free probability theory and random matricesweb.mit.edu/sea06/agenda/talks/speicher_survey.pdf ·...
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Free Probability Theory and Random
Matrices
Roland Speicher
Queen’s University
Kingston, Canada
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We are interested in the limiting eigenvalue distribution of
N × N random matrices
for
N → ∞.
Usually, large N distributions are close to the N → ∞ limit, and
asymptotic results give good predictions for finite N .
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We can consider the convergence for N → ∞ of
• the eigenvalue distribution of one ”typical” realization of the
N × N random matrix
• the averaged eigenvalue distribution over many realizations
of the N × N random matrices
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Consider (selfadjoint!) Gaussian N × N random matrix.
We have almost sure convergence (convergence of ”typical” re-
alization) of its eigenvalue distribution towards
Wigner’s semicircle.
−3 −2 −1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
N=300
Pro
babi
lity
−3 −2 −1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N=1000
Pro
babi
lity
−3 −2 −1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N=3000
Pro
babi
lity
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Convergence of the averaged eigenvalue distribution happens
usually much faster, very good agreement with asymptotic limit
for moderate N .
−3 −2 −1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N=5
Pro
babi
lity
−3 −2 −1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N=20
Pro
babi
lity
−3 −2 −1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N=50P
roba
bilit
y
trials=5000
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Consider Wishart random matrix A = XX∗, where X is N ×Mrandom matrix with independent Gaussian entries
Its eigenvalue distribution converges (averaged and almostsurely) towards Marchenko-Pastur distribution.
Example: M = 2N , 2000 trials
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
N=10
Pro
babi
lity
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
N=50
Pro
babi
lity
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We want to consider more complicated situations, built out of
simple cases (like Gaussian or Wishart) by doing operations like
• taking the sum of two matrices
• taking the product of two matrices
• taking corners of matrices
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Note: If different N × N random matrices A and B are involved
then the eigenvalue distribution of non-trivial functions f(A, B)
(like A+B or AB) will of course depend on the relation between
the eigenspaces of A and of B.
However: It turns out there is a deterministic and treatable result
if
• the eigenspaces are in ”generic” position and
• if N → ∞
This is the realm of free probability theory.
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Consider N × N random matrices A and C such that
• A has an asymptotic eigenvalue distribution for N → ∞ and
C has an asymptotic eigenvalue distribution for N → ∞
• A and C are independent (i.e., entries of A are independent
from entries of C)
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Then eigenspaces of A and of C might still be in special relation
(e.g., both A and C could be diagonal).
However, consider now
A and B := UCU∗,
where U is Haar unitary N × N random matrix.
Then, eigenspaces of A and of B are in ”generic” position and
the asymptotic eigenvalue distribution of A+B depends only on
the asymptotic eigenvalue distribution of A and the asymptotic
eigenvalue distribution of B (which is the same as the one of C).
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We can expect that the asymptotic eigenvalue distribution of
f(A, B) depends only on the asymptotic eigenvalue distribution
of A and the asymptotic eigenvalue distribution of B if
• A and B are independent
• one of them is unitarily invariant
(i.e., the joint distribution of the entries does not change
under unitary conjugation)
Note: Gaussian and Wishart random matrices are unitarily in-
variant
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Thus: the asymptotic eigenvalue distribution of
• the sum of random matrices in generic position
A + UCU∗
• the product of random matrices in generic position
AUCU∗
• corners of unitarily invariant matrices UCU∗
should only depend on the asymptotic eigenvalue distribution of
A and of C.
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Example: sum of independent Gaussian and Wishart (M = 2N)
random matrices, averaged over 10000 trials
−3 −2 −1 0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N=5−3 −2 −1 0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N=50
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Example: product of two independent Wishart (M = 5N) ran-
dom matrices, averaged over 10000 trials
0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N=50 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N=50
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Example: upper left corner of size N/2 × N/2 of a randomly
rotated N × N projection matrix,
with half of the eigenvalues 0 and half of the eigenvalues 1,
averaged over 10000 trials
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.5
1
1.5
N=80.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.5
1
1.5
N=3214
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Problems:
• Do we have a conceptual way of understanding
these asymptotic eigenvalue distributions?
• Is there an algorithm for actually calculating these
asymptotic eigenvalue distributions?
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How do we analyze the eigenvalue distributions?
eigenvalue distributionof matrix A
=̂
knowledge oftraces of powers,
tr(Ak)
1
N
(
λk1 + · · · + λk
N
)
= tr(Ak)
averaged eigenvaluedistribution of
random matrix A=̂
knowledge ofexpectations of
traces of powers,
E[tr(Ak)]
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Stieltjes inversion formula. If one knows the asymptotic mo-
ments
αk := limN→∞
E[tr(Ak)]
of a random matrix A, then one can get its asymptotic eigenvalue
distribution µ as follows:
Form Cauchy (or Stieltjes) transform
G(z) :=∞∑
k=0
αk
zk+1
Then:
dµ(t) = −1
πlimε→0
ℑG(t + iε)
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Consider random matrices A and B in generic position.
We want to understand A + B, i.e., for all k ∈ N
E[
tr(
(A + B)k)]
.
But
E[tr((A+B)6)] = E[tr(A6)]+· · ·+E[tr(ABAABA)]+· · ·+E[tr(B6)],
thus we need to understand
mixed moments in A and B
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Use following notation:
ϕ(A) := limN→∞
E[tr(A)].
Question: If A and B are in generic position, can we understand
ϕ (An1Bm1An2Bm2 · · · )
in terms of
(
ϕ(Ak))
k∈Nand
(
ϕ(Bk))
k∈N
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Example: Consider two Gaussian random matrices A, B which
are independent (and thus in generic position).
Then the asymptotic mixed moments in A and B
ϕ (An1Bm1An2Bm2 · · · )
are given by
#
{
non-crossing/planar pairings of the pattern
A · A · · ·A︸ ︷︷ ︸
n1-times
·B · B · · ·B︸ ︷︷ ︸
m1-times
·A · A · · ·A︸ ︷︷ ︸
n2-times
·B · B · · ·B︸ ︷︷ ︸
m2-times
· · · ,
which do not pair A with B
}
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Example: ϕ(AABBABBA) = 2 since there are two such non-
crossing pairings:
A A B B A B B A
and
A A B B A B B A
Note: each of the pairings connects at least one of the groups
An1, Bm1, An2, . . . only among itself!
and thus:
ϕ
((
A2−ϕ(A2)1)(
B2−ϕ(B2)1)(
A−ϕ(A)1)(
B2−ϕ(B2)1)(
A−ϕ(A)1))
= 0
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In general we have
ϕ((
An1 −ϕ(An1) ·1)·(Bm1 −ϕ(Bm1) ·1
)·(An2 −ϕ(An2) ·1
)· · ·
)
= #
{
non-crossing pairings which do not pair A with B,
and for which each group is connected with some other group
}
=0
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Actual equation for the calculation of the mixed moments
ϕ1 (An1Bm1An2Bm2 · · · )
is different for different random matrix ensembles.
However, the relation between the mixed moments,
ϕ
((
An1 − ϕ(An1) · 1)
·(
Bm1 − ϕ(Bm1) · 1)
· · ·
)
= 0
remains the same for matrix ensembles in generic position and
constitutes the definition of freeness.
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Definition [Voiculescu 1985]: A and B are free (with respect
to ϕ) if we have for all n1, m1, n2, · · · ≥ 1 that
ϕ
((
An1−ϕ(An1) ·1)
·(
Bm1−ϕ(Bm1) ·1)
·(
An2−ϕ(An2) ·1)
· · ·
)
= 0
ϕ
((
Bn1−ϕ(Bn1)·1)
·(
Am1−ϕ(Am1)·1)
·(
Bn2−ϕ(Bn2)·1)
· · ·
)
= 0
ϕ
(
alternating product in centered words in A and in B
)
= 0
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Note: freeness is a rule for calculating mixed moments in A and
B from the moments of A and the moments of B.
Example:
ϕ
((
An − ϕ(An)1)(
Bm − ϕ(Bm)1))
= 0,
thus
ϕ(AnBm)−ϕ(An·1)ϕ(Bm)−ϕ(An)ϕ(1·Bm)+ϕ(An)ϕ(Bm)ϕ(1·1) = 0,
and hence
ϕ(AnBm) = ϕ(An) · ϕ(Bm).
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Freeness is a rule for calculating mixed moments, analo-
gous to the concept of independence for random variables.
Thus freeness is also called free independence
Note: free independence is a different rule from classical indepen-
dence; free independence occurs typically for non-commuting
random variables.
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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Consider A, B free.
Then, by freeness, the moments of A+B are uniquely determined
by the moments of A and the moments of B.
Notation: We say the distribution of A + B is the
free convolution
of the distribution of A and the distribution of B,
µA+B = µA ⊞ µB.
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In principle, freeness determines this, but the concrete nature of
this rule is not clear.
Examples: We have
ϕ(
(A + B)1)
= ϕ(A) + ϕ(B)
ϕ(
(A + B)2)
= ϕ(A2) + 2ϕ(A)ϕ(B) + ϕ(B2)
ϕ(
(A + B)3)
= ϕ(A3) + 3ϕ(A2)ϕ(B) + 3ϕ(A)ϕ(B2) + ϕ(B3)
ϕ(
(A + B)4)
= ϕ(A4) + 4ϕ(A3)ϕ(B) + 4ϕ(A2)ϕ(B2)
+ 2(
ϕ(A2)ϕ(B)ϕ(B) + ϕ(A)ϕ(A)ϕ(B2)
− ϕ(A)ϕ(B)ϕ(A)ϕ(B))
+ 4ϕ(A)ϕ(B3) + ϕ(B4)
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To treat these formulas in general, linearize the free convolution
by going over from moments (ϕ(Am))m∈N to free cumulants
(κm)m∈N.
Those are defined by relations like:
ϕ(A1) = κ1
ϕ(A2) = κ2 + κ21
ϕ(A3) = κ3 + 3κ1κ2 + κ31
ϕ(A4) = κ4 + 4κ1κ3 + 2κ22 + 6κ2
1κ2 + κ41
...
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There is a combinatorial structure behind these formulas, the
sums are running over non-crossing partitions:
ϕ(A1) = ϕ(A2) = +
= κ1 = κ2 + κ1κ1
ϕ(A3) = + + + +
= κ3 + κ1κ2 + κ2κ1 + κ2κ1 + κ1κ1κ1
ϕ(A4) = + + + + + +
+ + + + + + +
= κ4 + 4κ1κ3 + 2κ22 + 6κ2
1κ2 + κ41
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This combinatorial relation between moments (ϕ(Am))m∈N and
cumulants (κm)m∈N can be translated into generating power se-
ries.
Put
G(z) =1
z+
∞∑
m=1
ϕ(Am)
zm+1Cauchy transform
and
R(z) =∞∑
m=1
κmzm−1R-transform
Then we have the relation
1
G(z)+ R(G(z)) = z.
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Theorem [Voiculescu 1986, Speicher 1994]:
Let A and B be free. Then one has
RA+B(z) = RA(z) + RB(z),
or equivalently
κA+Bm = κA
m + κBm ∀m.
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This, together with the relation between Cauchy transform and
R-transform and with the Stieltjes inversion formula, gives an
effective algorithm for calculating free convolutions, i.e., sums
of random matrices in generic position.
A GA RA
↓
RA +RB = RA+B GA+B A + B
↑
B GB RB
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Example: Wigner + Wishart (M = 2N), trials = 4000
−3 −2 −1 0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
N=10034
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One has similar analytic description for product.
Theorem [Voiculescu 1987, Haagerup 1997, Nica +
Speicher 1997]:
Put
MA(z) :=∞∑
m=0
ϕ(Am)zm
and define
SA(z) :=1 + z
zM<−1>
A (z) S-transform of A
Then: If A and B are free, we have
SAB(z) = SA(z) · SB(z).
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Example: Wishart x Wishart (M = 5N), trials=1000
0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N=10036
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upper left corner of size N/2 × N/2 of a projection matrix,
with N/2 eigenvalues 0 and N/2 eigenvalues 1; trials=5000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
N=6437
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• Free Calculator by Raj Rao and Alan Edelman
• A. Nica and R. Speicher: Lectures on the Combinatorics of
Free Probability.
To appear soon in the London Mathematical Society Lecture
Note Series, vol. 335, Cambridge University Press
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Outlook on other talks around free probability
• Anshelevich: ”free” orthogonal and Meixner polynomials
• Burda: free random Levy matrices
• Chatterjee: concentration of measures and free probability
• Demni: free stochastic processes
• Kargin: large deviations in free probability
• Mingo + Speicher: fluctuations of random matrices
• Rashidi Far: operator-valued free probability theory and block
matrices
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