FREE PROBABILITY OF TYPE B AND ASYMPTOTICS OF FINITE-RANK - PowerPoint PPT Presentation

FREE PROBABILITY OF TYPE B AND ASYMPTOTICS OF FINITE-RANK PERTURBATIONS OF RANDOM MATRICES Dima Shlyakhtenko, UCLA Free Probability and Large N Limit, V

Many B ’s around:

Many B ’s around: ◮ bi-free probability (Voiculescu’s talk)

Many B ’s around: ◮ bi-free probability (Voiculescu’s talk) ◮ B -valued probability (amalgamation over a subalgebra)

Many B ’s around: ◮ bi-free probability (Voiculescu’s talk) ◮ B -valued probability (amalgamation over a subalgebra) ◮ B -valued bi-free (Skoufranis’s talk)

Many B ’s around: ◮ bi-free probability (Voiculescu’s talk) ◮ B -valued probability (amalgamation over a subalgebra) ◮ B -valued bi-free (Skoufranis’s talk) ◮ type B free probability (this talk)

Many B ’s around: ◮ bi-free probability (Voiculescu’s talk) ◮ B -valued probability (amalgamation over a subalgebra) ◮ B -valued bi-free (Skoufranis’s talk) ◮ type B free probability (this talk)

Wigner’s Semicircle Law Let A ( N ) be an N × N random matrix so that { Re ( A ij ) , Im ( A ij ) : 1 ≤ i < j ≤ N } ∪ { A kk : 1 ≤ k ≤ N } are iid real Gaussians of variance N − 1 / 2 ( 1 + δ ij ) . Let λ A 1 ( N ) ≤ · · · ≤ λ A N ( N ) be the eigenvalues of A ( N ) , and let N = 1 � µ A δ λ A j ( N ) . N j Then as N → ∞ N ] → semicircle law = 1 � E [ µ A 2 − t 2 χ [ − √ √ 2 ] dt 2 , π

Voiculescu’s Asymptotic Freeness A ( N ) as before, B ( N ) diagonal matrix with eigenvalues λ B 1 ( N ) ≤ · · · ≤ λ B N ( N ) . Assume that N = 1 � µ B j ( N ) → µ B . δ λ B N j Then A ( N ) and B ( N ) are asymptoically freely independent. In particular, → µ A ⊞ µ B . µ A + B N

Voiculescu’s Asymptotic Freeness A ( N ) as before, B ( N ) diagonal matrix with eigenvalues λ B 1 ( N ) ≤ · · · ≤ λ B N ( N ) . Assume that N = 1 � µ B j ( N ) → µ B . δ λ B N j Then A ( N ) and B ( N ) are asymptoically freely independent. In particular, → µ A ⊞ µ B . µ A + B N Example: B = 3 P N with P N a projection of rank N / 2.

Analytic Subordination and Free Convolution [Biane,Voiculescu,...] To compute η = µ A ⊞ µ B define G ν = 1 ´ z − t d ν ( t ) . Then there exist analytic functions ω A , ω B : C + → C + uniquely determined by ◮ G µ A ( ω A ( z )) = G µ B ( ω B ( z )) = G η ( z ) ◮ ω A ( z ) + ω B ( z ) = z + 1 / G η ( z ) ◮ lim y ↑∞ ω A ( iy ) / ( iy ) = lim y →∞ ω ′ A ( iy ) = 1 and same for ω B .

Analytic Subordination and Free Convolution [Biane,Voiculescu,...] To compute η = µ A ⊞ µ B define G ν = 1 ´ z − t d ν ( t ) . Then there exist analytic functions ω A , ω B : C + → C + uniquely determined by ◮ G µ A ( ω A ( z )) = G µ B ( ω B ( z )) = G η ( z ) ◮ ω A ( z ) + ω B ( z ) = z + 1 / G η ( z ) ◮ lim y ↑∞ ω A ( iy ) / ( iy ) = lim y →∞ ω ′ A ( iy ) = 1 and same for ω B . Put F ν ( z ) = 1 / G ν ( z ) . Then R ν ( z ) = F − 1 ν ( z ) + z so that R µ A ( z ) + R µ B ( z ) = R µ A ⊞ µ B ( z ) becomes F − 1 µ A ( z ) + F − 1 z + F − 1 µ B ( z ) = µ A ⊞ µ B ( z ) .

Analytic Subordination and Free Convolution [Biane,Voiculescu,...] To compute η = µ A ⊞ µ B define G ν = 1 ´ z − t d ν ( t ) . Then there exist analytic functions ω A , ω B : C + → C + uniquely determined by ◮ G µ A ( ω A ( z )) = G µ B ( ω B ( z )) = G η ( z ) ⇔ F µ A ( ω A ( z )) = F µ B ( ω B ( z )) = F η ( z ) ◮ ω A ( z ) + ω B ( z ) = z + 1 / G η ( z ) ◮ lim y ↑∞ ω A ( iy ) / ( iy ) = lim y →∞ ω ′ A ( iy ) = 1 and same for ω B . Put F ν ( z ) = 1 / G ν ( z ) . Then R ν ( z ) = F − 1 ν ( z ) + z so that R µ A ( z ) + R µ B ( z ) = R µ A ⊞ µ B ( z ) becomes F − 1 µ A ( z ) + F − 1 z + F − 1 µ B ( z ) = µ A ⊞ µ B ( z ) . ω A = F − 1 ω B = F − 1 µ A ◦ F µ A ⊞ µ B µ B ◦ F µ A ⊞ µ B .

Analytic Subordination and Free Convolution [Biane,Voiculescu,...] To compute η = µ A ⊞ µ B define G ν = 1 ´ z − t d ν ( t ) . Then there exist analytic functions ω A , ω B : C + → C + uniquely determined by ◮ G µ A ( ω A ( z )) = G µ B ( ω B ( z )) = G η ( z ) ⇔ F µ A ( ω A ( z )) = F µ B ( ω B ( z )) = F η ( z ) ◮ ω A ( z ) + ω B ( z ) = z + 1 / G η ( z ) ⇔ ω A ( z ) + ω B ( z ) = z + F η ( z ) ◮ lim y ↑∞ ω A ( iy ) / ( iy ) = lim y →∞ ω ′ A ( iy ) = 1 and same for ω B . Put F ν ( z ) = 1 / G ν ( z ) . Then R ν ( z ) = F − 1 ν ( z ) + z so that R µ A ( z ) + R µ B ( z ) = R µ A ⊞ µ B ( z ) becomes F − 1 µ A ( z ) + F − 1 z + F − 1 µ B ( z ) = µ A ⊞ µ B ( z ) . ω A = F − 1 ω B = F − 1 µ A ◦ F µ A ⊞ µ B µ B ◦ F µ A ⊞ µ B .

Finite-rank perturbations [Ben Arous, Baik, Peche]. Let A ( N ) be as before but consider B ( N ) a finite rank matrix (e.g. B N = θ Q N ) with Q N rank 1 projection.

Finite-rank perturbations [Ben Arous, Baik, Peche]. Let A ( N ) be as before but consider B ( N ) a finite rank matrix (e.g. B N = θ Q N ) with Q N rank 1 projection. Semicircular limit for A ( N ) + B ( N ) but there may or may not be outlier eigenvalues: θ = 3 θ = 0 . 5

Finite rank perturbations and freeness? It was discovered (Capitaine, Belischi-Bercovici-Capitain-Fevrier) that the description of the outlier involves free subordination functions. For example, if A N is GUE and B N has 1 eigenvalue θ and the rest zero, then we set ( ω A , ω B ) = subordination functions for η ⊞ δ 0 with η = semicircle law , i.e., ω A ( z ) = F − 1 η ( z ) , ω B ( z ) = z , then there will be an outlier at θ ′ = ω A ( θ ) (i.e. G µ ( θ ′ ) = 1 /θ ).

Finite rank perturbations and freeness? It was discovered (Capitaine, Belischi-Bercovici-Capitain-Fevrier) that the description of the outlier involves free subordination functions. For example, if A N is GUE and B N has 1 eigenvalue θ and the rest zero, then we set ( ω A , ω B ) = subordination functions for η ⊞ δ 0 with η = semicircle law , i.e., ω A ( z ) = F − 1 η ( z ) , ω B ( z ) = z , then there will be an outlier at θ ′ = ω A ( θ ) (i.e. G µ ( θ ′ ) = 1 /θ ). Why?! Is there still some free independence involved?

Another look at laws of random matrices We consider the 1 / N expansion of the law of A N + B N : = µ A + B + 1 µ A + B + o ( N − 1 ) . µ A + B N ˙ N The idea is that moving 1 eigenvalue out of N gives a perturbation of µ A + B which is of order 1 / N . Our aim is to compute ˙ µ A + B . µ A + B and not just Thus we want to keep track of the pair µ A + B , ˙ µ A + B (ordinary free probability).

Infinitesimal free probability theory [Belinschi-D.S, 2012] To encode such questions we consider an infinitesimal probability space ( A , φ, φ ′ ) where A is a unital algebra, φ, φ ′ : A → C are linear functionals and φ ( 1 ) = 1, φ ′ ( 1 ) = 0. Example Let ( A , φ t ) be a family of probability spaces, and assume that φ t = φ + t φ ′ + o ( t ) . Then ( A , φ, φ ′ ) is an infinitesimal probability space.

Infinitesimal free probability theory [Belinschi-D.S, 2012] To encode such questions we consider an infinitesimal probability space ( A , φ, φ ′ ) where A is a unital algebra, φ, φ ′ : A → C are linear functionals and φ ( 1 ) = 1, φ ′ ( 1 ) = 0. Example Let ( A , φ t ) be a family of probability spaces, and assume that φ t = φ + t φ ′ + o ( t ) . Then ( A , φ, φ ′ ) is an infinitesimal probability space. Eg: X t family of random variables and you define φ t : C [ t ] → C by φ t ( p ) = E ( p ( X t )) .

Infinitesimal freeness, ctd. We say that A 1 , A 2 ⊂ A are infinitesimally free if the freeness condition in ( A , φ t = φ + t φ ′ ) holds to order o ( t ) .

Infinitesimal freeness, ctd. We say that A 1 , A 2 ⊂ A are infinitesimally free if the freeness condition in ( A , φ t = φ + t φ ′ ) holds to order o ( t ) . In other words, the following conditions holds whenever a 1 , . . . , a r ∈ A are such that a k ∈ A i k , i 1 � = i 2 , i 2 � = i 3 , . . . and φ ( a 1 ) = φ ( a 2 ) = · · · = φ ( a n ) = 0: φ ( a 1 · · · a r ) = 0 ; r � φ ′ ( a 1 · · · a r ) φ ( a 1 · · · a j − 1 φ ′ ( a j ) a j + 1 · · · a r ) . = j = 1

Free probability of type B [Biane-Goodman-Nica, 2003] We introduced infinitesimal free probability theory to get a better understanding of type B free probability introduced by Biane-Goodman-Nica. Their motivation was purely combinatorial: free probability is obtained from classical probability by replacing the lattice of all partitions by the lattice of (type A) non-crossing partitions:

Free probability of type B [Biane-Goodman-Nica, 2003] We introduced infinitesimal free probability theory to get a better understanding of type B free probability introduced by Biane-Goodman-Nica. Their motivation was purely combinatorial: free probability is obtained from classical probability by replacing the lattice of all partitions by the lattice of (type A) non-crossing partitions: Non-crossing partition (of type A): partition of ( 1 , . . . , n ) so that if i < j < k < l and i ∼ k , j ∼ l then i ∼ l . 1 2 3 4 5 6 • • • • • •