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, 2013
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Random Matrices
1
√
(x )
d ij
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Xd = Xd∗ =
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questions: eigenvalues, largest, smallest, gaps, density
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problem: {assumptions about distributions of xij }
{ {conclusions about eigenvalues of Xd }
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{xii }i ∪ {xij }i<j independent with mean 0 and E(|xij |2 ) = 1
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{xii }i real random variables identically distributed
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{xij }i<j real random variables identically distributed
with xij random variables
Wigner’s semi-circle law: eigenvalue
distribution converges to distribution
with density:
0.3
0.2
0.1
-1
0
1
2
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Using traces to find the eigenvalue distribution
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Xd is self-adjoint and the eigenvalues are λ1 , . . . , λd , we
P
make a random measure νd = d1 dk=1 δλk
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for a function f ,
R
P
f dν = 1d dk=1 f (λk ) = d1 Tr(f (Xd )) = tr(f (Xd ))
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thus we can study the eigenvalues by studying traces of
powers: i.e. moments {tr(Xdk )}k
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we only expect to get a simple answer in the large d limit
so we want to know for each k, limd tr(Xdk )
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for many examples the limit is not random and can be
found by finding limd E(tr(Xdk )), the moments of the
limiting eigenvalue distribution
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for many ensembles Tr(Xdk − E(tr(Xdk ))I) converges to a
random variable and so we have fluctuation moments
limd cov(Tr(Xdk ), Tr(Xdl )).
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Unitarily Invariant Ensembles
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X = √1 (xij ) is unitarily invariant if the joint distribution of
d
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 i1 , . . . , ik , j1 , . . . , jk we have
E(xi1 j1 · · · xik jk ) = E(yi1 j1 · · · yik jk )
examples
1
I if X = X∗ is the gue ensemble: X = X∗ = √
(x ) with
d ij
{xij }i<j ∪ {xii }i independent Gaussian random variables of
mean 0 and (complex) variance 1;
I X = 1 G∗ G with G = (gij ) i.i.d. complex Gaussian random
d
variables of mean 0 and variance 1 (complex Wishart);
I X = X∗ is distributed according to the law eTr(V(X)) dX,
√
2 /2 + · · · is a polynomial, x = s +
where V(x)
=
x
−1tij
ij
ij
Q
Q
and dX = i dsii i<j dsij dtij
(unitary ensembles in phys.).
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Asymptotic Freeness (unitary & orthogonal cases)
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let Xd 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, limd E(tr(Xdk )) = ϕ(xk ). Then
we say that Xd has a limit distribution
thm if Xd and Yd have limit distributions, are independent, and
one is unitarily invariant, then Xd and Yd are
asymptotically free(1)
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( )
( )
if limd E(tr(Xd 1 · · · Xd k )) = ϕ(x(1 ) · · · x(k ) ) for all
k = 1, 2, 3, . . . and all 1 , 2 , 3 , . . . then we sat that Xd has a
limit t-distribution
[X(−1) = Xt and x(−1) = xt ]
thm if Xd is has a limit t-distribution and is independent from
O, a Haar distributed random orthogonal matrix, then
{Xd , Xdt } and {O, Ot } are asymptotically free(2)
(1)
(2)
Voiculescu 1991, 1996, & M-Śniady-Speicher 2007
Collins-Śniady 2006
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Orthogonal Case
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O Haar distributed d × d orthogonal matrix,
U Haar distributed d × d unitary matrix,
A1 , A2 , A3 , A4 constant matrices
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E(Tr(OA1 O−1 A2 ) = d−1 Tr(A1 )Tr(A2 )
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E(Tr(UA1 U−1 A2 ) = d−1 Tr(A1 )Tr(A2 )
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E(Tr(UA1 UA2 )) = 0
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E(Tr(OAOB)) = d−1 Tr(ABt ) = tr(ABt )
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E(Tr(OA1 OA2 OA3 OA4 ))
= tr(A1 At4 )tr(A2 At3 )
+ d−1 tr(A1 At2 A3 At4 ) − tr(A1 At2 )tr(A3 At4 )
+ tr(A1 At4 A3 At2 ) − tr(A1 At4 )tr(A2 At3 )
− d−2 tr(A1 At2 A4 At3 ) + tr(A1 At4 A3 At2 )
+ tr(A1 At3 A2 At4 ) + tr(A1 At3 A4 At2 )
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Second Order Probability Spaces (c̄ Nica & Speicher)
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Xd random matrix ensemble with limit distribution
x ∈ (A, ϕ)
suppose for each m, n limd cov(Tr(Xdm ), Tr(Xdn )) exists then
we define ϕ2 (xm , xn ) to be this limit — these are the
fluctuation moments of Xd
ϕ2 : A ⊗ A → C is a bi-trace with ϕ2 (1, a) = ϕ2 (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
1
8
2
5
7
3
4
6
8
1
2
9
10
7
3
4
6
5
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Second Order Freeness (c̄ R. Speicher)
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A1 , A2 ⊂ (A, ϕ, ϕ2 ) are second order free if they are free in
Voiculescu’s sense and whenever we have centred
a1 , . . . , am , b1 , . . . , bn ∈ A with ai ∈ Aki and bj ∈ Alj with
k1 , k2 , · · · , km , k1 and l1 , l2 , · · · , ln , l1 then
I
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for m , n, ϕ2 (a1 · · · am , b1 · · · bn ) = 0
for m = n > 1
(indices of b are mod m)
ϕ2 (a1 · · · am , b1 · · · bn ) =
m Y
m
X
ϕ(ai bk−i )
k=1 i=1
b2
a3
a1
a1
b1
b1
b2
b3
a2
a3
a1
b1
b3
b2
a2
a3
b3
a2
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Real Second Order Freeness (Emily Redelmeier)
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(A, ϕ, ϕ2 , t) real second order non-commutative probability
space (as before but with addition of the transpose t)
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real second order freeness (same as before but also use
transposes) ϕ2 (a1 a2 a3 , b1 b2 b3 ) =
a1
b1
b2
b1
+
b3
a3
b2
a2
a3
b2
a3
a2
a3
b3
a2
a1
bt1
bt3
b1
+
b3
a1
+
a1
a1
bt1
+
bt2
a2
a1
bt3
a3
bt1
+
bt2
bt3
a2
a3
bt2
a2
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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
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Main Theorems (c̄ Popa & Redelmeier, arXiv:1210.6079)
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{Ad,1 , . . . , Ad,s }d ensemble of random matrices with real
second order limiting distribution
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Od Haar distributed random orthogonal matrix
independent from A’s
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thm: {Ad,1 , . . . , Ad,s } and Od are asymptotically real free of
second order
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thm: independent Haar distributed random orthogonal
matrices are asymptotically real free of second order
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thm: if {Ai }i and {Bj }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 {Ai } and {Bj } are asymptotically real
second order free.
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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
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if A = {A1 , . . . , Ar } and B = {B1 , . . . , Bs } are independent
and A is unitarily invariant then A and B are both
asymptotically real second order free and asymptotically
(complex) second order free
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thus lim E(tr(Ai Btj )) = 0
d
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Unitary Invariance and t-distributions: (c̄ M. Popa)
let U be a Haar distributed random unitary matrix and
U = (U∗ )t be the matrix with ij entry uij ;
U and U are Haar distributed random unitary matrices (by the
centrality of the Weingarten function)
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U = {U, U∗ , Ut , U } has a second order limit t-distribution
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U is orthogonally invariant (works because Ot = 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
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thm: if {A1 , . . . , As } has a second order limit distribution
and is unitarily invariant then it has a real second order
limit distribution
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Real and Complex Together
Suppose
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A1 is unitarily invariant and has a second order limit
distribution
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A2 is independent form A2 and has a second order limit
t-distribution
then 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, in
particular they are first order free (in the sense of Voiculescu)
thm if {A1 , . . . , An } are unitarily invariant and have a second
order limit distribution then {A1 , . . . , An } and {At1 , . . . , Atn }
are asymptotically second order free.
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Weingarten function (Collins & Śniady 2006)
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O = (oij ), d × d Haar distributed orthogonal matrix
E(oi1 i−1 oi2 i−2 · · · oin i−n ) = 0 for n odd
p, q ∈ P2 (n) (a pair of pairings) hϕ(p), qi = d#(p∨q) ,
ϕ : C[P2 (n)] → C[P2 (n)] is invertible, Wg = ϕ−1
δipδqδ = 1 only when ir = ip(r) & i−r = i−q(r) , ∀r
X
E(oi1 i−1 oi2 i−2 · · · oin i−n ) =
hWg(p), qiδipδqδ
p,q∈P2 (n)
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1 , 2 , . . . , n ∈ {−1, 1}
E(Trγ (O1 A1 , O2 A2 , . . . , On An ))
X
ηn
1
=
hWg(p), qiE(Trπp · q (Aη
1 , . . . , An ))
p,q∈P2 (n)
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(πp · q , ηp · q ) is the Kreweras complement of the pair of
pairings (p, q), πp · q ∈ Sn , η1 , . . . , ηn ∈ {−1, 1}
1 2
3
4 ∨ 1 2
3
4 = 1 2
3
4
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