − 1 δγδǫ ∨ σδσ − 1 ) ( “ ∨ ” means the sup of partitions and # means the number of blocks or cycles ) 9 / 20

Computing Moments via Permutations, Notation ◮ [ d 1 ] = { 1, 2, . . . , d 1 } , ◮ given i 1 , . . . , i n ∈ [ d 1 ] we think of this n -tuple as a function i : [ n ] → [ d 1 ] ◮ ker ( i ) ∈ P ( n ) is the partition of [ n ] such that i is constant on the blocks of ker ( i ) and assumes di ff erent values on di ff erent blocks ◮ if σ ∈ S n we also think of the cycles of σ as a partition and write σ � ker ( i ) to mean that i is constant on the cycles of σ ◮ given σ ∈ S n we extend σ to a permutation on [ ± n ] = { − n , . . . , − 1, 1, . . . , n } by setting σ (− k ) = − k for k > 0 ◮ γ = ( 1, 2, . . . , n ) , δ ( k ) = − k ◮ given ǫ 1 , . . . , ǫ n ∈ { − 1, 1 } let ǫ ∈ S ± n be given by ǫ ( k ) = ǫ | k | · k ◮ δγ − 1 δγδ = ( 1, − n )( 2, − 1 ) · · · ( n , −( n − 1 )) 10 / 20

Computing Moments via Permutations, II ◮ δγ − 1 δγδ = ( 1, − n )( 2, − 1 ) · · · ( n , −( n − 1 )) ◮ if A k = ( a ( k ) ij ) ij , a N × N matrix, then N � � a ( 1 ) i 1 i 2 a ( 2 ) i 2 i 3 · · · a ( n ) a ( 1 ) i 1 i − 1 a ( 2 ) i 2 i − 2 · · · a ( n ) Tr ( A 1 · · · A n ) = i n i 1 = i n i − n i 1 ,..., i n = 1 i ± 1 ,..., i ± n δγ − 1 δγδ � ker ( i ) W ( ǫ 1 , η 1 ) · · · W ( ǫ n , η n ) � � Tr � �� � d − n W ( ǫ 1 , η 1 ) � W ( ǫ n , η n ) ) i n i 1 � = Tr i 1 i 2 · · · 1 i 1 ,..., i n � �� � d − n W ( ǫ 1 , η 1 ) � � W ( ǫ n , η n ) ) i n i − n = Tr i 1 i − 1 · · · 1 i ± 1 ,..., i ± n � � W ( j 1 , j − 1 ) ( η 1 ) · · · W ( j n , j − n ) ( η n ) � d − n = Tr 1 j ± 1 ,..., j ± n where δγ − 1 δγδ � ker ( i ) , ǫδγ − 1 δγδǫ � ker ( j ) and j = i ◦ ǫ 11 / 20

Example of Twisting n = 5, ǫ = ( 1, 1, − 1, − 1, 1 ) δγ − 1 δγ = ( 1, − 5 )( 2, − 1 )( 3, − 2 )( 4, − 3 )( 5, − 4 ) j 1 j − 1 j 2 j − 2 j 3 j − 3 j 4 j − 4 j 5 j − 5 ǫδγ − 1 δγǫ = ( 1, − 5 )( 2, − 1 )( 3, − 4 )(− 3, − 2 )( 4, 5 ) j 1 j − 1 j 2 j − 2 j 3 j − 3 j 4 j − 4 j 5 j − 5 12 / 20

Computing Moments via Permutations, III � W ( ǫ 1 , η 1 ) · · · W ( ǫ n , η n ) � � W ( j 1 , j − 1 ) ( η 1 ) · · · W ( j n , j − n ) ( η n ) � = d − n � Tr Tr 1 j ± 1 ,..., j ± n with ǫδγ − 1 δγδǫ � ker ( j ) . Let s = r ◦ η then for δγ − 1 δγδ � ker ( r ) W ( j 1 , j − 1 ) ( η 1 ) · · · W ( j n , j − n ) ( η n ) � � Tr � W ( j 1 , j − 1 ) ( η 1 ) ) r 1 r − 1 · · · W ( j n , j − n ) ( η n ) � � � = r n r − n r ± 1 ,..., r ± n � � � � � = W ( j 1 , j − 1 ) s 1 s − 1 · · · W ( j n , j − n ) s n s − n s ± 1 ,..., s ± n � d − n G j 1 G ∗ G j n G ∗ � � � � = s 1 s − 1 · · · j − 1 j − n 2 s n s − n s ± 1 ,..., s ± n � � g ( j 1 ) s 1 t 1 g ( j − 1 ) s − 1 t 1 · · · g ( j n ) s n t n g ( j − n ) d − n = s − n t n 2 s ± 1 ,..., s ± n t 1 ,..., t n 13 / 20