Freeness and the Transpose ( Matrices Just Wanna be Free ) Jamie - - PowerPoint PPT Presentation

Freeness and the Transpose (Matrices Just Wanna be Free)

Jamie Mingo (Queen’s University)

with Mihai Popa

Recent Developments in Quantum Groups, Operator Algebras and Applications, February 6, 2015

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random variables and their distributions

◮ (A, ϕ) unital algebra with state; ◮ Cx1, . . . , xs is the unital algebra generated by the

non-commuting variables x1, . . . , xs

◮ the distribution of a1, . . . , as ∈ (A, ϕ) is the state

µ : Cx1, . . . , xs → C given by µ(p) = ϕ(p(a1, . . . , as))

◮ convergence in distribution of {a(N) 1

, . . . , a(N)

s

} ⊂ (AN, ϕN) to {a1, . . . , as} ⊂ (A, ϕ) means pointwise convergence of distributions: µN(p) → µ(p) for p ∈ Cx1, . . . , xs.

freeness

◮ A1, A2 ⊆ A unital subalgebras are free if given

◮ a1, · · · , an ∈ A with ϕ(ai) = 0 for all i, ◮ ai ∈ Aj1 with j1 · · · jn

we have ϕ(a1 · · · an) = 0,

◮ a1 and a2 are free if alg(1, a1) and alg(1, a2) are free,

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freeness and asymptotic freeness

◮ {a(N) 1

, . . . , a(N)

s

} ⊂ (AN, ϕN) are asymptotically free if µn → µ and x1, . . . , xs are free with respect to µ

◮ if a and b are free with respect to ϕ then

ϕ(abab) = ϕ(a2)ϕ(b)2 + ϕ(a)2ϕ(b2) − ϕ(a)2ϕ(b)2

◮ in general if a1, . . . , as are free then all mixed moments

ϕ(xi1 · · · xin) can be written as a polynomial in the moments

• f individual moments {ϕ(ak

i )}i,k. ◮ a(N) 1

, . . . , a(N)

s

∈ (An, ϕN) are asymptotically free if whenever we have b(N)

i

∈ alg(1, a(N)

ji

) is such that ϕN(b(N)

i

) = 0 and j1 j2 · · · jm we have ϕN(b(N)

1

· · · b(N)

m ) → 0

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simple distributions: Wigner and Marchenko-Pastur

◮ let f(t) = 1 √ 2πe−t2/2 be the density of the Gauss law ◮ then log(ˆ

f(is)) = s2 2 =

∞

• n=1

kn sn n! with k2 = 1 and kn = 0 for n 2, so the Gauss law is characterized by having all cumulants except k2 equal to 0

◮ µ a probability measure on R, z ∈ C+,

G(z) =

• (z − t)−1 dµ(t) is the Cauchy transform of µ and

R(z) = G−1(z) − 1

z = κ1 + κ2z + κ3z2 + · · · is the

R-transform of µ

◮ if dµ(t) = 1 2π

√ 4 − t2 dt is the semi-circle law we have κn = 0 except for κ2 = 1

◮ if 0 < c and a = (1 − √c)2 and b = (1 + √c)2 we let

dµ = √

(b−t)(t−a) 2πt

dt for c 1 and dµ = (1 − c)δ0 + √

(b−t)(t−a) 2πt

dt for 0 < c < 1, the Marchenko-Pastur distribution: κn = c for all n

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GUE random matrices and asymptotic freeness

◮ XN = X∗ N =

1 √ N (xij)ij a N × N self-adjoint random matrix with xij independent complex Gaussians with E(xij) = 0 and E(|xij|2) = 1 (modulo self-adjointness)

◮ λ1 λ2 · · · λN eigenvalues of XN,

µN = 1 N(δλ1 + · · · + δλN) is the spectral measure of XN,

• tk dµN(t) = tr(Xk

N) ◮

XN is the N × N GUE with limiting eigenvalue distribution given by Wigner’s semi-circle law

1 1 2 2 0.1 0.2 0.3

• ◮ YN another GUE with entries independent from those of

XN

◮ for large N mixed moments of XN and YN are close to those

• f freely independent semi-circular operators (thus

asymptotically free)

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Wishart Random Matrices

◮ Suppose G1, . . . , Gd1 are d2 × p random matrices where

Gi = (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. ◮

W = 1 d1d2    G1 . . . Gd1   

• G∗

1

· · · G∗

d1

• =

1 d1d2 (GiG∗

j )ij

is a d1d2 × d1d2 Wishart

• matrix. We write

W = d−1

1 (W(i, j))ij as d1 × d1

block matrix with each entry the d2 × d2 matrix d−1

2 GiG∗ j .

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Partial Transposes on Md1(C) ⊗ Md2(C)

· Gi a d2 × p matrix · W(i, j) = 1

d2 GiG∗ j , a d2 × d2 matrix,

· W = 1

d1 (W(i, j))ij is a d1 × d1 block matrix with entries W(i, j)

· WT = 1

d1 (W(j, i)T)ij is the “full” transpose

· W

Γ

= 1

d1 (W(j, i))ij is the “left” partial transpose

· WΓ = 1

d1 (W(i, j)T)ij is the “right” partial transpose

· we assume that p d1d2 → c, 0 < c < ∞ · eigenvalue distributions of W and WT converge to Marchenko-Pastur with parameter c · eigenvalues of W

Γ

and WΓ converge to a shifted semi-circular with mean c and variance c (Aubrun, 2012) · W and WT are asymptotically free (M. and Popa, 2014) · what about WΓ and W

Γ

?

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Semi-circle and Marchenko-Pastur Distributions

Suppose p d1d2 → c.

◮ limit eigenvalue distribution of W (Marchenko-Pastur)

lim E(tr(Wn)) = b

a

tn

• (b − t)(t − a)

2πt dt =

• σ∈NC(n)

c#(σ) #(σ) is the number of blocks of σ, a = (1 − √c)2, and b = (1 + √c)2

◮ limit eigenvalue distribution of WΓ (semi-circle)

lim E(tr((WΓ)n)) =

• σ∈NC1,2(n)

c#(σ) =

• π∈NC(n)

κπ NC1,2(n) is the set of non-crossing partitions with only blocks of size 1 and 2. (c.f. Fukuda and ´ Sniady (2013) and Banica and Nechita (2013))

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main theorem

◮ thm: The matrices {W, W

Γ

, WΓ, WT} form an asymptotically free family · let (ǫ, η) ∈ {−1, 1}2 = Z2

2.

· let W(ǫ,η) =        W if (ǫ, η) = (1, 1) W

Γ

if (ǫ, η) = (−1, 1) WΓ if (ǫ, η) = (1, −1) WT if (ǫ, η) = (−1, −1) · let (ǫ1, η1), . . . , (ǫn, ηn) ∈ Zn

2

E(Tr(W(ǫ1,η1) · · · W(ǫn,ηn))) =

• σ∈Sn

p d1d2 #(σ) d fǫ(σ)+#(σ)−n

1

d fη(σ)+#(σ)−n

2

where fǫ(σ) = #(ǫδγ−1δγδǫ ∨ σδσ−1) ( “∨” means the sup of partitions and # means the number of blocks or cycles)

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Computing Moments via Permutations, Notation

◮ [d1] = {1, 2, . . . , d1}, ◮ given i1, . . . , in ∈ [d1] we think of this n-tuple as a function

i : [n] → [d1]

◮ ker(i) ∈ P(n) is the partition of [n] such that i is constant on

the blocks of ker(i) and assumes different values on different blocks

◮ if σ ∈ Sn we also think of the cycles of σ as a partition and

write σ ker(i) to mean that i is constant on the cycles of σ

◮ given σ ∈ Sn we extend σ to a permutation on

[±n] = {−n, . . . , −1, 1, . . . , n} by setting σ(−k) = −k for k > 0

◮ γ = (1, 2, . . . , n), δ(k) = −k ◮ given ǫ1, . . . , ǫn ∈ {−1, 1} let ǫ ∈ S±n be given by

ǫ(k) = ǫ|k| · k

◮ δγ−1δγδ = (1, −n)(2, −1) · · · (n, −(n − 1))

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Computing Moments via Permutations, II

◮ δγ−1δγδ = (1, −n)(2, −1) · · · (n, −(n − 1)) ◮ if Ak = (a(k) ij )ij, a N × N matrix, then

Tr(A1 · · · An) =

N

• i1,...,in=1

a(1)

i1i2a(2) i2i3 · · · a(n) ini1 =

• i±1,...,i±n

δγ−1δγδker(i)

a(1)

i1i−1a(2) i2i−2 · · · a(n) ini−n

Tr

• W(ǫ1,η1) · · · W(ǫn,ηn)

= d−n

1

• i1,...,in

Tr

• W(ǫ1,η1)

i1i2 · · ·

• W(ǫn,ηn))ini1
• =

d−n

1

• i±1,...,i±n

Tr

• W(ǫ1,η1)

i1i−1 · · ·

• W(ǫn,ηn))ini−n
• =

d−n

1

• j±1,...,j±n

Tr

• W(j1, j−1)(η1) · · · W(jn, j−n)(ηn)

where δγ−1δγδ ker(i), ǫδγ−1δγδǫ ker(j) and j = i ◦ ǫ

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Example of Twisting

n = 5, ǫ = (1, 1, −1, −1, 1) δγ−1δγ = (1, −5)(2, −1)(3, −2)(4, −3)(5, −4)

j1 j−1 j2 j−2 j3 j−3 j4 j−4 j5 j−5

ǫδγ−1δγǫ = (1, −5)(2, −1)(3, −4)(−3, −2)(4, 5)

j1 j−1 j2 j−2 j3 j−3 j4 j−4 j5 j−5

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Computing Moments via Permutations, III

Tr

• W(ǫ1,η1) · · · W(ǫn,ηn)

= d−n

1

• j±1,...,j±n

Tr

• W(j1, j−1)(η1) · · · W(jn, j−n)(ηn)

with ǫδγ−1δγδǫ ker(j). Let s = r ◦ η then for δγ−1δγδ ker(r) Tr

• W(j1, j−1)(η1) · · · W(jn, j−n)(ηn)

=

• r±1,...,r±n
• W(j1, j−1)(η1))r1r−1 · · ·
• W(jn, j−n)(ηn)

rnr−n

=

• s±1,...,s±n
• W(j1, j−1)
• s1s−1 · · ·
• W(jn, j−n)
• sns−n

= d−n

2

• s±1,...,s±n
• Gj1G∗

j−1

• s1s−1 · · ·
• GjnG∗

j−n

• sns−n

= d−n

2

• s±1,...,s±n
• t1,...,tn

g(j1)

s1t1g(j−1) s−1t1 · · · g(jn) sntng(j−n) s−ntn

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Gaussian entries

E(Tr(W(ǫ1,η1) · · · W(ǫ1,η1))) = (d1d2)−n

• j±1,...,j±n
• s±1,...,s±n
• t1,...,tn

E(g(j1)

s1t1g(j−1) s−1t1 · · · g(jn) sntng(j−n) s−ntn)

= (d1d2)−n

• j±1,...,j±n
• s±1,...,s±n
• t1,...,tn

E(g(j1)

s1t1 · · · g(jn) sntn g(j−1) s−1t1 · · · g(j−n) s−ntn)

[subject to the condition that ǫδγ−1δγδǫ ker(j) and ηδγ−1δγδη ker(s)] = (d1d2)−n

• j±1,...,j±n
• s±1,...,s±n
• t1,...,tn

E(gα(1) · · · gα(n)gβ(1) · · · gβ(n)) where gα(k) = g(jk)

sktk and gβ(k) = g(j−k) s−ktk . Using

E(gα(1) · · · gα(n)gβ(1) · · · gβ(n)) = |{σ ∈ Sn | β = α ◦ σ}|

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Thus E(Tr(W(ǫ1,η1) · · · W(ǫ1,η1))) = (d1d2)−n

• j±1,...,j±n
• s±1,...,s±n
• t1,...,tn

|{σ ∈ Sn | “various conditions”}| where “various conditions” means

◮ ǫδγ−1δγδǫ ker(j) ◮ ηδγ−1δγδη ker(s) ◮ j−k = jσ(k) which is equivalent to σδσ−1 ker(j) ◮ s−k = sσ(k) which is equivalent to σδσ−1 ker(s) ◮ tk = tσ(k) which is equivalent to σ ker(t)

E(tr(W(ǫ1,η1) · · · W(ǫn,ηn))) =

• σ∈Sn

p d1d2 #(σ) d fǫ(σ)+#(σ)−(n+1)

1

d fη(σ)+#(σ)−(n+1)

2

. where fǫ(σ) = #(ǫδγ−1δγδǫ ∨ σδσ−1) ( “∨” means the sup of partitions)

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finding the highest order terms

◮ general fact: if p and q are pairings then #(p ∨ q) = 1 2#(pq).

In fact we can write the permutation pq as a product of cycles c1c′

1 · · · ckc′ k where c′ i = qc−1 i

q and the blocks of p ∨ q are ci ∪ c′

i ◮ #(ǫδγ−1δγδǫ ∨ σδσ−1) = 1 2#(δγ−1δγ · ǫδσδσ−1ǫ) ◮ if π, σ ∈ Sn and π, σ (the subgroup generated by π and σ)

has only one orbit then there is an integer g (the “genus”) such that #(π) + #(π−1σ) + #(σ) = n + 2(1 − g) and g = 0 only when π is planar or non-crossing with respect to σ.

◮ δγ−1δγ has two cycles so δγ−1δγ, ǫδσδσ−1ǫ can have

either 1 or 2 orbits

◮ if δγ−1δγ, ǫδσδσ−1ǫ has one orbit then

#(ǫδγ−1δγδǫ ∨ σδσ−1) + #(σ) n

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genus of a pairing

1 2 3 4 5 6

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E(tr(W(ǫ1,η1) · · · W(ǫn,ηn))) =

• σ∈Sn

p d1d2 #(σ) d fǫ(σ)+#(σ)−(n+1)

1

d fη(σ)+#(σ)−(n+1)

2

.

◮ σ will not contribute to the limit unless

δγ−1δγ, ǫδσδσ−1ǫ has two orbits, i.e. ǫ is constant on the cycles of σ (write ǫδσδσ−1ǫ = δǫσǫδ(ǫσǫ)−1)

◮ if ǫ is constant on the cycles of σ there is σǫ ∈ Sn such that

ǫδσδσ−1ǫ = δσǫδσ−1

ǫ

(if σ = c1c2 · · · ck then σǫ = cλ1

1 · · · cλk k

where λi is the sign of ǫ on ci)

◮ then 1 2#(δγ−1δγ · ǫδσδσ−1ǫ) = #(γσ−1 ǫ ) ◮ #(σ) + fǫ(σ) = #(σǫ) + #(γσ−1 ǫ ) n + 1 with equality only

if σǫ is non-crossing

◮ #(σ) + fη(σ) = #(ση) + #(γσ−1 η ) n + 1 with equality only

if ση is non-crossing

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E(tr(W(ǫ1,η1) · · · W(ǫn,ηn))) =

• σ

p d1d2 #(σ) + O 1 d1d2

• .

where the sum runs over σ ∈ Sn such that

◮ ǫ and η are constant on the cycles of σ and ◮ both σǫ and ση are both non-crossing. ◮ if ǫ η on a cycle of σ then this cycle must be either a fixed

point or a pair; σǫ = ση and so fǫ(σ) = fη(σ)

◮ σ can only connect W(1,1) to another W(1,1), a W(−1,1) to

another W(−1,1), a W(1,−1) to another W(1,−1), and a W(−1,−1) to another W(−1,−1)

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σ and σ−1

σ = (1, 3, 4)(2)(5)

1 2 3 4 5 1 2 3 4 5 −1 −2 −3 −4 −5

σ = (1, 4, 3)(2)(5)

1 2 3 4 5 1 2 3 4 5 −1 −2 −3 −4 −5

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