Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Two-parameter Noncommutative
Central Limit Theorem
Natasha Blitvić
Vanderbilt University
January 11, 2013
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
(Classical) Central Limit Theorem — CLT
(Classical) Probability space: (Ω, B, P).
Theorem (Classical CLT)
Let X1 , X2 , . . . be a sequence of independent and identically
distributed random variables with E(Xi ) = 0 and E(Xi2 ) = 1 for all
i ∈ N. Let
PN
i=1 Xi
SN = √
.
N
Then, as N → ∞,
SN ⇒
dµ1 (x) =
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√1 exp(−x 2 /2)
2π
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Free Central Limit Theorem — CLT
∗-probability space: (A, ϕ)
Theorem (Free CLT)
Let a1 , a2 , . . . be a sequence of freely independent and identically
distributed self-adjoint elements of A with ϕ(ai ) = 0 and
ϕ(ai2 ) = 1 for all i ∈ N. Let
PN
i=1 ai
.
SN = √
N
Then, as N → ∞,
SN ⇒
dµ0 (x) =
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1
2π
√
4 − x 2,
x ∈ [−2, 2]
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Speicher’s Non-commutative CLT
Condition (Speicher 1992)
Consider a ∗-algebra A, state ϕ : A → C and {ai }i∈N ∈ A
satisfying:
1. (vanishing means) for all i ∈ N,
ϕ(ai ) = ϕ(ai∗ ) = 0.
2. (normalized second moments) for all for all i, j ∈ N with i < j
and , 0 ∈ {1, ∗},
ϕ(ai ai∗ ) = 1
0
and ϕ(ai ai ) = 0 for (, 0 ) 6= (1, ∗).
3. (uniform moment bounds) for all n ∈ N and all
n
Y
(i)
j(1), . . . , j(n) ∈ N, (1), . . . , (n) ∈ {1, ∗}, |ϕ( aj(i) )| ≤ αn .
i=1
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
4. (“independence”) If Ai = ∗-alg(ai ) (i = 1, 2, . . .), then
ϕ(x1 x2 . . . xn ) = ϕ(x1 ) . . . ϕ(xn )
whenever
x1 ∈ Ai1 , . . . , xn ∈ Ain for i1 < . . . < in .
Assume additionally that for all i 6= j and all , 0 ∈ {1, ∗}, ai and
0
aj satisfy the commutation relation
0
0
ai aj = s(j, i) aj ai ,
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s(j, i) ∈ {−1, 1}.
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Theorem (Non-commutative CLT, Speicher 1992)
A non-commutative probability space (A, ϕ) and {ai }i∈N in A
satisfying Condition 1. Fix q ∈ [−1, 1], and let {s(i, j)}1≤i<j be
independent, identically distributed random variables taking values
in {−1, 1} with
E(s(i, j)) = q.
N
1 X
(ai + ai∗ ).
Let SN = √
N i=1
Then, for a.e. sign sequence {s(i, j)}1≤i<j , as N → ∞,
SN ⇒ q-Gaussian.
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
q-Gaussian
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Combinatorial View
Definition
Let P2 (2n) = {all pair-partitions of {1, . . . , 2n}}.
For V = {(w1 , z1 ), . . . , (wn , zn )} ∈ P2 (2n), pairs (wi , zi ) and
(wj , zj ) are said to cross if wi < wj < zi < zj .
Then,
lim ϕ(SN2n−1 ) = 0,
lim ϕ(SN2n ) =
N→∞
1
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2
3
4
N→∞
5
6
7
X
q cross(V ) ,
V ∈P2 (2n)
8
9
10
11
12
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
More general commutation structure?
Why s(i, j) ∈ {−1, 1}?
j
µ(i,j)
i
For µ(i, j) ∈ R,
• Limit exists?
• q-Gaussian?
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
More general combinatorics?
Definition
Let P2 (2n) = {all pair-partitions of {1, . . . , 2n}}.
For V = {(w1 , z1 ), . . . , (wn , zn )} ∈ P2 (2n), pairs (wi , zi ) and
(wj , zj ) are said to nest if wi < wj < zj < zi .
1
ei
...
ej
...
zi
...
zj
2n
1
ei
...
ej
...
zj
...
zi
2n
The number of nestings of V is:
nest(V ) = #{(wi , zi ), (wj , zj ) ∈ V | i < j, wi < wj < zj < zi }.
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Generalized Non-commutative CLT
Condition
Consider a ∗-algebra A, state ϕ : A → C and {ai }i∈N ∈ A
satisfying:
1. (vanishing means) for all i ∈ N,
ϕ(ai ) = ϕ(ai∗ ) = 0.
2. (normalized second moments) for all for all i, j ∈ N with i < j
and , 0 ∈ {1, ∗},
ϕ(ai ai∗ ) = 1
0
and ϕ(ai ai ) = 0 for (, 0 ) 6= (1, ∗).
3. (uniform moment bounds) for all n ∈ N and all
n
Y
(i)
j(1), . . . , j(n) ∈ N, (1), . . . , (n) ∈ {1, ∗}, |ϕ( aj(i) )| ≤ αn .
i=1
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
4. (“independence”) If Ai = ∗-alg(ai ) (i = 1, 2, . . .), then
ϕ(x1 x2 . . . xn ) = ϕ(x1 ) . . . ϕ(xn )
whenever
x1 ∈ Ai1 , . . . , xn ∈ Ain for i1 < . . . < in .
Assume additionally that for all i 6= j and all , 0 ∈ {1, ∗}, ai and
0
aj satisfy the commutation relations
for
ai aj = µ1,1 (j, i) aj ai ,
ai∗ aj∗ = µ∗,∗ (j, i) aj∗ ai∗
ai aj∗ = µ∗,1 (j, i) aj∗ ai ,
ai∗ aj = µ1,∗ (j, i) aj ai∗
µ1,1 (j, i), µ∗,∗ (j, i), µ∗,1 (j, i), µ1,∗ (j, i) ∈ R.
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Classical, Free, NC CLT
Moments and Combinatorics
0
0
ai aj = µ0 , (j, i) aj ai ,
Generalized NC CLT
Random Matrix Models
µ0 , (j, i) ∈ R
Lemma
Coefficients {µ0 , (i, j)} satisfy the consistency relations:
1
µ,0 (i, j)
1
µ1,1 (i, j) =
,
µ∗,∗ (i, j)
µ0 , (j, i) =
for all i 6= j, , 0 ∈ {1, ∗},
(1)
1
µ∗,1 (i, j)
(2)
µ1,∗ (i, j) =
In addition, when ϕ is assumed to be positive,
µ∗,∗ (i, j)
µ∗,1 (i, j)
=
.
| µ∗,1 (i, j) |
| µ∗,∗ (i, j) |
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(3)
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Theorem (Generalized non-commutative CLT, B. 2012)
Consider a ∗-probability space (A, ϕ) and {ai }i∈N in A
satisfying Condition 2.
In addition, suppose that {µ∗,∗ (i, j)}1≤i<j satisfy
µ∗,1 (i, j) = t µ∗,∗ (i, j)
(3’)
for some t > 0.
N
1 X
Let SN = √
(ai + ai∗ ).
N i=1
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
For fixed q ∈ R, let {µ∗,∗ (i, j)}1≤i<j be independent, identically
distributed random variables with
E(µ∗,∗ (i, j)) = q/t
and
E(µ∗,∗ (i, j)2 ) = 1,
and populate the remaining µ,0 (i, j), for , 0 ∈ {1, ∗} and i 6= j
(i, j ∈ N), by consistency conditions (1)-(3’).
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
For fixed q ∈ R, let {µ∗,∗ (i, j)}1≤i<j be independent, identically
distributed random variables with
E(µ∗,∗ (i, j)) = q/t
and
E(µ∗,∗ (i, j)2 ) = 1,
and populate the remaining µ,0 (i, j), for , 0 ∈ {1, ∗} and i 6= j
(i, j ∈ N), by consistency conditions (1)-(3’).
Then, for a.e. coefficient sequence {µ∗,∗ (i, j)}1≤i<j ,
Sn ⇒ (q, t)-Gaussian.
lim ϕ(SN2n−1 ) = 0,
N→∞
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lim ϕ(SN2n ) =
N→∞
X
q cross(V ) t nest(V )
V ∈P2 (2n)
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Proof sketch
Show that ZN =
a1 + . . . + aN
√
converges in mixed moments.
N
Fix k ∈ N and a ∗-pattern (1), . . . , (k) ∈ {1, ∗}, and look at the
mixed moment
X
1
(1)
(k)
(1)
(k)
ϕ ai(1) . . . ai(k) .
ϕ(ZN . . . ZN ) = k/2
N
i(1),...,i(k)∈[N]
(1)
(k)
Repetition patterns of indices in ϕ ai(1) . . . ai(k) induce set
partitions.
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Note:
• By standard arguments, only pair-partitions can contribute to
the limit. Set k = 2n.
(1)
(n)
• If limN→∞ E(ϕ(ZN
. . . ZN )) exists, Markov inequality +
more careful estimates yield convergence a.e.
Remains to factor
(1)
(2n)
ϕ ai(1) . . . ai(2n)
If i(1) = i(2) < i(3) = i(4) < . . . < i(2n − 1) = i(2n), then
(2n−1) (2n)
(1) (2)
(1)
(2n)
ϕ ai(1) . . . ai(2n) = ϕ ai(1) ai(2) . . . ϕ ai(2n−1) ai(2n)
e.g.
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ϕ
0
0
ai j ai j aji aj i
0
0
j j
= ϕ ai ai ϕ aji aj i
i <j
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Otherwise, use the commutation relation.
For i < j, need to factor
0
j j i 0i
ϕ aj aj ai ai
0
0
0
j i j 0i
i j i j
ϕ ai aj ai aj
and ϕ aj ai aj ai
0
0
0
j i 0i j
i j j i
ϕ ai aj aj ai
and ϕ aj ai ai aj
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Coefficients incurred:
µi ,0j (j, i)µi ,j (j, i)µ0i ,0j (j, i)µ0i ,j (j, i)
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µ0i ,j (i, j)
and
µ0j ,i (j, i)
µ0i ,j (i, j)µ0i ,0j (i, j)
and
µj ,i (j, i)µj ,0i (j, i)
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Random Matrix Models
0
If ϕ(ai ai∗ ) = 1 and ϕ(ai ai ) = 0 for (, 0 ) 6= (1, ∗),
µ1,∗ (j, i)µ1,1 (j, i)µ∗,∗ (j, i)µ∗,1 (j, i)
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µ∗,1 (i, j)
and
µ∗,1 (j, i)
µ∗,1 (i, j)µ∗,∗ (i, j)
and
µ1,1 (i, j)µ∗,1 (i, j)
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Recall:
0
0
ai aj = µ0 , (j, i) aj ai ,
µ0 , (j, i) ∈ R
And coefficients {µ0 , (i, j)} satisfy the consistency relations:
1
µ,0 (i, j)
1
µ1,1 (i, j) =
,
µ∗,∗ (i, j)
µ0 , (j, i) =
for all i 6= j, , 0 ∈ {1, ∗},
(1)
1
µ∗,1 (i, j)
(2)
µ1,∗ (i, j) =
µ∗,1 (i, j) = t µ∗,∗ (i, j)
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(3’)
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Applying consistency relations (1)-(3’) and taking E
1
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q
and
q
t
and
t
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Theorem (Extended Jordan-Wigner Transform, Biane 1997)
Non-commutative probability space (A, ϕ) with A = M2 (C)⊗n
and ϕ(a) = hae0⊗n , e0⊗n i.
Fix q ∈ [−1, 1] and s(i, j) ∈ {−1, 1} for all 1 ≤ i < j ≤ n. Let
1 0
1 0
0 1
σ1 =
,
σ−1 =
,
γ=
0 1
0 −1
0 0
and, for i = 1, . . . , n, let the element ai ∈ M2 (C)⊗n be given by
ai = σs(1,i) ⊗ σs(2,i) ⊗ . . . ⊗ σs(i−1,i) ⊗ γ ⊗ σ1 ⊗ . . . ⊗ σ1 .
|
{z
}
⊗(n−i)
=σ1
Then, for every n ∈ N, a1 , . . . , an satisfy Condition 1.
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Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Theorem (Two-parameter Jordan-Wigner Transform, B. 2012)
Non-commutative probability space (A, ϕ) with A = M2 (C)⊗n
and ϕ(a) = hae0⊗n , e0⊗n i.
Fix |q| ≤ t and µ,0 (i, j) ∈ R for all 1 ≤ i < j ≤ n, , 0 ∈ {1, ∗}.
Let
1 √0
0 1
,
γ=
σx =
0 0
tx
0
and, for i = 1, . . . , n, let the element ai ∈ M2 (C)⊗n be given by
ai = σµ(1,i) ⊗ σµ(2,i) ⊗ . . . ⊗ σµ(i−1,i) ⊗ γ ⊗ σ1 ⊗ . . . ⊗ σ1 .
|
{z
}
⊗(n−i)
=σ1
Then, for every n ∈ N, a1 , . . . , an satisfy Condition 2.
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Looking ahead
• Connections.
q = -1
free
q=0
anti-symmetric
q=1
symmetric
q=0<t<1
• More general commutation structure.
• Beyond i.i.d.
• Practical applications?
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Classical, Free, NC CLT
Moments and Combinatorics
Generalized NC CLT
Random Matrix Models
Taknh yuo
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