Block modified Random Matrices Octavio Arizmendi CIMAT joint work - - PowerPoint PPT Presentation

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Block modified Random Matrices

Octavio Arizmendi CIMAT

joint work with

• I. Nechita and C. Vargas

Bedlewo, 18.05.2017

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Probability Spaces

Definition

1 A non-commutative probability space is a pair (A, φ)

where A is a unitary complex algebra and φ is a linear functional φ : A → C such that φ(1A) = 1.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Probability Spaces

Definition

1 A non-commutative probability space is a pair (A, φ)

where A is a unitary complex algebra and φ is a linear functional φ : A → C such that φ(1A) = 1.

2 A non-commutative random variable (or random variable)

is just an element a ∈ A.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Probability Spaces

Definition

1 A non-commutative probability space is a pair (A, φ)

where A is a unitary complex algebra and φ is a linear functional φ : A → C such that φ(1A) = 1.

2 A non-commutative random variable (or random variable)

is just an element a ∈ A.

3 If there is a measure µa in C with compact support, such that

• C

zkzlµa(dz) = φ(ak(a∗)l), for all k, l ∈ N, we call µa the *-distribution of a.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Free Independence

Definition A family of subalgebras (Ai)i∈I of A is said to be free independent if for all a1 ∈ Ai(1), ..., an ∈ Ai(n) φ(a1a2...an) = 0 whenever φ(aj) = 0, aj ∈ Ai(j), and i(1) = i(2) = ... = i(n).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Random Matrices and Free Probability

Why is free probability useful for random matrices?

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Random Matrices and Free Probability

Why is free probability useful for random matrices? Example (Voiculescu 1991)

1 Let Ad and Bd deterministic matrices and Ud a Haar Unitary.

Then Ad and UdBdU∗

d are asymptotically free as d → ∞.

2 Let G (n)

1

and G (n)

2

hermitian Independent Gaussian Random Matrices of size n. Then G (n)

1

and G (n)

2

are asymptotically free as n → ∞.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Free Addition of Bernoullis

UB1U∗ + B2

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Marchenko-Pastur (free Poisson)

XX ∗

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Free Multiplication 1

W 2

1 W 2 2

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Free Multiplication 2

W1W 2

2

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Free Multiplication 3

(W 2

1 − I)W 2 2

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Free Multiplication 4

(UB1U∗ + B2)W 2

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Free Commutator

W1W2 − W2W1

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Random Matrices and Free Probability

How useful is free probability random matrices?

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Random Matrices and Free Probability

How useful is free probability random matrices? Many restrictions... on moments, definition of distribution, relation between entries, etc...

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Random Matrices and Free Probability

How useful is free probability random matrices? Many restrictions... on moments, definition of distribution, relation between entries, etc... One of the first extensions of free probability is Operator-Valued Free probability. It has helped in solving concrete problems in calculating limiting distributions in Random Matrix Theory for which Scalar Free Probability wasn’t enough:

1 Band Matrices (Shlyakhtenko 1996) 2 Block Matrices (MIMO) (Far-Oraby-Bryc-Speicher 2006) 3 Rectangular Matrices (Florent Benaych-Georges 2009) 4 Girko’s Deterministic Equivalents (Speicher, Vargas 2012,

2015)

5 Polynomials in independent random matrices. (Belinschi et al.

2014, Belinschi-Mai-Speicher, 2015)

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Block modified Matrices

Let W be a dn × dn self-adjoint random matrix

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Block modified Matrices

Let W be a dn × dn self-adjoint random matrix and let ϕ : Mn(C) → Mn(C) be a self-adjoint linear map.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Block modified Matrices

Let W be a dn × dn self-adjoint random matrix and let ϕ : Mn(C) → Mn(C) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := (idd ⊗ ϕ)(W ).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Block modified Matrices

Let W be a dn × dn self-adjoint random matrix and let ϕ : Mn(C) → Mn(C) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := (idd ⊗ ϕ)(W ). We study the asymptotic eigenvalue distribution of W ϕ.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Block modified Matrices

Let W be a dn × dn self-adjoint random matrix and let ϕ : Mn(C) → Mn(C) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := (idd ⊗ ϕ)(W ). We study the asymptotic eigenvalue distribution of W ϕ. W Wishart and ϕ(A) = At (Aubrun 2012). ϕ ”planar”

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Block modified Matrices

Let W be a dn × dn self-adjoint random matrix and let ϕ : Mn(C) → Mn(C) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := (idd ⊗ ϕ)(W ). We study the asymptotic eigenvalue distribution of W ϕ. W Wishart and ϕ(A) = At (Aubrun 2012). ϕ ”planar” W ϕ → compound free Poissons (Banica, Nechita 2012). ϕ(X) := XTr(X)I, W ϕ → compound free Poisson distribution (Jivulescu, Lupa, N. 2014,2015) ...

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Block modified Matrices

Let W be a dn × dn self-adjoint random matrix and let ϕ : Mn(C) → Mn(C) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := (idd ⊗ ϕ)(W ). We study the asymptotic eigenvalue distribution of W ϕ. W Wishart and ϕ(A) = At (Aubrun 2012). ϕ ”planar” W ϕ → compound free Poissons (Banica, Nechita 2012). ϕ(X) := XTr(X)I, W ϕ → compound free Poisson distribution (Jivulescu, Lupa, N. 2014,2015) ... All above results are based on the moment method by direct calculations expectation of traces.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Let us write ϕ(A) =

n

• i,j,k,l=1

αij

klEijAEkl,

where Eij ∈ Mn(C) are matrix units.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Let us write ϕ(A) =

n

• i,j,k,l=1

αij

klEijAEkl,

where Eij ∈ Mn(C) are matrix units. Then we can express W ϕ =

n

• i,j,k,l=1

αij

kl(Id ⊗ Eij)W (Id ⊗ Ekl),

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Let us write ϕ(A) =

n

• i,j,k,l=1

αij

klEijAEkl,

where Eij ∈ Mn(C) are matrix units. Then we can express W ϕ =

n

• i,j,k,l=1

αij

kl(Id ⊗ Eij)W (Id ⊗ Ekl),

As d → ∞, by Voiculescu’s asymptotic freeness results (Id ⊗ Eij)n

i,j=1 and W are asymptotically free

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Let us write ϕ(A) =

n

• i,j,k,l=1

αij

klEijAEkl,

where Eij ∈ Mn(C) are matrix units. Then we can express W ϕ =

n

• i,j,k,l=1

αij

kl(Id ⊗ Eij)W (Id ⊗ Ekl),

As d → ∞, by Voiculescu’s asymptotic freeness results (Id ⊗ Eij)n

i,j=1 and W are asymptotically free (W Wishart, Wigner,

• r randomly rotated).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

In the limit we replace:

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

In the limit we replace: (Id ⊗ Ejl)n

i,j=1 by an abstract collection (eij)i,j≤n of matrix units in

a non-commutative probability space (A, τ).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

In the limit we replace: (Id ⊗ Ejl)n

i,j=1 by an abstract collection (eij)i,j≤n of matrix units in

a non-commutative probability space (A, τ). i.e. eijekl = δjkeil, τ(eij) = n−1δij,

n

• i=1

eii = 1, e∗

ij = eji.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

In the limit we replace: (Id ⊗ Ejl)n

i,j=1 by an abstract collection (eij)i,j≤n of matrix units in

a non-commutative probability space (A, τ). i.e. eijekl = δjkeil, τ(eij) = n−1δij,

n

• i=1

eii = 1, e∗

ij = eji.

W by an element w ∈ A which is free from (eij)i,j≤n such that (dn)−1E ◦ Tr(W k) → τ(wk).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

In the limit we replace: (Id ⊗ Ejl)n

i,j=1 by an abstract collection (eij)i,j≤n of matrix units in

a non-commutative probability space (A, τ). i.e. eijekl = δjkeil, τ(eij) = n−1δij,

n

• i=1

eii = 1, e∗

ij = eji.

W by an element w ∈ A which is free from (eij)i,j≤n such that (dn)−1E ◦ Tr(W k) → τ(wk). Then W ϕ → wϕ :=

n

• i,j,k,l=1

αij

kleijwekl.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

In the limit we replace: (Id ⊗ Ejl)n

i,j=1 by an abstract collection (eij)i,j≤n of matrix units in

a non-commutative probability space (A, τ). i.e. eijekl = δjkeil, τ(eij) = n−1δij,

n

• i=1

eii = 1, e∗

ij = eji.

W by an element w ∈ A which is free from (eij)i,j≤n such that (dn)−1E ◦ Tr(W k) → τ(wk). Then W ϕ → wϕ :=

n

• i,j,k,l=1

αij

kleijwekl.

Note: The (eij) are not free among themselves!

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Example Restriction to the diagonal. k = 2 this is ˆ W = (Id ⊗ e11)W (Id ⊗ e11) + (Id ⊗ e22)W (Id ⊗ e22). So, the limiting distribution may be writen as e11we11+ e22we22 where e11 and e22 are orthogonal, φ(eii) = 1/2 and free from w. w w e11 e22 e11 e22

• =

w w e11 e22

• Then W and E are free in (M2(A), φ ⊗ tr). The distribution of

EE ∗ is 1/2(δ0 + δ1) so ˆ W converges to π ⊠ 1/2(δ0 + δ1).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Example Restriction to the diagonal. k = 2 this is ˆ W = (Id ⊗ e11)W (Id ⊗ e11) + (Id ⊗ e22)W (Id ⊗ e22). So, the limiting distribution may be writen as e11we11+ e22we22 where e11 and e22 are orthogonal, φ(eii) = 1/2 and free from w. w w e11 e22 e11 e22

• =

w w e11 e22

• Then W and E are free in (M2(A), φ ⊗ tr). The distribution of

EE ∗ is 1/2(δ0 + δ1) so ˆ W converges to π ⊠ 1/2(δ0 + δ1). The same calculation holds for k larger and a similar one for the case of the trace.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Example Restriction to the diagonal. k = 2 this is ˆ W = (Id ⊗ e11)W (Id ⊗ e11) + (Id ⊗ e22)W (Id ⊗ e22). So, the limiting distribution may be writen as e11we11+ e22we22 where e11 and e22 are orthogonal, φ(eii) = 1/2 and free from w. w w e11 e22 e11 e22

• =

w w e11 e22

• Then W and E are free in (M2(A), φ ⊗ tr). The distribution of

EE ∗ is 1/2(δ0 + δ1) so ˆ W converges to π ⊠ 1/2(δ0 + δ1). The same calculation holds for k larger and a similar one for the case of the trace. The general case really needs new tools:

• perator valued free multiplicative convolution

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Main Definitions

Definition An operator-valued non-commutative probability space is a triple (M, E, B), where

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Main Definitions

Definition An operator-valued non-commutative probability space is a triple (M, E, B), where

1 M is a C ∗-algebra, Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Main Definitions

Definition An operator-valued non-commutative probability space is a triple (M, E, B), where

1 M is a C ∗-algebra, 2 B ⊆ M is a C ∗-subalgebra containing the unit of M, Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Main Definitions

Definition An operator-valued non-commutative probability space is a triple (M, E, B), where

1 M is a C ∗-algebra, 2 B ⊆ M is a C ∗-subalgebra containing the unit of M, 3 E: M → B is a unit-preserving conditional expectation

(E(bab′) = bE(a)b′)

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Main Definitions

Definition An operator-valued non-commutative probability space is a triple (M, E, B), where

1 M is a C ∗-algebra, 2 B ⊆ M is a C ∗-subalgebra containing the unit of M, 3 E: M → B is a unit-preserving conditional expectation

(E(bab′) = bE(a)b′) Remark: If B = C, then we are in the usual (scalar) case.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Examples

If (A, τ) is a (scalar) C ∗ probability space, we may consider (M, E, B) , where M = M2(C) ⊗ A

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Examples

If (A, τ) is a (scalar) C ∗ probability space, we may consider (M, E, B) , where M = M2(C) ⊗ A, B = M2(C) and E : M → M2(C)

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Examples

If (A, τ) is a (scalar) C ∗ probability space, we may consider (M, E, B) , where M = M2(C) ⊗ A, B = M2(C) and E : M → M2(C) a1 a2 a3 a4

• →

τ(a1) τ(a2) τ(a3) τ(a4)

• Octavio Arizmendi CIMAT

Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Examples

If (A, τ) is a (scalar) C ∗ probability space, we may consider (M, E, B) , where M = M2(C) ⊗ A, B = M2(C) and E : M → M2(C) a1 a2 a3 a4

• →

τ(a1) τ(a2) τ(a3) τ(a4)

• r maybe B = D2(C) and

E : M → D2(C)

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Examples

If (A, τ) is a (scalar) C ∗ probability space, we may consider (M, E, B) , where M = M2(C) ⊗ A, B = M2(C) and E : M → M2(C) a1 a2 a3 a4

• →

τ(a1) τ(a2) τ(a3) τ(a4)

• r maybe B = D2(C) and

E : M → D2(C) a1 a2 a3 a4

• →

τ(a1) τ(a4)

• Octavio Arizmendi CIMAT

Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

B-Freeness

Definition Two algebras A1, A2 ⊆ M containing B are called free with amalgamation over B with respect to E (or just free over B) if for any tuple x1, . . . xn, such that xj ∈ Aij and ij = ij+1 E[¯ x1¯ x2 · · · ¯ xn] = 0 where ¯ x := x − E(x)

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Freeness vs B-Freeness

Freeness with amalgamation is less restrictive than scalar freeness.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Freeness vs B-Freeness

Freeness with amalgamation is less restrictive than scalar freeness. For example, if A1, A2 ⊂ A are free (w.r.t τ), then the algebras Mn(C) ⊗ A1, Mn(C) ⊗ A2 ⊂ Mn(C) ⊗ A are free with amalgamation over Mn(C) (w.r.t idn ⊗ τ).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Freeness vs B-Freeness

Freeness with amalgamation is less restrictive than scalar freeness. For example, if A1, A2 ⊂ A are free (w.r.t τ), then the algebras Mn(C) ⊗ A1, Mn(C) ⊗ A2 ⊂ Mn(C) ⊗ A are free with amalgamation over Mn(C) (w.r.t idn ⊗ τ). However, in general, they are NOT free over C (w.r.t. 1

nτ ◦ Tr).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

...back to our block modified matrices.

We express wϕ as a difference of completely positive maps α2eijwekl + ¯ α2elkweji = (αeij + ¯ αelk)w(¯ αeji + αekl) −α2eijweji − α2elkwekl

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

...back to our block modified matrices.

We express wϕ as a difference of completely positive maps α2eijwekl + ¯ α2elkweji = (αeij + ¯ αelk)w(¯ αeji + αekl) −α2eijweji − α2elkwekl so that wϕ =

• 1≤i,j,k,l≤n

(i,j)≤(l,k)

(f ij

kl )εij klw(f ij kl )∗

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

...back to our block modified matrices.

We express wϕ as a difference of completely positive maps α2eijwekl + ¯ α2elkweji = (αeij + ¯ αelk)w(¯ αeji + αekl) −α2eijweji − α2elkwekl so that wϕ =

• 1≤i,j,k,l≤n

(i,j)≤(l,k)

(f ij

kl )εij klw(f ij kl )∗

(1) From the elements f ij

kl , (i, j) ≤ (l, k) we build a vector

f = (f 11

11 , f 11 12 , . . . , f nn nn ) of size m := n2(n2 + 1)/2.

We consider also the diagonal matrix ˜ w = diag(ε11

11w, ε11 12w, . . . , εnn nnw), so that f ˜

wf ∗ = wϕ.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

The desired distribution is the same (modulo a dirac mass at zero

• f weight 1 − 1/m) as the distribution of f ∗f ˜

w in the C ∗-probability space (Mm(C) ⊗ A, trm ⊗ τ).

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

The desired distribution is the same (modulo a dirac mass at zero

• f weight 1 − 1/m) as the distribution of f ∗f ˜

w in the C ∗-probability space (Mm(C) ⊗ A, trm ⊗ τ). Moreover, since w and (eij

kl) are free, the matrices f ∗f and ˜

w are free with amalgamation over Mm(C) (with respect to the conditional expectation E := idm ⊗ τ). The problem is now reduced to calculate matrix-valued free multiplicative convolutions.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Operator-valued Cauchy Transform

To calculate free multiplicative convolutions we need the following analytic mappings, all defined on H+(B). For a fixed x ∈ M, we define The generalized Cauchy-Stieltjes transform: Gx(b) = E

• (b − x)−1

The eta transform (Boolean cumulant series): ηx(b) = 1 − bE

• (b−1 − x)−1−1 = 1 − bGx(b−1)−1;

An auxiliary “h transform:” hx(b) = b−1ηx(b) = b−1 − E

• (b−1 − x)−1−1 ;

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Subordination

Theorem (Belinschi et al. 2012) Let B be finite-dimensional. For any x ≥ 0, y = y∗ free over B, there exists a domain D ⊂ B containing C+ · 1 and an analytic map ω2 : D → H+(B) so that ηy(ω2(b)) = ηxy(b) and gb(ω2(b)) = ω2(b), b ∈ D. Moreover, if gb : H+(B) → H+(B), gb(w) = bhx(hy(w)b), then ω2(b) = lim

n→∞ g◦n b (w),

for any w ∈ H+(B), b ∈ D.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Figure: Block-modified Wigner matrix

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Figure: Block-modified Wishart matrix

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Figure: Block-modification of a rotated arcsine matrix

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Explicit Solutions

To any map f : Mn → A, associate its Choi matrix Cf =

n

• i,j=1

Eij ⊗ f (Eij), where Eij are the matrix units in Mn, then Theorem (Choi 72) A map f : Mn → A is CP iff its Choi matrix Cf is positive.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Explicit Solutions

Lemma The block-modified random variable wϕ has the following expression in terms of the eigenvalues and of the eigenvectors of the Choi matrix C: wϕ = v∗(X ⊗ C)v. where v =

n2

• s=1

b∗

s ⊗ as ∈ A ⊗ Mn2,

and as are the eigenvectors of C and the random variables bs ∈ A are defined by bs =

i,j < Ei ⊗ Ej, as > ei,j.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Explicit solutions

Definition The Choi matrix C is said to satisfy the unitarity condition if, for all t, there is some real constant dt such that [id ⊗ Tr](Pt) = dtIn, where Pt are the eigenprojectors of C. Theorem Let C : Mnm(C) → ⊗Mnm(C) be the Choi Matrix of φ . And suppose that C has tracially well behaved eigenspaces. Then the random variables w ⊗ C and ff ∗ are free with amalgamation over the commutative unital algebra B = 1 ⊗ C generated by C.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Corollary (based on Dykema’s Operator Valued S-transform) If the Choi matrix C satisfies the unitary condition, then the distribution of wϕ is given by ⊞s

i=1(Dpi/nµ)⊞ndi,

where pi are the different eigenvalues of C and ndi are the ranks of the corresponding eigenproyectors.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

The following are examples of maps satisfying the unitary condition, Transposition f (X) = X T. The trace and its dual ϕ(X) = Tr(X), ϕ(x) = xId Reduction map f (X) = InTr(X) − X Unitray conjugation ϕ(X) = UXU Linear combinations of the above Mixtures of orthogonal automorphisms ϕ(X) =

n2

• i=1

αiUiXU∗

i

for orthogonal unitary operators Ui, with Tr(UiU∗

j ) = ndij.

Octavio Arizmendi CIMAT Block modified Random Matrices

Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions

Thanks!!

Octavio Arizmendi CIMAT Block modified Random Matrices