FREE PROBABILITY AND RANDOM MATRICES Philippe Biane INRIA, - PowerPoint PPT Presentation

FREE PROBABILITY AND RANDOM MATRICES Philippe Biane INRIA, 28/02/2011 Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

Free probability, invented by D. Voiculescu, is a tool for understanding spectral properties of sets of large random matrices X =hermitan N × N matrix. X = UDU ∗ U =unitary (eigenvectors of X ); D =real diagonal (eigenvalues)   λ 1 0 0 · · · 0 0 λ 2 0 · · · 0     0 0 λ 3 · · · 0 D =    . . . .  ... . . . .   . . . .   0 0 · · · · · · λ N The geometry of X is specified, up to conjugation by a unitary matrix, by its spectrum. ⇒ 1 N Tr ( X n ) = 1 � N i =1 λ n Spec( X ) ⇐ i ; n = 1 , 2 , . . . N Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

X 1 , . . . , X n N × N hermitian matrices. In general they do not commute: no joint spectrum. Up to conjugation by a unitary X 1 , . . . , X n �→ UX 1 U ∗ , . . . , UX n U ∗ the n -tuple of matrices X 1 , . . . , X n can be recovered from their moments 1 N Tr ( X i 1 . . . X i k ); i 1 , . . . , i k ∈ { 1 , . . . , n } Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

A MODEL FOR INDEPENDENT MATRICES Take X i = U i D i U ∗ i where D i are fixed real diagonal, and U i are random unitaries (taken with Haar measure on U ( N )). Haar measure on U ( N ): U = ( V 1 V 2 . . . V N ) column vectors. Choose V 1 at random with norm 1. Then choose V 2 ⊥ V 1 at random with norm 1, then V 3 ⊥ V 1 , V 2 , etc... Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

Theorem (Voiculescu, 1990) When N → ∞ with probability almost one, 1 N Tr ( X i 1 . . . X i k ) can be expressed asymptotically, as polynomial functions, in terms of the moments 1 i ) = 1 N Tr ( D k N Tr ( X k i ) Examples: ( 1 N Tr = tr ) tr ( X 1 X 2 ) ∼ tr ( X 1 ) tr ( X 2 ) tr ( X 1 X 2 X 1 X 2 ) ∼ 1 ) tr ( X 2 ) 2 + tr ( X 1 ) 2 tr ( X 2 tr ( X 2 2 ) − tr ( X 1 ) 2 tr ( X 2 ) 2 Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

Corollary If we know the spectra of X 1 , . . . , X n then we can compute, with good approximation, and high probability, the spectrum of any combination of X i ’s (e.g. sum, product etc...). e.g. � tr (( X 1 + X 2 ) n ) = tr ( X i 1 . . . X i n ) i 1 ... i n can be computed from the values tr ( X k 1 ) , tr ( X k 2 ) , k = 1 , 2 , ... . Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

FREENESS A =algebra (of noncommutative random variables); 1 ∈ A , a + b , ab , λ a ∈ A if a , b ∈ A τ : A → C =linear functional (=expectation). τ (1) = 1 Definition (Voiculescu, 1983) { A i ; i ∈ I } =family of algebras are free in ( A , τ ) iff for all a 1 , . . . , a n ∈ A such that i) τ ( a j ) = 0 for all j, ii) a j ∈ A i j , i 1 � = i 2 , i 2 � = i 3 , . . . , i n − 1 � = i n , one has τ ( a 1 . . . a n ) = 0 Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

Example: a 1 ∈ A 1 , a 2 ∈ A 2 , free in ( A , τ ) a 1 = ¯ a 1 + τ ( a 1 )1; a 2 = ¯ a 2 + τ ( a 2 )1; τ (¯ a 1 ) = τ (¯ a 2 ) = 0 τ ( a 1 a 2 ) = τ ((¯ a 1 + τ ( a 1 ))(¯ a 2 + τ ( a 2 ))) by freeness assumption τ (¯ a 1 ¯ a 2 ) = 0 finally τ ( a 1 a 2 ) = τ ( a 1 ) τ ( a 2 ) Similarly 1 ) τ ( a 2 ) 2 + τ ( a 1 ) 2 τ ( a 2 τ ( a 1 a 2 a 1 a 2 ) = τ ( a 2 2 ) − τ ( a 1 ) 2 τ ( a 2 ) 2 In general τ ( a 1 . . . a n ) for a j ∈ A i j can be computed by a polynomial in moments τ ( a i 1 . . . a j r ) with a j 1 . . . a j r in the same algebra. Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

FREENESS AND RANDOM MATRICES Take X i = U i D i U ∗ i where D i are fixed real diagonal, and U i are random unitaries. Let a 1 , . . . , a n ∈ ( A , τ ) be free and such that τ ( a k i ) = tr ( X k i ) k = 1 , 2 , . . . then for N large tr ( X i 1 . . . X i k ) ∼ τ ( a i 1 . . . a i k ) with probability close to 1. As we saw τ ( a i 1 . . . a ik ) can be written as a polynomial in the moments τ ( a k i ) = tr ( X k i ). This solves the problem at the beginning. Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

COMBINATORICS OF FREENESS A combinatorial way of dealing with freeness has been devised by R. Speicher, using noncrossing partitions. A partition of { 1 , . . . , n } is noncrossing if there is no crossing. A crossing is a quadruple ( i , j , k , l ) with i < j < k < l and i ∼ k , k ∼ l and i , j not in the same part. Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

{ 1 , 4 , 5 } ∪ { 2 } ∪ { 3 } ∪ { 6 , 8 } ∪ { 7 } has no crossing 1 2 8 3 7 6 4 5 Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

NONCROSSING CUMULANTS On ( A , τ ) define R n , multilinear functionals by � τ ( a 1 . . . a n ) = R π ( a 1 , . . . , a n ) π ∈ NC ( n ) � R π ( a 1 , . . . , a n ) = R | p | ( a i 1 , . . . , a i | p | ) parts of π where p = { i 1 , . . . , i | p | } is a part of π . Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

Example: τ ( a 1 a 2 a 3 ) = R 3 ( a 1 , a 2 , a 3 ) { 1 , 2 , 3 } + R 1 ( a 1 ) R 2 ( a 2 , a 3 ) { 1 } ∪ { 2 , 3 } + R 2 ( a 1 , a 3 ) R 1 ( a 2 ) { 1 , 3 } ∪ { 2 } + R 2 ( a 1 , a 2 ) R 1 ( a 3 ) { 1 , 2 } ∪ { 3 } + R 1 ( a 1 ) R 1 ( a 2 ) R 1 ( a 3 ) { 1 } ∪ { 2 } ∪ { 2 } R 1 ( a ) = τ ( a ) R 2 ( a 1 , a 2 ) = τ ( a 1 a 2 ) − τ ( a 1 ) τ ( a 2 ) R 3 ( a 1 , a 2 , a 3 ) = τ ( a 1 a 2 a 3 ) − τ ( a 1 a 2 ) τ ( a 3 ) − τ ( a 1 a 3 ) τ ( a 2 ) − τ ( a 1 ) τ ( a 2 a 3 ) +2 τ ( a 1 ) τ ( a 2 ) τ ( a 3 ) Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

FREENESS AND FREE CUMULANTS Theorem (Speicher). If A i ⊂ A ; i ∈ I are free, and a 1 ∈ A i 1 , . . . , a n ∈ A i n , then R n ( a 1 , . . . , a n ) = 0 if there exists j , k such that i j � = i k . Remark If one uses all partitions instead of noncrossing partitions, this is Rota’s combinatorial approach to independence. Example: a , b free in ( A , τ ) then R n ( a + b , . . . , a + b ) = R n ( a , . . . , a ) + R n ( b , . . . , b ) Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

FREE CONVOLUTION A =*-algebra; τ =tracial state on A . Let X 1 , X 2 be free, selfadjoint in A . � � τ ( X n x n µ 1 ( dx ); τ ( X n x n µ 2 ( dx ) 1 ) = 2 ) = R R � τ (( X 1 + X 2 ) n ) = x n µ 1 ⊞ µ 2 ( dx ) R ∞ � z − x µ ( dx ) = 1 1 � � z − n − 1 x n µ ( dx ) G µ ( z ) = z + n =1 ∞ K µ ( z ) = 1 � R n ( µ ) z n K µ ( G µ ( z )) = G µ ( K µ ( z )); z + n =0 Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

Theorem (Voiculescu, 1986) R n ( µ 1 ⊞ µ 2 ) = R n ( µ 1 ) + R n ( µ 2 ) R n ( µ ) are called the free cumulants of µ . Compare with � � e itx µ ( dx ) = ( it ) n C n ( µ ) / n ! log n where C n are the cumulants of µ . C n ( µ 1 ∗ µ 2 ) = C n ( µ 1 ) + C n ( µ 2 ) . Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

Examples: 1 2( δ 0 + δ 1 ) ⊞ 1 2( δ 0 + δ 1 ) Random matrix model Π 1 + Π 2 where Π 1 , Π 2 = orthogonal projections on a random subspaces of dimension N / 2. 1 y = � π x (2 − x ) 1 2( δ 0 + δ 1 ) ⊞ 1 2( δ 0 + δ 1 ) ⊞ 1 2( δ 0 + δ 1 ) Random matrix model Π 1 + Π 2 + Π 3 � 8 − (2 x − 3) 2 y = � π x (3 − x ) Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

0*x 35 30 25 20 15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x Π 1 + Π 2 1 y = � π x (2 − x ) Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

(40*2^(1/2))*6*(2−(x−3/2)^2)^(1/2)/(pi*(9−4*(x−3/2)^2)) 26 24 22 20 18 16 14 12 10 0 0.5 1 1.5 2 2.5 3 x Π 1 + Π 2 + Π 3 � 8 − (2 x − 3) 2 y = � π x (3 − x ) Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

FREE CENTRAL LIMIT THEOREM Let X 1 , . . . , X n ∈ ( A , τ ) be free random variables, identically distributed. τ ( X 2 i ) = σ 2 τ ( X i ) = 0 Theorem (Voiculescu, 1983) As n → ∞ the distribution of X 1 + ... + X n converges to the semi-circular distribution with density √ n 1 � 4 σ 2 − x 2 x ∈ [ − 2 σ, 2 σ ] πσ This should be compared with Wigner’s theorem Let M be a random hermitian gaussian matrix such that E [ Tr ( M 2 )] = N then the distribution of eigenvalues of M converges to the semi-circular distribution as N → ∞ .Indeed one has M = M 1 + M 2 + . . . + M n √ n with independent random matrices M 1 , . . . , M n , which are asymptotically free. Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

YOUNG DIAGRAMS A Young diagram is a sequence of integers λ 1 ≥ λ 2 ≥ . . . ≥ λ n ≤ 0 Young diagrams label irreducible representations of symmetric groups. x 1 y 1 x 2 y 2 x 3 y 3 x 4 Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

A diagram may be identified with a function ω ( x ) such that | ω ( x ) | = | x | for x >> 1 | ω ( x ) − ω ( y ) | ≤ | x − y | . Philippe Biane FREE PROBABILITY AND RANDOM MATRICES

TRANSITION MEASURES Take ω as above, put σ ( u ) = ( ω ( u ) − | u | ) / 2 then (S.Kerov) there exists a unique probability measure m ω such that = 1 x − z σ ′ ( x ) dx 1 � G ω ( z ) z exp 1 � = z − x m ω ( dx ) Q n − 1 i =1 ( z − y k ) = Q n i =1 ( z − x k ) n � n − 1 i =1 ( x k − y i ) � m ω = µ k δ x k µ k = � i � = k ( x k − x i ) k =1 K ω = G �− 1 � ω Philippe Biane FREE PROBABILITY AND RANDOM MATRICES ∞