Introduction to Free Probability Theory:
Combinatorics, Random Matrices, and
Quantum Groups
Roland Speicher
Saarland University
Saarbrücken
Germany
Roland Speicher (Saarland University)
Free Probability Theory
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Free Probability: The Basics
Section 1
Free Probability: The Basics
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Free Probability Theory
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Free Probability: The Basics
Some History
1985 Voiculescu introduces "freeness" in the context of isomorphism
problem of free group factors
1991 Voiculescu discovers relation with random matrices (which leads,
among others, to deep results on free group factors)
1994 Speicher develops combinatorial theory of freeness, based on "free
cumulants"
2009 Köstler and Speicher discover free de Finetti theorem
.... ... many new results on operator algebras, eigenvalue distribution of
random matrices, and much more ....
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Free Probability Theory
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Free Probability: The Basics
Definition of Freeness
Definition (Voiculescu 1985)
Let (A, ϕ) be non-commutative probability space, i.e., A is a unital
algebra and ϕ : A → C is unital linear functional (i.e., ϕ(1) = 1)
Unital subalgebras Ai (i ∈ I) are free or freely independent, if
ϕ(a1 · · · an ) = 0 whenever
ai ∈ Aj(i) ,
j(i) ∈ I
∀i,
j(1) 6= j(2) 6= · · · =
6 j(n)
ϕ(ai ) = 0 ∀i
Random variables x1 , . . . , xn ∈ A are free, if their generated unital
subalgebras Ai := algebra(1, xi ) are so.
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Free Probability Theory
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Free Probability: The Basics
What is Freeness?
Freeness between A and B is an infinite set of equations relating various
moments in A and B:
ϕ p1 (A)q1 (B)p2 (A)q2 (B) · · · = 0
Basic Observation
Freeness between A and B is actually a rule for calculating mixed
moments in A and B from the moments of A and the moments of B:
ϕ An1 B m1 An2 B m2 · · · = polynomial ϕ(Ai ), ϕ(B j )
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Free Probability Theory
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Free Probability: The Basics
Freeness is a Rule for Calculating Mixed Moments
Example
ϕ An − ϕ(An )1 B m − ϕ(B m )1 = 0,
thus
ϕ(An B m ) − ϕ(An · 1)ϕ(B m ) − ϕ(An )ϕ(1 · B m ) + ϕ(An )ϕ(B m )ϕ(1 · 1) = 0,
and hence
ϕ(An B m ) = ϕ(An ) · ϕ(B m )
Freeness is a rule for calculating mixed moments, analogous to the concept
of independence for random variables.
Thus freeness is also called free independence
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Free Probability Theory
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Free Probability: The Basics
Freeness is a Rule for Calculating Mixed Moments
Freeness is analogous to the concept of classical independence for random
variables.
Note: free independence is a different rule from classical independence; free
independence occurs typically for non-commuting random variables, like
for operators on Hilbert spaces or (random) matrices
Example
ϕ A − ϕ(A)1 · B − ϕ(B)1 · A − ϕ(A)1 · B − ϕ(B)1 = 0,
which results in
ϕ(ABAB) = ϕ(AA) · ϕ(B) · ϕ(B) + ϕ(A) · ϕ(A) · ϕ(BB)
− ϕ(A) · ϕ(B) · ϕ(A) · ϕ(B)
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Free Probability Theory
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Free Probability and Random Matrices
Section 2
Free Probability and Random Matrices
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Free Probability Theory
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Free Probability and Random Matrices
Where Does Freeness Show Up?
generators of the free group in the corresponding free group von
Neumann algebras L(Fn )
creation and annihilation operators on full Fock spaces
for many classes of random matrices
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Free Probability Theory
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Free Probability and Random Matrices
Asymptotic Freeness of Random Matrices
Theorem (Voiculescu 1991)
Large classes of independent random matrices (like Gaussian or Wishart
matrices) become asymptoticially freely independent, with respect to
ϕ = tr := N1 Tr, if N → ∞.
Example
This means, for example: if XN and YN are independent N × N Wigner or
Wishart matrices, respectively, then we have almost surely:
2
lim tr(XN YN XN YN ) = lim tr(XN
) · lim tr(YN )2
N →∞
N →∞
2
+ lim tr(XN ) · lim
N →∞
Roland Speicher (Saarland University)
N →∞
tr(YN2 )
N →∞
− lim tr(XN )2 · lim tr(YN )2
N →∞
Free Probability Theory
N →∞
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Free Probability and Random Matrices
Asymptotic Freeness of Random Matrices
Theorem (Voiculescu 1991)
Large classes of independent random matrices (like Gaussian or Wishart
matrices) become asymptoticially freely independent, with respect to
ϕ = tr := N1 Tr, if N → ∞.
Example (tr(XN XN YN YN XN YN YN XN ) → 2 for XN , YN Gaussian)
4
3.5
tr(AABBABBA)
3
2.5
2
1.5
1
0.5
0
0
20
40
60
80
100
120
140
160
180
200
N
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Free Probability Theory
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Free Probability and Random Matrices
Asymptotic Freeness of Random Matrices
Theorem (Voiculescu 1991)
Large classes of independent random matrices (like Gaussian or Wishart
matrices) become asymptoticially freely independent, with respect to
ϕ = tr := N1 Tr, if N → ∞.
Example
This means, for example: if XN and YN are independent N × N Wigner or
Wishart matrices, respectively, then we have almost surely:
2
lim tr(XN YN XN YN ) = lim tr(XN
) · lim tr(YN )2
N →∞
N →∞
2
+ lim tr(XN ) · lim
N →∞
N →∞
tr(YN2 )
N →∞
− lim tr(XN )2 · lim tr(YN )2
N →∞
N →∞
Hence we have a rule for calculating asymptotically mixed moments of our
matrices with respect to the normalized trace tr.
Roland Speicher (Saarland University)
Free Probability Theory
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Free Probability and Random Matrices
Asymptotic Freeness of Random Matrices
Theorem (Voiculescu 1991)
Large classes of independent random matrices (like Gaussian or Wishart
matrices) become asymptoticially freely independent, with respect to
ϕ = tr := N1 Tr, if N → ∞.
Note that moments with respect to tr determine the eigenvalue distribution
of a matrix.
For an N × N matrix X = X ∗ with eigenvalues λ1 , . . . , λN its eigenvalue
distribution
1
µX := (δλ1 + · · · + δλN )
N
is determined by
Z
tk dµX (t) = tr(X k )
for all k = 0, 1, 2, . . .
R
.
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Free Probability Theory
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Free Probability and Random Matrices
Randomly Rotated Matrices are Asymptotically
Free
1/(π (4 − x2)1/2)
0.4
0.35
0.3
0.25
0.2
0.15
−1.5
−1
−0.5
0
x
0.5
1
1.5
2800 eigenvalues of A + U BU ∗ , where A and B are diagonal matrices with
1400 eigenvalues +1 and 1400 eigenvalues -1, and U is a randomly chosen
(with respect to the Haar measure) unitary matrix
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Free Probability Theory
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Cominatorics of Freeness
Section 3
The Combinatorics of Freeness: Non-Crossing
Partitions and Free Cumulants
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Free Probability Theory
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Cominatorics of Freeness
Understanding the Freeness Rule: the Idea of
Cumulants
write moments in terms of other quantities, which we call free
cumulants
freeness is much easier to describe on the level of free cumulants:
vanishing of mixed cumulants
relation between moments and cumulants is given by summing over
non-crossing or planar partitions
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Free Probability Theory
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Cominatorics of Freeness
Non-Crossing Partitions
Definition
A partition of {1, . . . , n} is a decomposition π = {V1 , . . . , Vr } with
[
Vi 6= ∅,
Vi ∩ Vj = ∅ (i 6= y),
Vi = {1, . . . , n}
i
The Vi are the blocks of π ∈ P(n).
1 2 3 4
1 2 3 4
1 2 3 4
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Free Probability Theory
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Cominatorics of Freeness
Non-Crossing Partitions
Definition
A partition of {1, . . . , n} is a decomposition π = {V1 , . . . , Vr } with
[
Vi 6= ∅,
Vi ∩ Vj = ∅ (i 6= y),
Vi = {1, . . . , n}
i
The Vi are the blocks of π ∈ P(n).
A partition π is non-crossing if we do not have
p1 < q1 < p2 < q2
such that p1 , p2 are in same block, q1 , q2 are in same block, but those two
blocks are different.
NC(n) := {non-crossing partitions of {1,. . . ,n}}
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Free Probability Theory
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Cominatorics of Freeness
Moments and Cumulants
Definition
For unital linear functional
ϕ:A→C
we define cumulant functionals kn (for all n ≥ 1)
k n : An → C
as multi-linear functionals by moment-cumulant relation
X
ϕ(a1 · · · an ) =
kπ [a1 , . . . , an ]
π∈N C(n)
Note: classical cumulants are defined by a similar formula, where only
N C(n) is replaced by P(n)
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Free Probability Theory
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Cominatorics of Freeness
Example (n = 1)
ϕ(a1 ) = k1 (a1 )
a1
Example (n = 2)
a1 a2
ϕ(a1 a2 ) = k2 (a1 , a2 )
+ k1 (a1 )k1 (a2 )
and thus
k2 (a1 , a2 ) = ϕ(a1 a2 ) − ϕ(a1 )ϕ(a2 )
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Free Probability Theory
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Cominatorics of Freeness
Example (n = 3)
a1 a2 a3
ϕ(a1 a2 a3 ) = k3 (a1 , a2 , a3 )
+ k1 (a1 )k2 (a2 , a3 )
+ k2 (a1 , a2 )k1 (a3 )
+ k2 (a1 , a3 )k1 (a2 )
+ k1 (a1 )k1 (a2 )k1 (a3 )
and thus
k3 (a1 , a2 , a3 ) = ϕ(a1 a2 a3 ) − ϕ(a1 )ϕ(a2 a3 ) − ϕ(a2 )ϕ(a1 a3 )
− ϕ(a3 )ϕ(a1 a2 ) + 2ϕ(a1 )ϕ(a2 )ϕ(a3 )
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Free Probability Theory
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Cominatorics of Freeness
Example (n = 4)
ϕ(a1 a2 a3 a4 ) =
+
+
+
+
+
+
+
+
+
+
+
+
+
= k4 (a1 , a2 , a3 , a4 ) + k1 (a1 )k3 (a2 , a3 , a4 )
+ k1 (a2 )k3 (a1 , a3 , a4 ) + k1 (a3 )k3 (a1 , a2 , a4 )
+ k3 (a1 , a2 , a3 )k1 (a4 ) + k2 (a1 , a2 )k2 (a3 , a4 )
+ k2 (a1 , a4 )k2 (a2 , a3 ) + k1 (a1 )k1 (a2 )k2 (a3 , a4 )
+ k1 (a1 )k2 (a2 , a3 )k1 (a4 ) + k2 (a1 , a2 )k1 (a3 )k1 (a4 )
+ k1 (a1 )k2 (a2 , a4 )k1 (a3 ) + k2 (a1 , a4 )k1 (a2 )k1 (a3 )
+ k2 (a1 , a3 )k1 (a2 )k1 (a4 ) + k1 (a1 )k1 (a2 )k1 (a3 )k1 (a4 )
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Free Probability Theory
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Cominatorics of Freeness
Freeness =
ˆ Vanishing of Mixed Cumulants
Theorem (Speicher 1994)
The fact that x1 , . . . , xm are free is equivalent to the fact that
kn (xi(1) , . . . , xi(n) ) = 0
whenever
1 ≤ i(1), . . . , i(n) ≤ m
there are p, q such that i(p) 6= i(q) (in particular, n ≥ 2)
Example
If x and y are free then: ϕ(xyxy) =
k1 (x)k1 (x)k2 (y, y) + k2 (x, x)k1 (y)k1 (y) + k1 (x)k1 (y)k1 (x)k1 (y)
xyxy
xyxy
xyxy
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Free Probability Theory
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Cominatorics of Freeness
Sum of Free Variables: Description via
R-Transform
Definition
Consider a random variable x ∈ A. We define its Cauchy transform
G = Gx and its R-transform R = Rx by
∞
G(z) =
1 X ϕ(xn )
+
,
z
z n+1
R(z) =
n=1
∞
X
kn (x, . . . , x)z n−1
n=1
Theorem (Voiculescu 1986, Speicher 1994)
Then we have
1
G(z)
+ R(G(z)) = z
Rx+y (z) = Rx (z) + Ry (z) if x and y are free
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Free Probability Theory
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Cominatorics of Freeness
Eigenvalues of the Sum of Independent Gaussian
and Wishart 3000 × 3000 Random Matrices
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−2
−1
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0
1
2
Free Probability Theory
3
4
5
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Cominatorics of Freeness
Product of Free Variables: Description via
S-Transform
Theorem (Voiculescu 1987; Haagerup 1997; Nica, Speicher 1997)
Put
Mx (z) :=
∞
X
ϕ(xm )z m
m=1
and define
Sx (z) :=
1 + z <−1>
Mx
(z)
z
S-transform of x
Then: If x and y are free, we have
Sxy (z) = Sx (z) · Sy (z).
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Free Probability Theory
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Cominatorics of Freeness
Eigenvalues of the Product of Two Independent
Wishart 2000 × 2000 Random Matrices
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
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1
1.5
Free Probability Theory
2
2.5
3
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Cominatorics of Freeness
Polynomials in Free Variables: Linearisation and
Operator-Valued Free Probability
Theorem (Belinschi, Mai, Speicher 2013)
The following algorithm allows the calculation of the distribution of any
selfadjoint polynomial p(x, y) in two non-commuting variables x and y,
given the distribution of x and the distribution of y:
Linearize p(x, y) to p̂ = x̂ + ŷ.
Calculate Gx̂ (b) out of Gx (z) and Gŷ (b) out of Gy (z)
Get w1 (b) as the fixed point of the iteration
w 7→ Gŷ (b + Gx̂ (w)−1 − w)−1 − (Gx̂ (w)−1 − w)
Calculate Gp̂ (b)
x̂ (ω1 (b)) and recover Gp (z) as one entry of
= G
z 0
Gp̂ (b) for b =
0 0
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Free Probability Theory
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Cominatorics of Freeness
Polynomials in Free Variables: Linearisation and
Operator-Valued Free Probability
Example
P (X, Y ) = XY + Y X + X 2
for independent X, Y ; X is Gaussian and Y is Wishart
0.35
0.3

0
p̂ =  x
y+
x
2
x
0
−1

x
y+ 2
−1 
0
0.25
0.2
0.15
0.1
0.05
0
−5
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Free Probability Theory
0
5
10
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Freeness and Quantum Groups
Section 4
Freeness and Quantum Groups:
Non-Commutative de Finetti Theorem
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Free Probability Theory
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Freeness and Quantum Groups
Classical Exchangeable Random Variables
Consider probability space (Ω, A, P ). Denote expectation by ϕ,
Z
ϕ(Y ) =
Y (ω)dP (ω).
Ω
Definition
We say that random variables X1 , X2 , . . . are exchangeable if their joint
distribution is invariant under finite permutations, i.e. if
ϕ(Xi(1) · · · Xi(n) ) = ϕ(Xπ(i(1)) · · · Xπ(i(n)) )
for all n ∈ N, all i(1), . . . , i(n) ∈ N, and all permutations π
Example
ϕ(X1n ) = ϕ(X7n ),
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ϕ(X13 X37 X4 ) = ϕ(X83 X27 X9 )
Free Probability Theory
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Freeness and Quantum Groups
Classical de Finetti Theorem
Definition
Atail :=
\
σ(Xk | k ≥ i)
i∈N
E : L∞ (Ω, A, P ) → L∞ (Ω, Atail , P )
Theorem (de Finetti 1931, Hewitt, Savage 1955)
The following are equivalent for an infinite sequence of random variables:
the sequence is exchangeable
the sequence is independent and identically distributed with respect to
the conditional expectation E onto the tail σ-algebra of the sequence
m(1)
E[X1
m(2)
X2
m(1)
· · · Xnm(n) ] = E[X1
Roland Speicher (Saarland University)
m(2)
] · E[X2
Free Probability Theory
] · · · E[Xnm(n) ]
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Freeness and Quantum Groups
Quantum Permutation Group
Definition (Wang 1998)
The quantum permutation group As (k) is given by the universal unital
C ∗ -algebra generated by uij (i, j = 1, . . . , k) subject to the relations
u2ij = uij = u∗ij for all i, j = 1, . . . , k
each row and column of u = (uij )ki,j=1 is a partition of unity:
k
X
uij = 1 ∀i
and
j=1
k
X
uij = 1 ∀j
i=1
(note: elements within a row or within a column are orthogonal)
As (k) is a compact quantum group in the sense of Woronowicz.
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Free Probability Theory
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Freeness and Quantum Groups
Quantum Exchangeability
Definition
A sequence x1 , . . . , xk in (A, ϕ) is quantum exchangeable if its
distribution does not change under the action of quantum permutations
Sk+ , i.e., if we have:
Let a quantum permutation u = (uij ) ∈ C(Sk+ ) act on (x1 , . . . , xk ) by
yi :=
X
uij ⊗ xj
∈
C(Sk+ ) ⊗ A
j
Then
(x1 , . . . , xk ) ∈ (A, ϕ)
(y1 , . . . , yk ) ∈ (C(Sk+ ) ⊗ A, id ⊗ ϕ)
have the same distribution, i.e.,
ϕ(xi(1) · · · xi(n) ) · 1C(S + ) = id ⊗ ϕ(yi(1) · · · yi(n) )
k
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Free Probability Theory
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Freeness and Quantum Groups
Non-commutative de Finetti Theorem
Definition
Define the tail algebra of the sequence:
\
Atail :=
vN(xk | k ≥ i),
i∈N
then there exists conditional expectation E : vN(xi | i ∈ N) → Atail .
Theorem (Köstler, Speicher 2009)
The following are equivalent for an infinite sequence of non-commutative
random variables:
the sequence is quantum exchangeable
the sequence is free and identically distributed with respect to the
conditional expectation E onto the tail-algebra of the sequence
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Free Probability Theory
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The End
Section 5
The End
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Free Probability Theory
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The End
Summary
Free independence is the basic probabilistic structure in a maximal
non-commutative world
It shows up in quite different contextes; like operator algebras, but ...
... in particular, it describes also the asymptotic large N -regime of
random matrices
Its combinatorial structure is governed by non-crossing partitions and
free cumulants
The symmetries of freeness are given by liberated quantum groups,
like quantum permutation and quantum orthogonal groups
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Free Probability Theory
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