Second Order Freeness and RandomOrthogonal Matrices

Jamie Mingo (Queen’s University)

(joint work with Mihai Popa and Emily Redelmeier)

AMS San Diego Meeting, January 11, 2013

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Random Matrices

I Xd = X∗d = 1√d(xij) with xij random variables

I questions: eigenvalues, largest, smallest, gaps, density

I problem: {assumptions about distributions of xij}

{ {conclusions about eigenvalues of Xd}

I {xii}i ∪ {xij}i<j independent with mean 0 and E(|xij|2) = 1

I {xii}i real random variables identically distributedI {xij}i<j real random variables identically distributed

Wigner’s semi-circle law: eigenvaluedistribution converges to distributionwith density:

-1 0 1 2

0.1

0.2

0.3

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Using traces to find the eigenvalue distribution

I Xd is self-adjoint and the eigenvalues are λ1, . . . , λd, wemake a random measure νd = 1

d∑d

k=1 δλk

I for a function f ,∫f dν = 1

d∑d

k=1 f (λk) =1d Tr(f (Xd)) = tr(f (Xd))

I thus we can study the eigenvalues by studying traces ofpowers: i.e. moments {tr(Xk

d)}kI we only expect to get a simple answer in the large d limit

so we want to know for each k, limd tr(Xkd)

I for many examples the limit is not random and can befound by finding limd E(tr(Xk

d)), the moments of thelimiting eigenvalue distribution

I for many ensembles Tr(Xkd − E(tr(Xk

d))I) converges to arandom variable and so we have fluctuation momentslimd cov(Tr(Xk

d), Tr(Xld)).

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Unitarily Invariant EnsemblesI X = 1√

d(xij) is unitarily invariant if the joint distribution of

its entries is unchanged when we conjugate X by a unitarymatrix; this means that if U is a d× d unitary matrix andY = UXU∗ then for all i1, . . . , ik, j1, . . . , jk we have

E(xi1j1 · · · xikjk) = E(yi1j1 · · · yikjk)

examples

I if X = X∗ is the gue ensemble: X = X∗ = 1√d(xij) with

{xij}i<j ∪ {xii}i independent Gaussian random variables ofmean 0 and (complex) variance 1;

I X = 1d G∗G with G = (gij) i.i.d. complex Gaussian random

variables of mean 0 and variance 1 (complex Wishart);I X = X∗ is distributed according to the law eTr(V(X)) dX,

where V(x) = x2/2 + · · · is a polynomial, xij = sij +√−1tij

and dX =∏

i dsii∏

i<j dsij dtij (unitary ensembles in phys.).

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Asymptotic Freeness (unitary & orthogonal cases)

I let Xd be an ensemble of random matrices and supposethat there is a non-commutative probability space (A,ϕ)and x ∈ A such that for all k, limd E(tr(Xk

d)) = ϕ(xk). Then

we say that Xd has a limit distributionthm if Xd and Yd have limit distributions, are independent, and

one is unitarily invariant, then Xd and Yd areasymptotically free(1)

I if limd E(tr(X(ε1)d · · ·X(εk)

d )) = ϕ(x(ε1) · · · x(εk)) for allk = 1, 2, 3, . . . and all ε1, ε2, ε3, . . . then we sat that Xd has alimit t-distribution [X(−1) = Xt and x(−1) = xt]

thm if Xd is has a limit t-distribution and is independent fromO, a Haar distributed random orthogonal matrix, then{Xd, Xt

d} and {O, Ot} are asymptotically free(2)

(1)Voiculescu 1991, 1996, & M-Sniady-Speicher 2007(2)Collins-Sniady 2006

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Orthogonal Case

I O Haar distributed d× d orthogonal matrix,U Haar distributed d× d unitary matrix,A1, A2, A3, A4 constant matrices

I E(Tr(OA1O−1A2) = d−1Tr(A1)Tr(A2)

I E(Tr(UA1U−1A2) = d−1Tr(A1)Tr(A2)

I E(Tr(UA1UA2)) = 0I E(Tr(OAOB)) = d−1Tr(ABt) = tr(ABt)

I E(Tr(OA1OA2OA3OA4))

= tr(A1At4)tr(A2At

3)

+ d−1{tr(A1At2A3At

4) − tr(A1At2)tr(A3At

4)

+ tr(A1At4A3At

2) − tr(A1At4)tr(A2At

3)}

− d−2{tr(A1At2A4At

3) + tr(A1At4A3At

2)

+ tr(A1At3A2At

4) + tr(A1At3A4At

2)}

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Second Order Probability Spaces (c Nica & Speicher)I Xd random matrix ensemble with limit distribution

x ∈ (A,ϕ)I suppose for each m, n limd cov(Tr(Xm

d ), Tr(Xnd)) exists then

we define ϕ2(xm, xn) to be this limit — these are thefluctuation moments of Xd

I ϕ2 : A⊗A→ C is a bi-trace with ϕ2(1, a) = ϕ2(1, a) = 0 forall a

I (A,ϕ,ϕ) is a second order probability spaceI fluctuation moments exist for many random matrix

models and are described by planar objects1

2

3

48

6

7

51

3

5

7

2

46

8

910

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Second Order Freeness (c R. Speicher)

I A1,A2 ⊂ (A,ϕ,ϕ2) are second order free if they are free inVoiculescu’s sense and whenever we have centreda1, . . . , am, b1, . . . , bn ∈ A with ai ∈ Aki and bj ∈ Alj withk1 , k2 , · · · , km , k1 and l1 , l2 , · · · , ln , l1 then

I for m , n, ϕ2(a1 · · · am, b1 · · · bn) = 0I for m = n > 1 (indices of b are mod m)

ϕ2(a1 · · · am, b1 · · · bn) =

m∑k=1

m∏i=1

ϕ(aibk−i)

a1

a2a3

b1

b2 b3

a1

a2a3

b1

b2 b3

a1

a2a3

b1

b2b3

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Real Second Order Freeness (Emily Redelmeier)

I (A,ϕ,ϕ2, t) real second order non-commutative probabilityspace (as before but with addition of the transpose t)

I real second order freeness (same as before but also usetransposes) ϕ2(a1a2a3, b1b2b3) =

a1

a2a3

b1

b2 b3

+

a1

a2a3

b1

b2 b3

+

a1

a2a3

b1

b2b3

+

a1

a2a3

bt1

bt2bt

3

+

a1

a2a3

bt1

bt2bt

3

+

a1

a2a3

bt1

bt2bt

3

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Example: Covariance of O’s and A’sSuppose O is a Haar distributed d× d orthogonal matrix andA1, . . . , A6 are constant matrices

(1 + d−1− 2d−2)cov(Tr(OA1O−1A2), Tr(OA3O−1A4))

= d−4{Tr(A1)Tr(A2)Tr(A3)Tr(A4) + Tr(A1)Tr(A2)Tr(At3)Tr(At

4)}

− d−3{Tr(A1A3)Tr(A2)Tr(A4) + Tr(A1At3)Tr(A2)Tr(At

4)

+ Tr(A1)Tr(A2A4)Tr(A3) + Tr(A1)Tr(A2At4)Tr(At

3)}

+ (d−2 + d−3){Tr(A1A3)Tr(A2A4) + Tr(A1At3)Tr(A2At

4)}

− d−3{Tr(A1At3)Tr(A2A4) + Tr(A1A3)Tr(A2At

4)}.

subleading termscan producenon-orientable maps

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Main Theorems (c Popa & Redelmeier, arXiv:1210.6079)

I {Ad,1, . . . , Ad,s}d ensemble of random matrices with realsecond order limiting distribution

I Od Haar distributed random orthogonal matrixindependent from A’s

I thm: {Ad,1, . . . , Ad,s} and Od are asymptotically real free ofsecond order

I thm: independent Haar distributed random orthogonalmatrices are asymptotically real free of second order

I thm: if {Ai}i and {Bj}j have a real second order limitdistribution and are independent and the joint distributionof the entries of A’s is invariant under conjugation by aorthogonal matrix then {Ai} and {Bj} are asymptotically realsecond order free.

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Orthogonal versus unitary

if we put together the main theorems of M-Sniady-Speicherwith M-Popa-Redelmeier we get; as a unitarily invariantensemble is orthogonally invariantI if A = {A1, . . . , Ar} and B = {B1, . . . , Bs} are independent

and A is unitarily invariant then A and B are bothasymptotically real second order free and asymptotically(complex) second order free

I thus limd

E(tr(AiBtj)) = 0

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Unitary Invariance and t-distributions: (c M. Popa)

let U be a Haar distributed random unitary matrix andU = (U∗)t be the matrix with ij entry uij;U and U are Haar distributed random unitary matrices (by thecentrality of the Weingarten function)

I U = {U, U∗, Ut, U } has a second order limit t-distributionI U is orthogonally invariant (works because Ot = O−1) but

not unitarily invariant; e.g. E((U)1,2(U)1,2) = d−1 butE((VUV−1)1,2(VUV−1)1,2) = −d−1 whereV = diag(i, 1, . . . , 1) is a unitary (but not orthogonal) matrix

I thm: if {A1, . . . , As} has a second order limit distributionand is unitarily invariant then it has a real second orderlimit distribution

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Real and Complex Together

SupposeI A1 is unitarily invariant and has a second order limit

distributionI A2 is independent form A2 and has a second order limit

t-distributionthen A1 and A2 are asymptotically real second order free.

thm: if U is a Haar distributed random unitary matrix then{U, U∗} and {Ut, U } are asymptotically real second order free, inparticular they are first order free (in the sense of Voiculescu)

thm if {A1, . . . , An} are unitarily invariant and have a secondorder limit distribution then {A1, . . . , An} and {At

1, . . . , Atn}

are asymptotically second order free.

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Weingarten function (Collins & Sniady 2006)I O = (oij), d× d Haar distributed orthogonal matrixI E(oi1i−1oi2i−2 · · · oini−n) = 0 for n oddI p, q ∈ P2(n) (a pair of pairings) 〈ϕ(p), q〉 = d#(p∨q),ϕ : C[P2(n)]→ C[P2(n)] is invertible, Wg = ϕ−1

I δipδqδ = 1 only when ir = ip(r) & i−r = i−q(r), ∀r

I E(oi1i−1oi2i−2 · · · oini−n) =∑

p,q∈P2(n)

〈Wg(p), q〉δipδqδ

I ε1, ε2, . . . , εn ∈ {−1, 1}I E(Trγ(Oε1A1, Oε2A2, . . . , OεnAn))

=∑

p,q∈P2(n)

〈Wg(p), q〉E(Trπp ·εq(Aη11 , . . . , Aηn

n ))

I (πp ·εq,ηp ·εq) is the Kreweras complement of the pair ofpairings (p, q), πp ·εq ∈ Sn, η1, . . . ,ηn ∈ {−1, 1}

1 2 3 4 ∨ 1 2 3 4 = 1 2 3 4

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