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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

  1. 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

  2. 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 ( 1 A ) = 1 . Octavio Arizmendi CIMAT Block modified Random Matrices

  3. 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 ( 1 A ) = 1 . 2 A non-commutative random variable (or random variable) is just an element a ∈ A . Octavio Arizmendi CIMAT Block modified Random Matrices

  4. 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 ( 1 A ) = 1 . 2 A non-commutative random variable (or random variable) is just an element a ∈ A . 3 If there is a measure µ a in C with compact support, such that � z k z l µ a ( dz ) = φ ( a k ( a ∗ ) l ) , for all k , l ∈ N , C we call µ a the *-distribution of a . Octavio Arizmendi CIMAT Block modified Random Matrices

  5. Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions Free Independence Definition A family of subalgebras ( A i ) i ∈ I of A is said to be free independent if for all a 1 ∈ A i (1) , ..., a n ∈ A i ( n ) φ ( a 1 a 2 ... a n ) = 0 whenever φ ( a j ) = 0 , a j ∈ A i ( j ) , and i (1) � = i (2) � = ... � = i ( n ). Octavio Arizmendi CIMAT Block modified Random Matrices

  6. 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

  7. 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 A d and B d deterministic matrices and U d a Haar Unitary. Then A d and U d B d U ∗ d are asymptotically free as d → ∞ . 2 Let G ( n ) and G ( n ) hermitian Independent Gaussian Random 1 2 Matrices of size n . Then G ( n ) and G ( n ) are asymptotically 1 2 free as n → ∞ . Octavio Arizmendi CIMAT Block modified Random Matrices

  8. Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions Free Addition of Bernoullis UB 1 U ∗ + B 2 Octavio Arizmendi CIMAT Block modified Random Matrices

  9. 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

  10. 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

  11. Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions Free Multiplication 2 W 1 W 2 2 Octavio Arizmendi CIMAT Block modified Random Matrices

  12. 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

  13. Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions Free Multiplication 4 ( UB 1 U ∗ + B 2 ) W 2 Octavio Arizmendi CIMAT Block modified Random Matrices

  14. Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions Free Commutator W 1 W 2 − W 2 W 1 Octavio Arizmendi CIMAT Block modified Random Matrices

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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 : M n ( C ) → M n ( C ) be a self-adjoint linear map. Octavio Arizmendi CIMAT Block modified Random Matrices

  20. 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 : M n ( C ) → M n ( C ) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := ( id d ⊗ ϕ )( W ) . Octavio Arizmendi CIMAT Block modified Random Matrices

  21. 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 : M n ( C ) → M n ( C ) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := ( id d ⊗ ϕ )( W ) . We study the asymptotic eigenvalue distribution of W ϕ . Octavio Arizmendi CIMAT Block modified Random Matrices

  22. 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 : M n ( C ) → M n ( C ) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := ( id d ⊗ ϕ )( W ) . We study the asymptotic eigenvalue distribution of W ϕ . W Wishart and ϕ ( A ) = A t (Aubrun 2012). ϕ ”planar” Octavio Arizmendi CIMAT Block modified Random Matrices

  23. 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 : M n ( C ) → M n ( C ) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := ( id d ⊗ ϕ )( W ) . We study the asymptotic eigenvalue distribution of W ϕ . W Wishart and ϕ ( A ) = A t (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

  24. 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 : M n ( C ) → M n ( C ) be a self-adjoint linear map. We consider the block-modified matrix W ϕ := ( id d ⊗ ϕ )( W ) . We study the asymptotic eigenvalue distribution of W ϕ . W Wishart and ϕ ( A ) = A t (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

  25. Preliminaries in Free Probability Block modified matrices Operator-Valued free probability Explicit solutions Let us write n � α ij ϕ ( A ) = kl E ij AE kl , i , j , k , l =1 where E ij ∈ M n ( C ) are matrix units. Octavio Arizmendi CIMAT Block modified Random Matrices

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