Second Order Freeness and Random Orthogonal Matrices

Transcription

1 Second Order Freeness and Random Orthogonal Matrices Jamie Mingo (Queen s University) (joint work with Mihai Popa and Emily Redelmeier) AMS San Diego Meeting, January 11, / 15

2 Random Matrices X d = X d = 1 d (x ij ) with x ij random variables questions: eigenvalues, largest, smallest, gaps, density problem: {assumptions about distributions of x ij } {conclusions about eigenvalues of X d } {x ii } i {x ij } i<j independent with mean 0 and E( x ij 2 ) = 1 {x ii } i real random variables identically distributed {x ij } i<j real random variables identically distributed Wigner s semi-circle law: eigenvalue distribution converges to distribution with density: / 15

3 Using traces to find the eigenvalue distribution X d is self-adjoint and the eigenvalues are λ 1,..., λ d, we make a random measure ν d = 1 d d k=1 δ λ k for a function f, f dν = 1 d d k=1 f (λ k) = 1 d Tr(f (X d)) = tr(f (X d )) thus we can study the eigenvalues by studying traces of powers: i.e. moments {tr(x k d )} k we only expect to get a simple answer in the large d limit so we want to know for each k, lim d tr(x k d ) for many examples the limit is not random and can be found by finding lim d E(tr(Xd k )), the moments of the limiting eigenvalue distribution for many ensembles Tr(Xd k E(tr(Xk d ))I) converges to a random variable and so we have fluctuation moments lim d cov(tr(xd k), Tr(Xl d )). 3 / 15

4 Unitarily Invariant Ensembles X = 1 (x d ij ) is unitarily invariant if the joint distribution of its entries is unchanged when we conjugate X by a unitary matrix; this means that if U is a d d unitary matrix and Y = UXU then for all i 1,..., i k, j 1,..., j k we have E(x i1 j 1 x ik j k ) = E(y i1 j 1 y ik j k ) examples if X = X is the gue ensemble: X = X = 1 (x d ij ) with {x ij } i<j {x ii } i independent Gaussian random variables of mean 0 and (complex) variance 1; X = 1 d G G with G = (g ij ) i.i.d. complex Gaussian random variables of mean 0 and variance 1 (complex Wishart); X = X is distributed according to the law e Tr(V(X)) dx, where V(x) = x 2 /2 + is a polynomial, x ij = s ij + 1t ij and dx = i ds ii i<j ds ij dt ij (unitary ensembles in phys.). 4 / 15

5 Asymptotic Freeness (unitary & orthogonal cases) let X d be an ensemble of random matrices and suppose that there is a non-commutative probability space (A, ϕ) and x A such that for all k, lim d E(tr(X k d )) = ϕ(xk ). Then we say that X d has a limit distribution thm if X d and Y d have limit distributions, are independent, and one is unitarily invariant, then X d and Y d are asymptotically free (1) if lim d E(tr(X (ɛ 1) d X (ɛ k) d )) = ϕ(x (ɛ 1) x (ɛk) ) for all k = 1, 2, 3,... and all ɛ 1, ɛ 2, ɛ 3,... then we sat that X d has a limit t-distribution [X ( 1) = X t and x ( 1) = x t ] thm if X d is has a limit t-distribution and is independent from O, a Haar distributed random orthogonal matrix, then {X d, X t d } and {O, Ot } are asymptotically free (2) (1) Voiculescu 1991, 1996, & M-Śniady-Speicher 2007 (2) Collins-Śniady / 15

6 Orthogonal Case O Haar distributed d d orthogonal matrix, U Haar distributed d d unitary matrix, A 1, A 2, A 3, A 4 constant matrices E(Tr(OA 1 O 1 A 2 ) = d 1 Tr(A 1 )Tr(A 2 ) E(Tr(UA 1 U 1 A 2 ) = d 1 Tr(A 1 )Tr(A 2 ) E(Tr(UA 1 UA 2 )) = 0 E(Tr(OAOB)) = d 1 Tr(AB t ) = tr(ab t ) E(Tr(OA 1 OA 2 OA 3 OA 4 )) = tr(a 1 A t 4 )tr(a 2A t 3 ) + d 1{ tr(a 1 A t 2 A 3A t 4 ) tr(a 1A t 2 )tr(a 3A t 4 ) + tr(a 1 A t 4 A 3A t 2 ) tr(a 1A t 4 )tr(a 2A t 3 )} d 2{ tr(a 1 A t 2 A 4A t 3 ) + tr(a 1A t 4 A 3A t 2 ) + tr(a 1 A t 3 A 2A t 4 ) + tr(a 1A t 3 A 4A t 2 )} 6 / 15

7 Second Order Probability Spaces ( c Nica & Speicher) X d random matrix ensemble with limit distribution x (A, ϕ) suppose for each m, n lim d cov(tr(xd m), Tr(Xn d )) exists then we define ϕ 2 (x m, x n ) to be this limit these are the fluctuation moments of X d ϕ 2 : A A C is a bi-trace with ϕ 2 (1, a) = ϕ 2 (1, a) = 0 for all a (A, ϕ, ϕ) is a second order probability space fluctuation moments exist for many random matrix models and are described by planar objects / 15

8 Second Order Freeness ( c R. Speicher) A 1, A 2 (A, ϕ, ϕ 2 ) are second order free if they are free in Voiculescu s sense and whenever we have centred a 1,..., a m, b 1,..., b n A with a i A ki and b j A lj with k 1 k 2 k m k 1 and l 1 l 2 l n l 1 then for m n, ϕ 2 (a 1 a m, b 1 b n ) = 0 for m = n > 1 (indices of b are mod m) ϕ 2 (a 1 a m, b 1 b n ) = m k=1 i=1 m ϕ(a i b k i ) b1 b1 b1 b2 b3 b2 b3 b2 b3 8 / 15

9 Real Second Order Freeness (Emily Redelmeier) (A, ϕ, ϕ 2, t) real second order non-commutative probability space (as before but with addition of the transpose t) real second order freeness (same as before but also use transposes) ϕ 2 (a 1 a 2 a 3, b 1 b 2 b 3 ) = b2 b1 b3 + b2 b1 b3 + b2 b1 b3 + b t 3 b t 1 b t 2 + b t 3 b t 1 b t 2 + b t 3 b t 1 b t 2 9 / 15

10 Example: Covariance of O s and A s Suppose O is a Haar distributed d d orthogonal matrix and A1,..., A6 are constant matrices (1 + d 1 2d 2 )cov(tr(oa1 O 1 A2 ), Tr(OA3 O 1 A4 )) = d 4 {Tr(A1 )Tr(A2 )Tr(A3 )Tr(A4 ) + Tr(A1 )Tr(A2 )Tr(At3 )Tr(At4 )} d 3 {Tr(A1 A3 )Tr(A2 )Tr(A4 ) + Tr(A1 At3 )Tr(A2 )Tr(At4 ) + Tr(A1 )Tr(A2 A4 )Tr(A3 ) + Tr(A1 )Tr(A2 At4 )Tr(At3 )} + (d 2 + d 3 ){Tr(A1 A3 )Tr(A2 A4 ) + Tr(A1 At3 )Tr(A2 At4 )} d 3 {Tr(A1 At3 )Tr(A2 A4 ) + Tr(A1 A3 )Tr(A2 At4 )}. subleading terms can produce non-orientable maps 10 / 15

11 Main Theorems ( c Popa & Redelmeier, arxiv: ) {A d,1,..., A d,s } d ensemble of random matrices with real second order limiting distribution O d Haar distributed random orthogonal matrix independent from A s thm: {A d,1,..., A d,s } and O d are asymptotically real free of second order thm: independent Haar distributed random orthogonal matrices are asymptotically real free of second order thm: if {A i } i and {B j } j have a real second order limit distribution and are independent and the joint distribution of the entries of A s is invariant under conjugation by a orthogonal matrix then {A i } and {B j } are asymptotically real second order free. 11 / 15

12 Orthogonal versus unitary if we put together the main theorems of M-Śniady-Speicher with M-Popa-Redelmeier we get; as a unitarily invariant ensemble is orthogonally invariant if A = {A 1,..., A r } and B = {B 1,..., B s } are independent and A is unitarily invariant then A and B are both asymptotically real second order free and asymptotically (complex) second order free thus lim d E(tr(A i B t j )) = 0 12 / 15

13 Unitary Invariance and t-distributions: ( c M. Popa) let U be a Haar distributed random unitary matrix and U = (U ) t be the matrix with ij entry u ij ; U and U are Haar distributed random unitary matrices (by the centrality of the Weingarten function) U = {U, U, U t, U } has a second order limit t-distribution U is orthogonally invariant (works because O t = O 1 ) but not unitarily invariant; e.g. E((U) 1,2 (U) 1,2 ) = d 1 but E((VUV 1 ) 1,2 (VUV 1 ) 1,2 ) = d 1 where V = diag(i, 1,..., 1) is a unitary (but not orthogonal) matrix thm: if {A 1,..., A s } has a second order limit distribution and is unitarily invariant then it has a real second order limit distribution 13 / 15

14 Real and Complex Together Suppose A 1 is unitarily invariant and has a second order limit distribution A 2 is independent form A 2 and has a second order limit t-distribution then A 1 and A 2 are asymptotically real second order free. thm: if U is a Haar distributed random unitary matrix then {U, U } and {U t, U } are asymptotically real second order free, in particular they are first order free (in the sense of Voiculescu) thm if {A 1,..., A n } are unitarily invariant and have a second order limit distribution then {A 1,..., A n } and {A t 1,..., At n} are asymptotically second order free. 14 / 15

15 Weingarten function (Collins & Śniady 2006) O = (o ij ), d d Haar distributed orthogonal matrix E(o i1 i 1 o i2 i 2 o in i n ) = 0 for n odd p, q P 2 (n) (a pair of pairings) ϕ(p), q = d #(p q), ϕ : C[P 2 (n)] C[P 2 (n)] is invertible, Wg = ϕ 1 δ i pδqδ = 1 only when i r = i p(r) & i r = i q(r), r E(o i1 i 1 o i2 i 2 o in i n ) = Wg(p), q δ i pδqδ ɛ 1, ɛ 2,..., ɛ n { 1, 1} p,q P 2 (n) E(Tr γ (O ɛ 1 A 1, O ɛ 2 A 2,..., O ɛ n A n )) = Wg(p), q E(Tr πp ɛq (A η 1 1,..., Aη n n )) p,q P 2 (n) (π p ɛq, η p ɛq) is the Kreweras complement of the pair of pairings (p, q), π p ɛq S n, η 1,..., η n { 1, 1} = / 15