Freeness and Graph Sums Jamie Mingo (Queens University) based on - PowerPoint PPT Presentation

Freeness and Graph Sums Jamie Mingo (Queen’s University) based on joint work with Roland Speicher and Mihai Popa An´ alise funcional e sistemas dinˆ amicos Universidade Federal de Santa Catarina February 23, 2015 1 / 15

GUE random matrices ◮ ( Ω , P ) is a probability space ◮ X N : Ω → M N ( C ) is a random matrix 1 ◮ X N = X ∗ √ ( x ij ) ij a N × N self-adjoint random matrix N = 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 2 / 15

Wigner and Universality ◮ in the physics literature universality refers to the fact that the limiting eigenvalue distribution is semi-circular even if we don’t assume the entries are Gaussian 0.3 0.2 0.1 - - - 3 2 1 0 1 2 3 3 / 15

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 � . 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 1 and k 2 equal to 0 4 / 15

Moments and Cumulants ◮ a 1 , . . . , a s ∈ ( A , ϕ ) random variables ◮ a partition, π = { V 1 , . . . , V k } , of [ n ] = { 1, 2, 3, . . . , n } is a decomposition of [ n ] into a disjoint union of subsets: V i ∩ V j = ∅ for i � j and [ n ] = V 1 ∪ · · · ∪ V k . ◮ P ( n ) is set of all partitions of [ n ] ◮ given a family of maps { k 1 , k 2 , k 3 , . . . , } with k n : A ⊗ n → C we define � k π ( a 1 , . . . , a n ) = k j ( a i 1 , . . . , a i j ) V ∈ π V = { i 1 ,..., i j } ◮ in general moments are defined by the moment-cumulant formula � ϕ ( a 1 · · · a n ) = k π ( a 1 , . . . , a n ) π ∈ P ( n ) ◮ k 1 ( a 1 ) = ϕ ( a 1 ) and ϕ ( a 1 a 2 ) = k 2 ( a 1 , a 2 ) + k 1 ( a 1 ) k 1 ( a 2 ) 5 / 15

cumulants and independence ◮ a ∈ A , n th cumulant of a is k ( a ) = k n ( a , . . . , a ) n ◮ if a 1 and a 2 are (classically) independent then k ( a 1 + a 2 ) = k ( a 1 ) + k ( a 2 ) for all n n n n ◮ if k n ( a i 1 , . . . , a i n ) = 0 unless i 1 = · · · i n we say mixed cumulants vanish ◮ if mixed cumulants vanish then a 1 and a 2 are independent free cumulants and free independence ( R. Speicher ) ◮ partition with a crossing: 1 2 3 4 ◮ non-crossing partition: 1 2 3 4 ◮ NC ( n ) = { non-crossing partitions of [ n ] } � ◮ ϕ ( a 1 · · · a n ) = κ π ( a 1 , . . . , a n ) defines the free π ∈ NC ( n ) cumulants : same rules apply as for classical independence. 6 / 15

freeness and asymptotic freeness ◮ 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 µ n → µ s 1 and x 1 , . . . , x s are free with respect to µ ◮ in practice this means: a ( N ) , . . . , a ( N ) ∈ ( A n , ϕ N ) are s 1 asymptotically free if whenever we have b ( N ) ∈ alg ( 1, a ( N ) ) i j i is such that ϕ N ( b ( N ) ) = 0 and j 1 � j 2 � · · · � j m we have i ϕ N ( b ( N ) · · · b ( N ) m ) → 0 1 7 / 15

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 = n ! with k 2 = 1 and k n = 0 for k n n = 1 n � 2, so the Gauss law is characterized by having all cumulants except k 1 and 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 1 < c and a = ( 1 − √ c ) 2 and b = ( 1 + √ c ) 2 we let √ ( b − t )( t − a ) d µ = dt , µ is the Marchenko-Pastur distribution: 2 π t κ n = c for all n 8 / 15

random matrices and asymptotic freeness 1 ◮ X N = X ∗ √ ( x ij ) ij a N × N self-adjoint random matrix N = N with x ij independent complex Gaussians with E ( x ij ) = 0 and E ( | x ij | 2 ) = 1 ( modulo self-adjointness) ◮ Voiculescu’s big theorem: for large N mixed moments of X N and Y N are close to those of freely independent semi-circular operators (thus asymptotically free ) 0.3 0.5 0.4 0.2 0.3 0.2 0.1 0.1 � 1 0 1 2 3 4 5 6 � 2 � 1 0 1 2 3 4 X 1000 + X 2 X 1000 + ( X T 1000 ) 2 1000 ◮ ( with M. Popa ) transposing a matrix can free it from itself 9 / 15

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 . 10 / 15

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) Γ , W Γ , W T } form an ◮ ( main theorem ) the matrices { W , W asymptotically free family 11 / 15

graphs and graphs sums ( with Roland Speicher ) ◮ a graph means a finite oriented graph with possibly loops and multiple edges ◮ a graph sum means attach a matrix to each edge and sum over vertices j T 1 T 2 T i T i j i k T 3 � � � i , j , k t ( 1 ) ij t ( 2 ) jk t ( 3 ) i , j t ij i t ii ki 12 / 15

graph sums and their growth ◮ given G = ( V , E ) a graph and an assignment e �→ T e ∈ M N ( C ) we have a graph sum � � t ( e ) S G ( T ) = i t ( e ) i s ( e ) e ∈ E i : V → [ N ] ◮ problem find “best” r ( G ) ∈ R + such that for all T we have | S G ( T ) | � N r ( G ) � � T e � e ∈ E ◮ for example: | S G ( T 1 , T 2 , T 3 ) | � N 3 / 2 � T 1 � � T 2 � � T 3 � when k T 2 j T 1 G = i T 3 l 13 / 15

finding the growth ( J.F.A. 2012 ) T 4 i 4 i 4 T 3 i 3 T 2 r = 3 i 2 i 2 = i 3 T 1 � ∴ T 5 i 1 = i 5 = i 6 2 T 6 i 5 i 1 T 7 i 7 = i 8 i 6 T 11 T 10 T 8 T 9 i 8 i 7 T 12 ◮ a edge is cutting is its removal disconnects the graph ◮ a graph is two-edge connected if it has no cutting edge ◮ a two-edge connected component is a two-edge connected subgraph which is maximal ◮ we make a quotient graph whose vertices are the two-edge connected components on the old graph and the edges are the cutting edges of the old graph ◮ r ( G ) is 1 2 the number of leaves on the quotient graph ( always a union of trees ) 14 / 15

Conclusion: traces and graph sums Γ ◮ X = W is the partially transposed Wishart matrix, but now we no longer assume entries are Gaussian ◮ we let A 1 , A 2 , . . . , A n be d 1 d 2 × d 1 d 2 constant matrices ◮ compute E ( Tr ( XA 1 XA 2 · · · XA n )) ; when A i = I we get the n th moment of the eigenvalue distribution ◮ integrating out the X ’s leaves a sum of graph sums, one for each partition π ∈ P ( n ) a ( 1 ) i 1 A i − 1 1 i 1 i − 1 X X ( 1, − 3 ) (− 1, 3 ) π = i 2 i − 4 a ( 2 ) ( 1, − 3 )(− 1, 3 ) i 2 i − 2 A A 2 ( 3 ) 4 ( 2, − 2 )( 4, − 4 ) i − 2 i 4 ( 2, − 2 ) a i 3 i − 3 X X a ( 4 ) i − 3 i 3 ( 4, − 4 ) A i 4 i − 4 3 thm : the only π ’s for which r ( G π ) is large enough ( n / 2 + 1 in this case) are non-crossing partitions with blocks of size 1 or 2 (corresponding to the free cumulants κ 1 and κ 2 ) Γ , W Γ , W T } ass. free thm : method extends to showing that { W , W 15 / 15