[PPT] - The Ginibre ensemble GinUE ( N ) is the space Mat N ( C ) circular PowerPoint Presentation

SLIDE 1

1 / 11

Resolvent Behaviour of R-diagonal operators

Todd Kemp CLE Moore Instructor MIT

Banf and only Banf January 17, 2008

SLIDE 2

The Ginibre Ensemble

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

2 / 11

The Ginibre ensemble GinUE(N) is the space MatN(C) equipped with the probability measure

αN e−X∗X dX.

Alternatively, it is the set of N × N random matrices whose entries are all i.i.d. complex normals (Re,Im i.i.d. N(0, 1)).

SLIDE 3

The Ginibre Ensemble

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

2 / 11

The Ginibre ensemble GinUE(N) is the space MatN(C) equipped with the probability measure

αN e−X∗X dX.

Alternatively, it is the set of N × N random matrices whose entries are all i.i.d. complex normals (Re,Im i.i.d. N(0, 1)). Renormalize it by 1/

√ 2N, and . . .

Matlab code:

X=randn(4000); Y=randn(4000); C=(X+iY)/sqrt(8000); E=eig(C); plot(E,’b.’);

SLIDE 4

The Ginibre Ensemble

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

2 / 11

The Ginibre ensemble GinUE(N) is the space MatN(C) equipped with the probability measure

αN e−X∗X dX.

Alternatively, it is the set of N × N random matrices whose entries are all i.i.d. complex normals (Re,Im i.i.d. N(0, 1)). Renormalize it by 1/

√ 2N, and . . .

Matlab code:

X=randn(4000); Y=randn(4000); C=(X+iY)/sqrt(8000); E=eig(C); plot(E,’b.’);

−1 −0.5 0.5 1 −1 −0.5 0.5 1

SLIDE 5

Voiculescu’s Circular Operator

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

3 / 11

The circular operator c is the limit (in the sense of free probability) of the renormalized GinUE(N) as N → ∞. It can also be realized as c =

1 √ 2(s1 + is2), where s1, s2 are free

semicircular operators.

SLIDE 6

Voiculescu’s Circular Operator

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

3 / 11

The circular operator c is the limit (in the sense of free probability) of the renormalized GinUE(N) as N → ∞. It can also be realized as c =

1 √ 2(s1 + is2), where s1, s2 are free

semicircular operators. It is my favourite operator.

SLIDE 7

Voiculescu’s Circular Operator

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

3 / 11

The circular operator c is the limit (in the sense of free probability) of the renormalized GinUE(N) as N → ∞. It can also be realized as c =

1 √ 2(s1 + is2), where s1, s2 are free

semicircular operators. It is my favourite operator. From a combinatorial standpoint, it is characterized by its extremely simple free cumulants: among all cumulants in c, c∗, the only non-zero ones are

κ2[c, c∗] = κ2[c∗, c] = 1.

SLIDE 8

Voiculescu’s Circular Operator

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

3 / 11

The circular operator c is the limit (in the sense of free probability) of the renormalized GinUE(N) as N → ∞. It can also be realized as c =

1 √ 2(s1 + is2), where s1, s2 are free

semicircular operators. It is my favourite operator. From a combinatorial standpoint, it is characterized by its extremely simple free cumulants: among all cumulants in c, c∗, the only non-zero ones are

κ2[c, c∗] = κ2[c∗, c] = 1.

Quick advertisement: pick up Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher for everything you need to know about free cumulants.

SLIDE 9

R-Diagonal Operators

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

4 / 11

Let u be a Haar unitary operator (renormalized limit of Haar unitary ensemble). As Jamie Mingo showed us on Monday, the only non-zero free cumulants of u, u∗ are of the form

κ2n[u, u∗, . . . , u, u∗] = κ2n[u∗, u, . . . , u∗, u] = (−1)nCn−1.

SLIDE 10

R-Diagonal Operators

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

4 / 11

Let u be a Haar unitary operator (renormalized limit of Haar unitary ensemble). As Jamie Mingo showed us on Monday, the only non-zero free cumulants of u, u∗ are of the form

κ2n[u, u∗, . . . , u, u∗] = κ2n[u∗, u, . . . , u∗, u] = (−1)nCn−1.

Definition. a is R-diagonal if its only non-zero free cumulants are
f the forms

κ2n[a, a∗, . . . , a, a∗] κ2n[a∗, a, . . . , a∗, a].

Alternate characterization. a is R-diagonal if, given u Haar unitary ∗-free from a,

ua ∼ a.

SLIDE 11

Properties of R-Diagonal Operators

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

5 / 11

Matrix models: ensembles of the form αN e−V (X∗X) dX.
If a, b are R-diagonal and ∗-free, then a + b and an are

R-diagonal; if x is anything ∗-free from a, ax is R-diagonal.

Never normal except scalar multiples of Haar unitaries.
Brown measure of a can be computed explicitly from the

S -transform of a∗a; rotationally-invariant, analytic density.

Have continuous families of invariant subspaces.
Maximize free entropy (χ and χ∗) under distribution constraints.
Satisfy a strong Haagerup inequality.

SLIDE 12

Strong Haagerup Inequality

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

6 / 11

Theorem. Let a1, . . . , ad be ∗-free R-diagonal operators. If T is

spanned by words of length n in a1, . . . , ad (and not a∗

1, . . . , a∗ d),

then

T ≤ α √n T2,

where α is a constant depending on sup(aj/aj2).

SLIDE 13

Strong Haagerup Inequality

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

6 / 11

Theorem. Let a1, . . . , ad be ∗-free R-diagonal operators. If T is

spanned by words of length n in a1, . . . , ad (and not a∗

1, . . . , a∗ d),

then

T ≤ α √n T2,

where α is a constant depending on sup(aj/aj2). E.g. for Haar unitaries u1, . . . , ud, this is a statement about the free group: if f : Fd → C is supported on words of length n in the generators (and not their inverses), then

f∗ ≤ √e √ n + 1 f2.

(If the inverses are included, the constant is (n + 1); this is the classical Haagerup inequality.)

SLIDE 14

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u−1

5

u−1

3

u4 u5 u4 u−1

4

u−1

5

u−1

4

u3 u3 u1 u−1

1

u−1

2

u−1

1

u1 u2 u3 u−1

3

u−1

3

u−1

2

u2 u5 u1

SLIDE 15

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u4 u3 u−1

2

u1 u−1

2

u1 u2 u5 u−1

4

u−1

4

u1 u3 u−1

5

u−1

3

u4 u3 u−1

3 u−1 3

u2 u−1

1

u−1

1

u−1

5

u5

SLIDE 16

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u4 u3 u−1

2

u1 u−1

2

u2 u1 u2 u5 u−1

4

u−1

4

u−1

5

u−1

3

u−1

1

u−1

1 u1 u3

u−1

5

u−1

3

u5 u4 u3 u−1

3

SLIDE 17

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u4 u3 u−1

2

u1 u−1

2

u2 u4 u1 u2 u5 u5 u−1

4

u−1

4

u−1

5

u−1

3 u−1 3

u−1

1

u−1

1 u1 u3

u−1

5

u−1

3

u3

SLIDE 18

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u4 u3 u−1

2

u1 u−1

2

u2 u4 u1 u2 u5 u−1

3

u5 u−1

4

u−1

4

u−1

5

u−1

3 u−1 3

u−1

1

u−1

1 u1 u3 u3

u−1

5

SLIDE 19

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u−1

5

u4 u3 u−1

2

u1 u−1

2

u2 u4 u1 u2 u5 u−1

3

u5 u−1

4

u−1

4

u−1

5

u−1

3 u−1 3

u−1

1

u−1

1 u1 u3 u3

SLIDE 20

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u4 u3 u−1

2

u1 u−1

2

u2 u4 u1 u2 u5 u−1

3

u5 u−1

4

u−1

4

u−1

5

u−1

3 u−1 3

u−1

1

u−1

1 u1 u3 u3

u−1

5

SLIDE 21

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u4 u3 u−1

2

u1 u−1

2

u2 u4 u1 u2 u5 u−1

3

u5 u−1

4

u−1

4

u−1

5

u−1

3 u−1 3

u−1

1

u−1

1 u1 u3 u3

u−1

5

We get a non-crossing pairing π.

SLIDE 22

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u4 u3 u−1

2

u1 u−1

2

u2 u4 u1 u2 u5 u−1

3

u5 u−1

4

u−1

4

u−1

5

u−1

3 u−1 3

u−1

1

u−1

1 u1 u3 u3

u−1

5

We get a non-crossing pairing π. More precisely,

π is non-crossing
π matches 1s to ∗s in the string 1 · · · 1

n

∗ · · · ∗

n

· · · 1 · · · 1

n

∗ · · · ∗

n

SLIDE 23

Non-Crossing ∗-Pairings

7 / 11

E.g. with n = 3, r = 4:

u−1

1

u4 u3 u−1

2

u1 u−1

2

u2 u4 u1 u2 u5 u−1

3

u5 u−1

4

u−1

4

u−1

5

u−1

3 u−1 3

u−1

1

u−1

1 u1 u3 u3

u−1

5

We get a non-crossing pairing π. More precisely,

π is non-crossing
π matches 1s to ∗s in the string 1 · · · 1

n

∗ · · · ∗

n

· · · 1 · · · 1

n

∗ · · · ∗

n

The set of such ∗-pairings is counted by the Fuss-Catalan numbers

C(n)

r

= 1 nr + 1 (n + 1)r r

∼

√e √ n + 1 2r .

SLIDE 24

A Resolvent Upper Bound

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

8 / 11

For a R-diagonal, consider the resolvent function

ρa(λ) = 1 λ − a λ / ∈ spec a.

SLIDE 25

A Resolvent Upper Bound

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

8 / 11

For a R-diagonal, consider the resolvent function

ρa(λ) = 1 λ − a λ / ∈ spec a.

Write this as a geometric series

ρa(1/λ) =

λn+1 an.

Apply the strong Haagerup inequality term-by-term, use the Cauchy-Schwarz inequality, and arrive at

Proposition. There is a constant α(a) > 0 so that, for 1 < λ < 2,

ρa(λ) ≤ α(a) (λ − 1)3/2.

SLIDE 26

A Resolvent Upper Bound

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

8 / 11

For a R-diagonal, consider the resolvent function

ρa(λ) = 1 λ − a λ / ∈ spec a.

Write this as a geometric series

ρa(1/λ) =

λn+1 an.

Apply the strong Haagerup inequality term-by-term, use the Cauchy-Schwarz inequality, and arrive at

Proposition. There is a constant α(a) > 0 so that, for 1 < λ < 2,

ρa(λ) ≤ α(a) (λ − 1)3/2.

Question. Is this optimal?

SLIDE 27

Answer = YES

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

9 / 11

Theorem. Let a be R-diagonal, and suppose that a2 = 1 and

a4 > 1 (i.e. a is not Haar unitary). Then ρa(λ) ∼

27

32 v(a)

1 (λ − 1)3/2 ,

where v(a) =

a4

4 − 1.

SLIDE 28

Answer = YES

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

9 / 11

Theorem. Let a be R-diagonal, and suppose that a2 = 1 and

a4 > 1 (i.e. a is not Haar unitary). Then ρa(λ) ∼

27

32 v(a)

1 (λ − 1)3/2 ,

where v(a) =

a4

4 − 1. The proof uses both the combinatorics

f planar diagrams, and complex analytic methods like those

discussed by Hari Bercovici on Monday. Key idea: the spectral radius of ρa(λ) = (λ − a)−1 is the infimum

f the spectrum of |λ − a|.

SLIDE 29

Answer = YES

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

9 / 11

Theorem. Let a be R-diagonal, and suppose that a2 = 1 and

a4 > 1 (i.e. a is not Haar unitary). Then ρa(λ) ∼

27

32 v(a)

1 (λ − 1)3/2 ,

where v(a) =

a4

4 − 1. The proof uses both the combinatorics

f planar diagrams, and complex analytic methods like those

discussed by Hari Bercovici on Monday. Key idea: the spectral radius of ρa(λ) = (λ − a)−1 is the infimum

f the spectrum of |λ − a|.

Therefore need to calculate (or estimate) R|λ−a|.

SLIDE 30

Answer = YES

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

9 / 11

Theorem. Let a be R-diagonal, and suppose that a2 = 1 and

a4 > 1 (i.e. a is not Haar unitary). Then ρa(λ) ∼

27

32 v(a)

1 (λ − 1)3/2 ,

where v(a) =

a4

4 − 1. The proof uses both the combinatorics

f planar diagrams, and complex analytic methods like those

discussed by Hari Bercovici on Monday. Key idea: the spectral radius of ρa(λ) = (λ − a)−1 is the infimum

f the spectrum of |λ − a|.

Therefore need to calculate (or estimate) R|λ−a|. The trick is to symmetrize.

SLIDE 31

Answer = YES

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

9 / 11

Theorem. Let a be R-diagonal, and suppose that a2 = 1 and

a4 > 1 (i.e. a is not Haar unitary). Then ρa(λ) ∼

27

32 v(a)

1 (λ − 1)3/2 ,

where v(a) =

a4

4 − 1. The proof uses both the combinatorics

f planar diagrams, and complex analytic methods like those

discussed by Hari Bercovici on Monday. Key idea: the spectral radius of ρa(λ) = (λ − a)−1 is the infimum

f the spectrum of |λ − a|.

Therefore need to calculate (or estimate) R|λ−a|. The trick is to symmetrize. E.g.

R|λ−c|2(z) = 1 1 − z + λ2 (1 − z)2 .

SLIDE 32

Related Techniques: Negative Moments

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

10 / 11

The same techniques allow us to calculate (to leading order, at least) the negative moments of µλ.

Theorem. For λ ց 1 and k ≥ 0,
t−2k−2 dµλ(t) ∼ C(2)

k

v(a)k (λ2 − 1)3k+1 , λ ց 1.

SLIDE 33

Related Techniques: Negative Moments

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

10 / 11

The same techniques allow us to calculate (to leading order, at least) the negative moments of µλ.

Theorem. For λ ց 1 and k ≥ 0,
t−2k−2 dµλ(t) ∼ C(2)

k

v(a)k (λ2 − 1)3k+1 , λ ց 1.

Fuss–Catalan number.

SLIDE 34

Related Techniques: Negative Moments

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

10 / 11

The same techniques allow us to calculate (to leading order, at least) the negative moments of µλ.

Theorem. For λ ց 1 and k ≥ 0,
t−2k−2 dµλ(t) ∼ C(2)

k

v(a)k (λ2 − 1)3k+1 , λ ց 1.

Fuss–Catalan number. In terms of complex analytic techniques, shows up in the Lagrange inversion formula for a cubic polynomial.

SLIDE 35

Related Techniques: Negative Moments

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

10 / 11

The same techniques allow us to calculate (to leading order, at least) the negative moments of µλ.

Theorem. For λ ց 1 and k ≥ 0,
t−2k−2 dµλ(t) ∼ C(2)

k

v(a)k (λ2 − 1)3k+1 , λ ց 1.

Fuss–Catalan number. In terms of complex analytic techniques, shows up in the Lagrange inversion formula for a cubic polynomial. But we have a much nicer explanation for its appearance: pairing structure diagrams!

SLIDE 36

Related Techniques: Negative Moments

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

10 / 11

The same techniques allow us to calculate (to leading order, at least) the negative moments of µλ.

Theorem. For λ ց 1 and k ≥ 0,
t−2k−2 dµλ(t) ∼ C(2)

k

v(a)k (λ2 − 1)3k+1 , λ ց 1.

Fuss–Catalan number. In terms of complex analytic techniques, shows up in the Lagrange inversion formula for a cubic polynomial. But we have a much nicer explanation for its appearance: pairing structure diagrams! Let’s look at the circular case.

SLIDE 37

References

GinUE(N)
circular
R-diagonal
Properties
Haagerup Ineq.
∗-Pairings
Resolvent
Blow-Up
Moments
References

11 / 11

Kemp, T.; Speicher, R.: Strong Haagerup inequalities for free

R-diagonal elements. J. Funct. Anal. 251 (2007) 141–173.

Kemp, T.: R-diagonal dilation semigroups. Preprint.
Haagerup, U.; Kemp, T.; Speicher, R.: Resolvents and norms for

R-diagonal operators. (1 − ǫ)Preprint.