Products of Non-Hermitian Random Matrices David Renfrew Department of Mathematics University of California, Los Angeles March 26, 2014 Joint work with S. O’Rourke, A. Soshnikov, V. Vu David Renfrew Products
Non-Hermitian random matrices C N is an N × N real random matrix with i.i.d entries such that E [ C 2 E [ C ij ] = 0 ij ] = 1 / N We study in the large N limit of the empirical spectral measure: N µ N ( z ) = 1 � δ λ i ( z ) N i = 1 David Renfrew Products
Circular law Girko, Bai, . . . , Tao-Vu. As N → ∞ , µ N ( z ) converges a.s. in distribution to µ c , the uniform law on the unit disk, d µ c ( z ) = 1 2 π 1 | z |≤ 1 , dz David Renfrew Products
1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure : Eigenvalues of a 1000 × 1000 iid random matrix David Renfrew Products
Products of iid random matrices Let m ≥ 2 be a fixed integer. Let C N , 1 , C N , 2 , . . . , C N , m be an independent family of random matrices each with iid entries. Götze-Tikhomirov and O’Rourke-Soshnikov computed the limiting distribution of the product C N , 1 C N , 2 · · · C N , m as N goes to infinity. Limiting density is given by the m th power of the circular law. d µ m ( z ) 1 2 m − 2 1 | z |≤ 1 . = m π | z | dz David Renfrew Products
1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Left: eigenvalues of the product of two independent 1000 × 1000 iid random matrices Right: eigenvalues of the product of four independent 1000 × 1000 iid random matrices David Renfrew Products
Products of iid random matrices Studied in physics, either non-rigorously or in Gaussian case. Z. Burda, R. A. Janik,and B. Waclaw Akemann G, Ipsen J, Kieburg M Akemann G, Kieburg M, Wei L David Renfrew Products
Elliptical Random matrices A generalization of the iid model, that interpolates between iid and Wigner. X N is an N × N real random matrix such that E [ X 2 E [ | X ij | 2 + ǫ ] < ∞ E [ X ij ] = 0 ij ] = 1 / N For i � = j , − 1 ≤ ρ ≤ 1 E [ X ij X ji ] = ρ/ N Entries are otherwise independent. Simplest case is weighted sum of GOE and real Ginibre. X N = √ ρ W N + � 1 − ρ C N David Renfrew Products
Elliptical Law The limiting distribution of X N for general ρ is an ellipse. (Girko; Naumov; Nguyen-O’Rourke) and µ ρ is the uniform probability measure on the ellipsoid z ∈ C : Re ( z ) 2 ( 1 + ρ ) 2 + Im ( z ) 2 � � E ρ = ( 1 − ρ ) 2 < 1 . David Renfrew Products
1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.5 −1 −0.5 0 0.5 1 1.5 Figure : Eigenvalues of a 1000 × 1000 Elliptic random matrix, with ρ = . 5 David Renfrew Products
Products of random matrices Theorem (O’Rourke,R,Soshnikov,Vu) Let X 1 N , X 2 N , . . . , X m N be independent elliptical random matrices. Each with parameter − 1 < ρ i < 1 , for 1 ≤ i ≤ m. Almost surely the empirical spectral measure of the product X 1 N X 2 N · · · X m N converges to µ m , the m th power of the circular law. David Renfrew Products
1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Left: eigenvalues of the product of two identically distributed elliptic random matrices with Gaussian entries when ρ 1 = ρ 2 = 1 / 2 Right: eigenvalues of the product of a Wigner matrix and an independent iid random matrix David Renfrew Products
Linearization Let 0 X N , 1 0 0 0 X N , 2 0 ... ... Y N := 0 0 X N , m − 1 X N , m 0 David Renfrew Products
Linearization Note that raising Y N to the m th power leads to Z N , 1 0 0 0 Z N , 2 0 0 ... ... Y m N := 0 0 Z N , m − 1 0 0 Z N , m Where Z N , k = X N , k X N , k + 1 · · · X N , k − 1 So λ is an eigenvalue of Y N iff λ m is an eigenvalue of X N , 1 X N , 2 · · · X N , m . David Renfrew Products
Hermitization The log potential allows one to connect eigenvalues of a non-Hermitian matrix to those of a family of Hermitian matrices. � ∞ � log | z − s | d µ N ( s ) = 1 N log ( | det ( Y N − z ) | ) = log ( x ) ν N , z ( x ) 0 Where ν N , z ( x ) is the empirical spectral measure of � � 0 X N − z . ( X N − z ) ∗ 0 The spectral measure can be recovered from the log potential. � 2 πµ N ( z ) = ∆ log | z − s | d µ N ( s ) David Renfrew Products
Hermitization First step is to show ν N , z → ν z Show that log ( x ) can be integrated by bounding singular values. David Renfrew Products
Circular law In order to compute ν N , z , we use the Stieltjes transform. � d ν N , z ( x ) a N ( η, z ) := x − η which is also the normalized trace of the resolvent. � − 1 � − η C N − z R ( η, z ) = ( C N − z ) ∗ − η It is useful to keep the block structure of R N and define � a N ( η, z ) � b N ( η, z ) Γ N ( η, z ) = ( I 2 ⊗ tr N ) R N ( η, z ) = c N ( η, z ) a N ( η, z ) David Renfrew Products
Circular law The Stietljes transform corresponding to the circular law is characterized as the unique Stieltjes transform that solves the equation a ( η, z ) + η a ( η, z ) = | z | 2 − ( a ( η, z ) + η ) 2 for each z ∈ C , η ∈ C + . Our goal is to show a N ( η, z ) approximately satisfies this equation. David Renfrew Products
Circular law Let � − 1 � − ( a ( η, z ) + η ) − z Γ( η, z ) := . − ¯ z − ( a ( η, z ) + η ) By the defining equation of a , � z � a ( η, z ) ( a ( η, z )+ η ) 2 −| z | 2 Γ( η, z ) = . z a ( η, z ) ( a ( η, z )+ η ) 2 −| z | 2 David Renfrew Products
Circular law Letting � η � z q := . z η and Σ( A ) := diag ( A ) This relationship can compactly be written Γ( η, z ) = − ( q + Σ(Γ( η, z ))) − 1 . So we can instead show Γ N is close to Γ . David Renfrew Products
Resolvent Schur’s complement � − 1 � A B = ( A − BD − 1 C ) − 1 C D 11 � R 1 , 1 � R 1 , N + 1 R N + 1 , 1 R N + 1 , N + 1 �� η � � � R ( 1 ) 11 � � �� C ( 1 ) C ( 1 ) � R ( 1 ) 12 z 0 0 1 · · 1 = − + C ( 1 ) ∗ R ( 1 ) 21 R ( 1 ) 22 C ( 1 ) ∗ z η 0 0 1 · 1 · �� − 1 � tr ( R 22 ) �� η � z 0 ≈ − + tr ( R 11 ) η z 0 David Renfrew Products
Resolvent So Γ N ( η, z ) ≈ − ( q + Σ(Γ N ( η, z ))) − 1 David Renfrew Products
Products It will suffice to prove the circular law for 0 X N , 1 0 0 0 X N , 2 0 ... ... Y N = (1) 0 0 X N , m − 1 X N , m 0 Let � 0 � Y N H N = Y ∗ 0 N Once again we study the hermitized resolvent �� 0 �� − 1 � � η I mN Y N zI mN R N ( η, z ) = − Y ∗ 0 zI mN η I mN N David Renfrew Products
Block Resolvent As before we keep the block structure of R N and let Γ N ( η, z ) = ( I 2 m ⊗ tr N ) R N ( η, z ) Let R N ; 11 be the 2 m × 2 m matrix whose entries are the ( 1 , 1 ) entry of each block of the resolvent. Let H ( 1 ) N ; 1 be a 2 m × 2 m matrix with N − 1 dimensional vectors � − 1 � q ⊗ I m + H ( 1 ) ∗ N ; 1 R ( 1 ) N H ( 1 ) R N ; 11 = − N ; 1 David Renfrew Products
0 X N , 1 0 ... ... 0 0 0 X N , m − 1 X N , m 0 H N = X ∗ 0 0 N , m ... ... X ∗ N , 1 0 ... 0 0 X ∗ 0 0 N , m − 1 David Renfrew Products
Block Resolvent So Γ N ( η, z ) ≈ ( q ⊗ I m − Σ(Γ N ( η, z )) − 1 where Σ being a linear operator on 2 m × 2 m matrices defined by: 2 m � Σ( A ) ab = σ ( a , c ; d , b ) A cd c , d = 1 σ ( a , c ; d , b ) = N E [ H ac 12 H db 12 ] Σ( A ) ab = A a ′ a ′ δ ab + ρ a A a ′ a δ aa ′ , David Renfrew Products
Fixed point equation In the limit Γ = − ( q ⊗ I m + Σ(Γ)) − 1 This equation has a unique solution that is a matrix valued Stietljes transform (J. Helton, R. Far, R. Speicher) As η → ∞ , � η I m − 1 � − zI m Γ ∼ . η 2 − | z | 2 − ¯ zI m η I M Since Σ leaves main diagonal invariant and sets diagonals of the upper blocks to zero, Γ is of this form. David Renfrew Products
So Γ actually satisfies the equation: Γ( η, z ) = − ( q ⊗ I m + diag (Γ( η, z ))) − 1 This means for 1 ≤ i ≤ 2 m , the diagonal entries of the matrix valued Stieltjes transform are given by the Stieltjes transform corresponding to the circular law. Γ( η, z ) ii = a ( η, z ) David Renfrew Products
Smallest singular value Theorem (Nguyen, O’Rourke) Let X N be an elliptical random matrix with − 1 < ρ < 1 and F N be deterministic matrix, for any B > 0, there exists A > 0 � σ N ( X N + F N ) ≤ N − A � = O ( N − B ) . P David Renfrew Products
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