Complex Analytic Methods in Free vvb Probability Theory Hari - PowerPoint PPT Presentation

Cauchy transforms f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • µ probability distribution on R Regularity Extensions • z ∈ C + upper half-plane Omissions vvb • � ∞ d µ ( t ) G µ ( z ) = z − t −∞ • F µ ( z ) = 1 / G µ ( z ) ; F µ : C + → C + • Function inverse F < − 1 > defined in µ D r ,ε = { z : | z − ir | < ( 1 − ε ) r } for large r • ϕ µ ( z ) = R µ ( 1 / z ) = F < − 1 > ( z ) − z V-transform of µ µ

Cauchy transforms f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • µ probability distribution on R Regularity Extensions • z ∈ C + upper half-plane Omissions vvb • � ∞ d µ ( t ) G µ ( z ) = z − t −∞ • F µ ( z ) = 1 / G µ ( z ) ; F µ : C + → C + • Function inverse F < − 1 > defined in µ D r ,ε = { z : | z − ir | < ( 1 − ε ) r } for large r • ϕ µ ( z ) = R µ ( 1 / z ) = F < − 1 > ( z ) − z V-transform of µ µ • ϕ µ ⊞ ν = ϕ µ + ϕ ν

Cauchy transforms f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • µ probability distribution on R Regularity Extensions • z ∈ C + upper half-plane Omissions vvb • � ∞ d µ ( t ) G µ ( z ) = z − t −∞ • F µ ( z ) = 1 / G µ ( z ) ; F µ : C + → C + • Function inverse F < − 1 > defined in µ D r ,ε = { z : | z − ir | < ( 1 − ε ) r } for large r • ϕ µ ( z ) = R µ ( 1 / z ) = F < − 1 > ( z ) − z V-transform of µ µ • ϕ µ ⊞ ν = ϕ µ + ϕ ν • There are corresponding results for ⊠

A basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • ϕ µ ( z ) = F < − 1 > ( z ) − z µ

A basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • ϕ µ ( z ) = F < − 1 > ( z ) − z µ • ϕ µ ( z ) ≈ z − F µ ( z ) in D r ,ε as r → ∞

A basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • ϕ µ ( z ) = F < − 1 > ( z ) − z µ • ϕ µ ( z ) ≈ z − F µ ( z ) in D r ,ε as r → ∞ • The approximation is uniformly better when µ is concentrated near zero.

A basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • ϕ µ ( z ) = F < − 1 > ( z ) − z µ • ϕ µ ( z ) ≈ z − F µ ( z ) in D r ,ε as r → ∞ • The approximation is uniformly better when µ is concentrated near zero. • meaning: ratio closer to one

A basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • ϕ µ ( z ) = F < − 1 > ( z ) − z µ • ϕ µ ( z ) ≈ z − F µ ( z ) in D r ,ε as r → ∞ • The approximation is uniformly better when µ is concentrated near zero. • meaning: ratio closer to one • larger r • smaller ε

Some results f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • µ n → µ equivalent to ϕ µ n → ϕ µ in D r ,ε , ε fixed, r large, vvb with some uniformity at ∞

Some results f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • µ n → µ equivalent to ϕ µ n → ϕ µ in D r ,ε , ε fixed, r large, vvb with some uniformity at ∞ • { µ n } n ≥ 1 , ν n = µ 1 ⊞ µ 2 ⊞ · · · ⊞ µ n , ρ n = µ 1 ∗ µ 2 ∗ · · · ∗ µ n

Some results f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • µ n → µ equivalent to ϕ µ n → ϕ µ in D r ,ε , ε fixed, r large, vvb with some uniformity at ∞ • { µ n } n ≥ 1 , ν n = µ 1 ⊞ µ 2 ⊞ · · · ⊞ µ n , ρ n = µ 1 ∗ µ 2 ∗ · · · ∗ µ n • ν n → ν ⇔ ρ n → ρ (three series theorem)

Some results f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • µ n → µ equivalent to ϕ µ n → ϕ µ in D r ,ε , ε fixed, r large, vvb with some uniformity at ∞ • { µ n } n ≥ 1 , ν n = µ 1 ⊞ µ 2 ⊞ · · · ⊞ µ n , ρ n = µ 1 ∗ µ 2 ∗ · · · ∗ µ n • ν n → ν ⇔ ρ n → ρ (three series theorem) • n -divisibility: µ = ν ⊞ ν ⊞ · · · ⊞ ν � �� � n times

Some results f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • µ n → µ equivalent to ϕ µ n → ϕ µ in D r ,ε , ε fixed, r large, vvb with some uniformity at ∞ • { µ n } n ≥ 1 , ν n = µ 1 ⊞ µ 2 ⊞ · · · ⊞ µ n , ρ n = µ 1 ∗ µ 2 ∗ · · · ∗ µ n • ν n → ν ⇔ ρ n → ρ (three series theorem) • n -divisibility: µ = ν ⊞ ν ⊞ · · · ⊞ ν � �� � n times • infinite-divisibility: all n

Some results f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • µ n → µ equivalent to ϕ µ n → ϕ µ in D r ,ε , ε fixed, r large, vvb with some uniformity at ∞ • { µ n } n ≥ 1 , ν n = µ 1 ⊞ µ 2 ⊞ · · · ⊞ µ n , ρ n = µ 1 ∗ µ 2 ∗ · · · ∗ µ n • ν n → ν ⇔ ρ n → ρ (three series theorem) • n -divisibility: µ = ν ⊞ ν ⊞ · · · ⊞ ν � �� � n times • infinite-divisibility: all n • µ is ∞ -divisible ⇔ ϕ µ : C + → C −

Limit laws f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • { µ nj : n ≥ 1 , 1 ≤ j ≤ k n } distributions on R Regularity Extensions Omissions vvb

Limit laws f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • { µ nj : n ≥ 1 , 1 ≤ j ≤ k n } distributions on R Regularity Extensions • Infinitesimal if for all ε > 0 Omissions vvb n →∞ min lim 1 ≤ j ≤ k n µ nj (( − ε, ε )) = 1

Limit laws f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • { µ nj : n ≥ 1 , 1 ≤ j ≤ k n } distributions on R Regularity Extensions • Infinitesimal if for all ε > 0 Omissions vvb n →∞ min lim 1 ≤ j ≤ k n µ nj (( − ε, ε )) = 1 • µ n = µ n 1 ⊞ µ n 2 ⊞ · · · ⊞ µ nk n , ν n = µ n 1 ∗ µ n 2 ∗ · · · ∗ µ nk n

Limit laws f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • { µ nj : n ≥ 1 , 1 ≤ j ≤ k n } distributions on R Regularity Extensions • Infinitesimal if for all ε > 0 Omissions vvb n →∞ min lim 1 ≤ j ≤ k n µ nj (( − ε, ε )) = 1 • µ n = µ n 1 ⊞ µ n 2 ⊞ · · · ⊞ µ nk n , ν n = µ n 1 ∗ µ n 2 ∗ · · · ∗ µ nk n • If µ n → µ then µ is ∞ -divisible

Limit laws f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • { µ nj : n ≥ 1 , 1 ≤ j ≤ k n } distributions on R Regularity Extensions • Infinitesimal if for all ε > 0 Omissions vvb n →∞ min lim 1 ≤ j ≤ k n µ nj (( − ε, ε )) = 1 • µ n = µ n 1 ⊞ µ n 2 ⊞ · · · ⊞ µ nk n , ν n = µ n 1 ∗ µ n 2 ∗ · · · ∗ µ nk n • If µ n → µ then µ is ∞ -divisible • µ n → µ ⇔ ν n → ν , where ν is uniquely determined by µ . (Ex.: µ semicircle corresponds with ν normal; CLT)

Limit laws f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • { µ nj : n ≥ 1 , 1 ≤ j ≤ k n } distributions on R Regularity Extensions • Infinitesimal if for all ε > 0 Omissions vvb n →∞ min lim 1 ≤ j ≤ k n µ nj (( − ε, ε )) = 1 • µ n = µ n 1 ⊞ µ n 2 ⊞ · · · ⊞ µ nk n , ν n = µ n 1 ∗ µ n 2 ∗ · · · ∗ µ nk n • If µ n → µ then µ is ∞ -divisible • µ n → µ ⇔ ν n → ν , where ν is uniquely determined by µ . (Ex.: µ semicircle corresponds with ν normal; CLT) • almost analogous results for ⊠

Limit laws f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • { µ nj : n ≥ 1 , 1 ≤ j ≤ k n } distributions on R Regularity Extensions • Infinitesimal if for all ε > 0 Omissions vvb n →∞ min lim 1 ≤ j ≤ k n µ nj (( − ε, ε )) = 1 • µ n = µ n 1 ⊞ µ n 2 ⊞ · · · ⊞ µ nk n , ν n = µ n 1 ∗ µ n 2 ∗ · · · ∗ µ nk n • If µ n → µ then µ is ∞ -divisible • µ n → µ ⇔ ν n → ν , where ν is uniquely determined by µ . (Ex.: µ semicircle corresponds with ν normal; CLT) • almost analogous results for ⊠ • differences: the correspondence µ ↔ ν not bijective

Limit laws f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems • { µ nj : n ≥ 1 , 1 ≤ j ≤ k n } distributions on R Regularity Extensions • Infinitesimal if for all ε > 0 Omissions vvb n →∞ min lim 1 ≤ j ≤ k n µ nj (( − ε, ε )) = 1 • µ n = µ n 1 ⊞ µ n 2 ⊞ · · · ⊞ µ nk n , ν n = µ n 1 ∗ µ n 2 ∗ · · · ∗ µ nk n • If µ n → µ then µ is ∞ -divisible • µ n → µ ⇔ ν n → ν , where ν is uniquely determined by µ . (Ex.: µ semicircle corresponds with ν normal; CLT) • almost analogous results for ⊠ • differences: the correspondence µ ↔ ν not bijective • for the circle, there are no ⊠ -idempotents

Another basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • f , g : C + → C analytic

Another basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • f , g : C + → C analytic • f ≺ g if f ( z ) = g ( h ( z )) for some h : C + → C + analytic (Littlewood subordination)

Another basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • f , g : C + → C analytic • f ≺ g if f ( z ) = g ( h ( z )) for some h : C + → C + analytic (Littlewood subordination) • µ, ν distributions on R

Another basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • f , g : C + → C analytic • f ≺ g if f ( z ) = g ( h ( z )) for some h : C + → C + analytic (Littlewood subordination) • µ, ν distributions on R • Then F µ ⊞ ν ≺ F µ (and F µ ⊞ ν ≺ F ν )

Another basic tool f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • f , g : C + → C analytic • f ≺ g if f ( z ) = g ( h ( z )) for some h : C + → C + analytic (Littlewood subordination) • µ, ν distributions on R • Then F µ ⊞ ν ≺ F µ (and F µ ⊞ ν ≺ F ν ) • Note: subordination functions F < − 1 > ◦ F µ ⊞ ν obviously µ exist at ∞ ; the important point is they continue to C + .

Regularity consequences f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • If d µ/ dt ∈ L p , same is true for µ ⊞ ν

Regularity consequences f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • If d µ/ dt ∈ L p , same is true for µ ⊞ ν • µ ⊞ ν has only finitely many atoms, with total mass < 1

Regularity consequences f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • If d µ/ dt ∈ L p , same is true for µ ⊞ ν • µ ⊞ ν has only finitely many atoms, with total mass < 1 • µ ⊞ ν has no singular continuous component

Regularity consequences f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • If d µ/ dt ∈ L p , same is true for µ ⊞ ν • µ ⊞ ν has only finitely many atoms, with total mass < 1 • µ ⊞ ν has no singular continuous component • the density of µ ⊞ ν is locally analytic a.e.

Regularity consequences f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • If d µ/ dt ∈ L p , same is true for µ ⊞ ν • µ ⊞ ν has only finitely many atoms, with total mass < 1 • µ ⊞ ν has no singular continuous component • the density of µ ⊞ ν is locally analytic a.e. • But: the density of µ ⊞ ν may have points of nondifferentiability even when those of µ and ν don’t

Ignorance f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • µ distribution of R

Ignorance f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • µ distribution of R • ν symmetric to µ ( µ = µ x , ν = µ − x )

Ignorance f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • µ distribution of R • ν symmetric to µ ( µ = µ x , ν = µ − x ) • ρ = µ ∗ ν , λ = µ ⊞ ν ; ρ and λ are symmetric

Ignorance f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • µ distribution of R • ν symmetric to µ ( µ = µ x , ν = µ − x ) • ρ = µ ∗ ν , λ = µ ⊞ ν ; ρ and λ are symmetric • tails of ρ : 1 − ρ (( − t , t )) , t → ∞

Ignorance f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • µ distribution of R • ν symmetric to µ ( µ = µ x , ν = µ − x ) • ρ = µ ∗ ν , λ = µ ⊞ ν ; ρ and λ are symmetric • tails of ρ : 1 − ρ (( − t , t )) , t → ∞ • are comparable to those of µ

Ignorance f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb • µ distribution of R • ν symmetric to µ ( µ = µ x , ν = µ − x ) • ρ = µ ∗ ν , λ = µ ⊞ ν ; ρ and λ are symmetric • tails of ρ : 1 − ρ (( − t , t )) , t → ∞ • are comparable to those of µ • what about λ ?

Operator-valued f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity • A algebra, B ⊂ A unital subalgebra Extensions Omissions vvb

Operator-valued f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity • A algebra, B ⊂ A unital subalgebra Extensions Omissions vvb • τ : A → B conditional expectation

Operator-valued f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity • A algebra, B ⊂ A unital subalgebra Extensions Omissions vvb • τ : A → B conditional expectation • projection so τ ( bab ′ ) = b τ ( a ) b ′ for b , b ′ ∈ B

Operator-valued f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity • A algebra, B ⊂ A unital subalgebra Extensions Omissions vvb • τ : A → B conditional expectation • projection so τ ( bab ′ ) = b τ ( a ) b ′ for b , b ′ ∈ B • Freeness can be defined relative to such τ

Operator-valued f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity • A algebra, B ⊂ A unital subalgebra Extensions Omissions vvb • τ : A → B conditional expectation • projection so τ ( bab ′ ) = b τ ( a ) b ′ for b , b ′ ∈ B • Freeness can be defined relative to such τ • x ∈ A has a combinatorial analogue of a distribution τ ( ab 1 ab 2 a · · · b n − 1 a )

Operator-valued f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity • A algebra, B ⊂ A unital subalgebra Extensions Omissions vvb • τ : A → B conditional expectation • projection so τ ( bab ′ ) = b τ ( a ) b ′ for b , b ′ ∈ B • Freeness can be defined relative to such τ • x ∈ A has a combinatorial analogue of a distribution τ ( ab 1 ab 2 a · · · b n − 1 a ) • This is a “Fourier coefficient” of order n

Operator-valued f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity • A algebra, B ⊂ A unital subalgebra Extensions Omissions vvb • τ : A → B conditional expectation • projection so τ ( bab ′ ) = b τ ( a ) b ′ for b , b ′ ∈ B • Freeness can be defined relative to such τ • x ∈ A has a combinatorial analogue of a distribution τ ( ab 1 ab 2 a · · · b n − 1 a ) • This is a “Fourier coefficient” of order n • µ x + y = µ x ⊞ µ y where ⊞ is defined combinatorially

Operator-valued f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity • A algebra, B ⊂ A unital subalgebra Extensions Omissions vvb • τ : A → B conditional expectation • projection so τ ( bab ′ ) = b τ ( a ) b ′ for b , b ′ ∈ B • Freeness can be defined relative to such τ • x ∈ A has a combinatorial analogue of a distribution τ ( ab 1 ab 2 a · · · b n − 1 a ) • This is a “Fourier coefficient” of order n • µ x + y = µ x ⊞ µ y where ⊞ is defined combinatorially • Subordination survives

Analytic despair f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • vvb G x ( z ) = τ (( z − x ) − 1 )

Analytic despair f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • vvb G x ( z ) = τ (( z − x ) − 1 ) • To be viewed as a function of z ∈ B . (If x = x ∗ , z can be anything with positive invertible imaginary part; Siegel half-plane.)

Analytic despair f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • vvb G x ( z ) = τ (( z − x ) − 1 ) • To be viewed as a function of z ∈ B . (If x = x ∗ , z can be anything with positive invertible imaginary part; Siegel half-plane.) • Function theory not sufficiently well developed to undertake the analysis available when B = C .

Analytic despair f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions • vvb G x ( z ) = τ (( z − x ) − 1 ) • To be viewed as a function of z ∈ B . (If x = x ∗ , z can be anything with positive invertible imaginary part; Siegel half-plane.) • Function theory not sufficiently well developed to undertake the analysis available when B = C . • Fully matricial analytic functions may be needed for full understanding.

⊞ λ f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • X n ( ω ) an n × n random matrix, X n = X ∗ Analytic apparatus n Limit theorems Regularity Extensions Omissions vvb

⊞ λ f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • X n ( ω ) an n × n random matrix, X n = X ∗ Analytic apparatus n Limit theorems • With E expected value, τ n = Tr / n , X n has distribution Regularity Extensions given by Omissions vvb G µ Xn ( z ) = E τ n (( zI − X n ) − 1 ) z ∈ C +

⊞ λ f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • X n ( ω ) an n × n random matrix, X n = X ∗ Analytic apparatus n Limit theorems • With E expected value, τ n = Tr / n , X n has distribution Regularity Extensions given by Omissions vvb G µ Xn ( z ) = E τ n (( zI − X n ) − 1 ) z ∈ C + • Asymptotic (eigenvalue) distribution of X n : lim n µ X n

⊞ λ f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • X n ( ω ) an n × n random matrix, X n = X ∗ Analytic apparatus n Limit theorems • With E expected value, τ n = Tr / n , X n has distribution Regularity Extensions given by Omissions vvb G µ Xn ( z ) = E τ n (( zI − X n ) − 1 ) z ∈ C + • Asymptotic (eigenvalue) distribution of X n : lim n µ X n • X n , Y n independent (classically) with asymptotic distributions µ, ν

⊞ λ f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • X n ( ω ) an n × n random matrix, X n = X ∗ Analytic apparatus n Limit theorems • With E expected value, τ n = Tr / n , X n has distribution Regularity Extensions given by Omissions vvb G µ Xn ( z ) = E τ n (( zI − X n ) − 1 ) z ∈ C + • Asymptotic (eigenvalue) distribution of X n : lim n µ X n • X n , Y n independent (classically) with asymptotic distributions µ, ν • Then (spoonful of salt here): X n + Y n has asymptotic distribution µ ⊞ ν .

⊞ λ f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • X n ( ω ) an n × n random matrix, X n = X ∗ Analytic apparatus n Limit theorems • With E expected value, τ n = Tr / n , X n has distribution Regularity Extensions given by Omissions vvb G µ Xn ( z ) = E τ n (( zI − X n ) − 1 ) z ∈ C + • Asymptotic (eigenvalue) distribution of X n : lim n µ X n • X n , Y n independent (classically) with asymptotic distributions µ, ν • Then (spoonful of salt here): X n + Y n has asymptotic distribution µ ⊞ ν . • Assume now X n , Y n are n × λ n matrices, and | X n | = ( X ∗ n X n ) 1 / 2 , | Y n | have asymptotic distributions µ, ν

⊞ λ f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • X n ( ω ) an n × n random matrix, X n = X ∗ Analytic apparatus n Limit theorems • With E expected value, τ n = Tr / n , X n has distribution Regularity Extensions given by Omissions vvb G µ Xn ( z ) = E τ n (( zI − X n ) − 1 ) z ∈ C + • Asymptotic (eigenvalue) distribution of X n : lim n µ X n • X n , Y n independent (classically) with asymptotic distributions µ, ν • Then (spoonful of salt here): X n + Y n has asymptotic distribution µ ⊞ ν . • Assume now X n , Y n are n × λ n matrices, and | X n | = ( X ∗ n X n ) 1 / 2 , | Y n | have asymptotic distributions µ, ν • Then: | X n + Y n | has asymptotic distribution µ ⊞ λ ν

⊞ λ f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • X n ( ω ) an n × n random matrix, X n = X ∗ Analytic apparatus n Limit theorems • With E expected value, τ n = Tr / n , X n has distribution Regularity Extensions given by Omissions vvb G µ Xn ( z ) = E τ n (( zI − X n ) − 1 ) z ∈ C + • Asymptotic (eigenvalue) distribution of X n : lim n µ X n • X n , Y n independent (classically) with asymptotic distributions µ, ν • Then (spoonful of salt here): X n + Y n has asymptotic distribution µ ⊞ ν . • Assume now X n , Y n are n × λ n matrices, and | X n | = ( X ∗ n X n ) 1 / 2 , | Y n | have asymptotic distributions µ, ν • Then: | X n + Y n | has asymptotic distribution µ ⊞ λ ν • many results extend to this operation, questions remain

Boolean f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus • ( A , τ ) probability space, B , C ⊂ A subalgebras not Limit theorems Regularity containing the unit. Extensions Omissions vvb

Boolean f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus • ( A , τ ) probability space, B , C ⊂ A subalgebras not Limit theorems Regularity containing the unit. Extensions Omissions vvb • B and C are Boolean independent if for b j ∈ B , c j ∈ C , τ ( · b 1 c 1 b 2 c 2 · · · b n c n · ) = · τ ( b 1 ) τ ( c 1 ) · · · τ ( b n ) τ ( c n ) ·

Boolean f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus • ( A , τ ) probability space, B , C ⊂ A subalgebras not Limit theorems Regularity containing the unit. Extensions Omissions vvb • B and C are Boolean independent if for b j ∈ B , c j ∈ C , τ ( · b 1 c 1 b 2 c 2 · · · b n c n · ) = · τ ( b 1 ) τ ( c 1 ) · · · τ ( b n ) τ ( c n ) · • µ b + c = µ b ⊎ µ c for b ∈ B , c ∈ C

Boolean f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus • ( A , τ ) probability space, B , C ⊂ A subalgebras not Limit theorems Regularity containing the unit. Extensions Omissions vvb • B and C are Boolean independent if for b j ∈ B , c j ∈ C , τ ( · b 1 c 1 b 2 c 2 · · · b n c n · ) = · τ ( b 1 ) τ ( c 1 ) · · · τ ( b n ) τ ( c n ) · • µ b + c = µ b ⊎ µ c for b ∈ B , c ∈ C • analytic calculation: F µ ⊎ ν ( z ) − z = [ F µ ( z ) − z ] + [ F ν ( z ) − z ]

Boolean f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus • ( A , τ ) probability space, B , C ⊂ A subalgebras not Limit theorems Regularity containing the unit. Extensions Omissions vvb • B and C are Boolean independent if for b j ∈ B , c j ∈ C , τ ( · b 1 c 1 b 2 c 2 · · · b n c n · ) = · τ ( b 1 ) τ ( c 1 ) · · · τ ( b n ) τ ( c n ) · • µ b + c = µ b ⊎ µ c for b ∈ B , c ∈ C • analytic calculation: F µ ⊎ ν ( z ) − z = [ F µ ( z ) − z ] + [ F ν ( z ) − z ] • much of the limit theory is preserved, but

Boolean f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus • ( A , τ ) probability space, B , C ⊂ A subalgebras not Limit theorems Regularity containing the unit. Extensions Omissions vvb • B and C are Boolean independent if for b j ∈ B , c j ∈ C , τ ( · b 1 c 1 b 2 c 2 · · · b n c n · ) = · τ ( b 1 ) τ ( c 1 ) · · · τ ( b n ) τ ( c n ) · • µ b + c = µ b ⊎ µ c for b ∈ B , c ∈ C • analytic calculation: F µ ⊎ ν ( z ) − z = [ F µ ( z ) − z ] + [ F ν ( z ) − z ] • much of the limit theory is preserved, but • generally δ s ⊎ δ t � = δ s + t

Boolean f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions Analytic apparatus • ( A , τ ) probability space, B , C ⊂ A subalgebras not Limit theorems Regularity containing the unit. Extensions Omissions vvb • B and C are Boolean independent if for b j ∈ B , c j ∈ C , τ ( · b 1 c 1 b 2 c 2 · · · b n c n · ) = · τ ( b 1 ) τ ( c 1 ) · · · τ ( b n ) τ ( c n ) · • µ b + c = µ b ⊎ µ c for b ∈ B , c ∈ C • analytic calculation: F µ ⊎ ν ( z ) − z = [ F µ ( z ) − z ] + [ F ν ( z ) − z ] • much of the limit theory is preserved, but • generally δ s ⊎ δ t � = δ s + t • There is a multiplicative analogue

Monotonic f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • ( A , τ ) probability space, B , C ⊂ A subalgebras not Analytic apparatus Limit theorems containing the unit. Regularity Extensions Omissions vvb

Monotonic f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • ( A , τ ) probability space, B , C ⊂ A subalgebras not Analytic apparatus Limit theorems containing the unit. Regularity Extensions • B and C are monotone independent if for b j ∈ B , c j ∈ C , Omissions vvb τ ( b 1 c 1 ) = τ ( c 1 b 1 ) = τ ( b 1 ) τ ( c 1 ) , τ ( c 1 b 1 c 2 ) = τ ( c 1 ) τ ( b 1 ) τ ( c 2 ) , b 1 c 1 b 2 = τ ( c 1 ) b 1 b 2 , and

Monotonic f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • ( A , τ ) probability space, B , C ⊂ A subalgebras not Analytic apparatus Limit theorems containing the unit. Regularity Extensions • B and C are monotone independent if for b j ∈ B , c j ∈ C , Omissions vvb τ ( b 1 c 1 ) = τ ( c 1 b 1 ) = τ ( b 1 ) τ ( c 1 ) , τ ( c 1 b 1 c 2 ) = τ ( c 1 ) τ ( b 1 ) τ ( c 2 ) , b 1 c 1 b 2 = τ ( c 1 ) b 1 b 2 , and • µ b + c = µ b ⊲ µ c b ∈ B , c ∈ C

Monotonic f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • ( A , τ ) probability space, B , C ⊂ A subalgebras not Analytic apparatus Limit theorems containing the unit. Regularity Extensions • B and C are monotone independent if for b j ∈ B , c j ∈ C , Omissions vvb τ ( b 1 c 1 ) = τ ( c 1 b 1 ) = τ ( b 1 ) τ ( c 1 ) , τ ( c 1 b 1 c 2 ) = τ ( c 1 ) τ ( b 1 ) τ ( c 2 ) , b 1 c 1 b 2 = τ ( c 1 ) b 1 b 2 , and • µ b + c = µ b ⊲ µ c b ∈ B , c ∈ C • analytic calculation: F µ⊲ν = F µ ◦ F ν

Monotonic f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • ( A , τ ) probability space, B , C ⊂ A subalgebras not Analytic apparatus Limit theorems containing the unit. Regularity Extensions • B and C are monotone independent if for b j ∈ B , c j ∈ C , Omissions vvb τ ( b 1 c 1 ) = τ ( c 1 b 1 ) = τ ( b 1 ) τ ( c 1 ) , τ ( c 1 b 1 c 2 ) = τ ( c 1 ) τ ( b 1 ) τ ( c 2 ) , b 1 c 1 b 2 = τ ( c 1 ) b 1 b 2 , and • µ b + c = µ b ⊲ µ c b ∈ B , c ∈ C • analytic calculation: F µ⊲ν = F µ ◦ F ν • some of the limit theory is preserved

Monotonic f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • ( A , τ ) probability space, B , C ⊂ A subalgebras not Analytic apparatus Limit theorems containing the unit. Regularity Extensions • B and C are monotone independent if for b j ∈ B , c j ∈ C , Omissions vvb τ ( b 1 c 1 ) = τ ( c 1 b 1 ) = τ ( b 1 ) τ ( c 1 ) , τ ( c 1 b 1 c 2 ) = τ ( c 1 ) τ ( b 1 ) τ ( c 2 ) , b 1 c 1 b 2 = τ ( c 1 ) b 1 b 2 , and • µ b + c = µ b ⊲ µ c b ∈ B , c ∈ C • analytic calculation: F µ⊲ν = F µ ◦ F ν • some of the limit theory is preserved • δ s ⊲ δ t � = δ s + t

Monotonic f ( z ) , µ ⊞ ν , etc. H. Bercovici Free convolutions • ( A , τ ) probability space, B , C ⊂ A subalgebras not Analytic apparatus Limit theorems containing the unit. Regularity Extensions • B and C are monotone independent if for b j ∈ B , c j ∈ C , Omissions vvb τ ( b 1 c 1 ) = τ ( c 1 b 1 ) = τ ( b 1 ) τ ( c 1 ) , τ ( c 1 b 1 c 2 ) = τ ( c 1 ) τ ( b 1 ) τ ( c 2 ) , b 1 c 1 b 2 = τ ( c 1 ) b 1 b 2 , and • µ b + c = µ b ⊲ µ c b ∈ B , c ∈ C • analytic calculation: F µ⊲ν = F µ ◦ F ν • some of the limit theory is preserved • δ s ⊲ δ t � = δ s + t • there is a multiplicative version of the convolution