On Links Between the Random Matrix and Random Operator Theories L. - PowerPoint PPT Presentation

L. Pastur Links between ROT and RMT On Links Between the Random Matrix and Random Operator Theories L. Pastur Institute for Low Temperatures of Ac. of Sci. of Ukraine Kharkiv, Ukraine St.-Petersbourg, 15 July 2010 Slide 1

Outline • Introduction • Most Widely Known Random Matrices – Description – Basic Results • ”Corresponding” Families of Random Operators – Description – Integrated Density of States (IDS) – Asymptotic Results on IDS • Comments 2

1 Introduction Let { M n } be the sequence of n × n random hermitian matrices with ν n non-zero entries, all are on the principal and adjacent diagonals and independent (ergodic) modulo symmetry. Then { M n } determines • random operator, if ν n /n → 2 p + 1 , n → ∞ , p ∈ Z ; • random matrix, if ν n /n → ∞ , n → ∞ . 3

1 Introduction Let { M n } be the sequence of n × n random hermitian matrices with ν n non-zero entries, all are on the principal and adjacent diagonals and independent (ergodic) modulo symmetry. Then { M n } determines • random operator, if ν n /n → 2 p + 1 , n → ∞ , p ∈ Z ; • random matrix, if ν n /n → ∞ , n → ∞ . Random Operator Theory (ROT) is mostly on spectral types of ”limiting” selfadjoint ergodic operators in l 2 ( Z ) more generally in l 2 ( Z d ) , defined by the double infinite ”limit” of the corresponding finite size matrices). 3

Random Matrix Theory (RMT) is mostly on the eigenvalue distribution as n → ∞ (no ”limiting” operators but still well defined limiting and asymptotic spectral characteristics, cf statistical mechanics). Common topics • Integrated Density of States, the n → ∞ limit of the Normalized Counting Measure of Eigenvalues of M n . • Asymptotics of spacing between adjacent eigenvalues. • Eigenvectors 4

Random Matrix Theory (RMT) is mostly on the eigenvalue distribution as n → ∞ (no ”limiting” operators but still well defined limiting and asymptotic spectral characteristics, cf statistical mechanics). Common topics • Integrated Density of States, the n → ∞ limit of the Normalized Counting Measure of Eigenvalues of M n . • Asymptotics of spacing between adjacent eigenvalues. • Eigenvectors Subject of the talk: families of random ergodic operators, possessing certain properties of random matrices in certain asymptotic regimes and vice versa. 4

2 Most Widely Known Random Matrices 2.1 Description 2.1.1 Gaussian Unitary Ensemble (GUE) M n = n − 1 / 2 W n , W n = { W jk } n j,k =1 , W jk = W kj ∈ C W jk , 1 ≥ j ≥ k ≥ n are independent complex Gaussian and E { W jk } = E { W 2 jk } = 0 , E {| W jk | 2 } = w 2 (1 + δ jk ) / 2 . 5

(i) Band Version b − 1 / 2 ( M n,b ) j,k = ϕ ( | j − k | /β n ) W j,k , b n = 2 β n + 1 , β n ∈ N , n � ϕ 2 ( t ) dt = 1 . supp ϕ = [0 , 1] , R 6

(i) Band Version b − 1 / 2 ( M n,b ) j,k = ϕ ( | j − k | /β n ) W j,k , b n = 2 β n + 1 , β n ∈ N , n � ϕ 2 ( t ) dt = 1 . supp ϕ = [0 , 1] , R (ii) Deformed Version M n = M (0) + n − 1 / 2 W n , n where M (0) is either non-random or random and independent of W n . n 6

2.1.2 ”Wishart” Matrices m,n X m,n , X m,n = { X αj } m,n M n = n − 1 X ∗ α,j , where { X αj } m,n α,j are i.i.d. complex Gaussian and E { X αj } = E { X 2 αj } = 0 , E {| X αj | 2 } = a 2 In statistics one calls white (or null) Wishart matrices those with real Gaussian X ’s (sample covariance matrix of Gaussian population). The above case is known in the RMT as the Laguerre Ensemble . Deformed Versions (both additively and multiplicatively): M n = M (0) + n − 1 X ∗ m,n T m X m,n n 7

and (signal+noise) M n = ( A (0) m,n + n − 1 / 2 X m,n ) ∗ T m ( A (0) m,n + X m,n ) , where M (0) , A m,n , and T m are either non-random or random and n independent of X m,n and one of another. 8

and (signal+noise) M n = ( A (0) m,n + n − 1 / 2 X m,n ) ∗ T m ( A (0) m,n + X m,n ) , where M (0) , A m,n , and T m are either non-random or random and n independent of X m,n and one of another. 2.1.3 Law of Addition (Free Probability) M n = A n + U ∗ n B n U n , where U n is Haar distributed over U ( n ) and A n and B n are n × n either non-random or random hermitian and independent of U n and one of another. 8

2.1.4 Wigner Matrices Replace W jk , 1 ≤ j ≤ k ≤ n in the GUE and its band and deformed versions by arbitrary random variables (double array) with the same first and second moment. 9

2.1.4 Wigner Matrices Replace W jk , 1 ≤ j ≤ k ≤ n in the GUE and its band and deformed versions by arbitrary random variables (double array) with the same first and second moment. 2.1.5 Sample Covariance Matrices Replace { X αj } m,n α,j in ”Wishart” and its deformed versions by arbitrary random variables with the same first and second moment. 9

2.2 Basic Results Introduce the Normalized Counting Measure (NCM) N n of eigenvalues { λ ( n ) } n l =1 of M n l N n (∆) = ♯ { l = 1 , ..., n : λ ( n ) ∈ ∆ } /n, ∆ ⊂ R l and assume that the NCM’s N (0) for H (0) n , σ m of T m , N A n of A n , and n B n have weak limits (with probability 1 if random) as m, n → ∞ , m/n → c ∈ [0 , ∞ ) . 10

Then in all above cases N n converges weakly with probability 1 to a non-random limit N . The limit can be found via its Stieltjes transform � N ( dλ ) f ( z ) = λ − z , ℑ z � = 0 , that solves the functional equations below, and the inversion formula � 1 N (∆) = lim ℑ f ( λ + i 0) dλ. π ε → 0 ∆ 11

2.2.1 Deformed GUE (i) The Stieltjes transform of the limiting NCM solves the equation f ( z ) = f (0) ( z + w 2 f ( z )) , that has a unique solution in the class of Nevanlinna functions, i.e., analytic for non-real z and such that ℑ f ℑ z ≥ 0 , f ( z ) = − z − 1 + o ( z − 1 ) , z → ∞ . The corresponding limiting measure is known as the deformed semicircle (or Wigner) law. N is absolutely continuous and has continuous density ρ . The same limit is for Wigner matrices (macroscopic universality) P . 72 . 12

In particular, if M (0) = 0 (GUE, Wigner), then we have the semicircle law n by Wigner �� � 1 z 2 − 4 w 2 − z f ( z ) = , 2 w 2 N ( dλ ) = ρ ( λ ) dλ, (2 πw 2 ) − 1 / 2 � 4 w 2 − λ 2 1 [ − 2 w, 2 w ] ( λ ) . ρ ( λ ) = The same limit is for band matrices if b n /n → 0 , n → ∞ . Khorunzhy, Molchanov, P . 92 . 13

(ii) If λ 0 belongs to the interior (bulk) of the support of N and ∈ λ ( n ) E n ( s ) = P { [ λ 0 , λ 0 + s/ρ ( λ 0 )] / , l = 1 , ..., n } l is the gap probability. Then we have for the deformed GUE the Gaudin law for E ( s ) = lim n →∞ E n ( s ) = det(1 − S ( s )) , where � s sin π ( x − y ) ( S ( s ) f )( x ) = f ( y ) dy. π ( x − y ) 0 14

In particular, we have for the limiting probability density p ( s ) = E ′′ ( s ) of spacing between adjacent eigenvalues: p ( s ) = π 36 s 2 (1 + o (1)) , s → 0 , i.e., the eigenvalue (level) repulsion . ( Gaudin 61, Brezin-Hikami 96, Johansson 01, 09, T. Shcherbina 09 ). (iii). Eigenvectors of the GUE are uniformly (Haar) distributed over U ( n ) , an analog of pure absolutely continuous spectrum (complete delocalization). 15

2.2.2 Deformed Wishart and Sample Covariance Matrices The Stieltjes transform solves � � � τσ ( dτ ) f ( z ) = f (0) z − a 2 c , 1 + a 2 τf ( z ) where c = lim n →∞ m/n . Marchenko, P . 67 In particular, for M (0) = 0 , T m = Id n N MP ( dλ ) = (1 − c ) + δ ( λ ) dλ + ρ MP ( λ ) dλ, ρ MP ( λ ) = (2 πa 2 λ ) − 1 / 2 � ( a + − λ )( λ − a − ) 1 [ a + ,a − ] and a ± = a 2 (1 ± √ c ) 2 , x + = max { x, 0 } . 16

2.2.3 Law of Addition The Stieltjes transform of the limiting NCM solves the system, determined by the Stieltjes transforms f A and f B of limiting NCM of { A n } and { B n } : f ( z ) = f A ( h B ( z )) f ( z ) = f B ( h A ( z )) f − 1 ( z ) = z − h A ( z ) − h B ( z ) , and uniquely soluble in the Nevanlinna class for f and h A , h B analytic for non-real z and satisfying h A,B ( z ) = z + O (1) , z → ∞ . P ., Vasilchuk, 00, 07 17

3 ”Corresponding” Random Operators. 3.1 Description Define symmetric random operators: (i). H R G in l 2 ( Z d ) , d ≥ 1 by matrix { H R G ( x, y ) } x,y ∈ Z d as H R G ( x, y ) = h ( x − y )+ R − d/ 2 x, y ∈ Z d , ϕ (( x − y ) /R G ) W ( x, y ) , G where h : Z d → C , � h ( − x ) = h ( x ) , | h ( x ) | < ∞ , x ∈ Z d 18

R G > 0 , ϕ : R d → R is piece-wise continuous, � R d ϕ 2 ( t ) dt = 1 , max t ∈ R | ϕ ( t ) | ≤ ϕ 0 < ∞ , ϕ ( t ) = 0 , | t | > 1 , and x, y ∈ Z d , W ( x, y ) = W ( y, x ) , are independent (modulo the above symmetry condition) complex Gaussian random variables: E { W ( x, y ) } = E { W ( x, y ) 2 } = 0 , E {| W ( x, y ) | 2 } = 1 , In the case d = 1 the random part of H R G is an infinite matrix having nonzero entries only inside the band of width (2 R G + 1) around the principal diagonal, i.e., an analog of band matrix. 19

(ii) H d = { H d ( x, y ) } x,y ∈ Z d in l 2 ( Z d ) by H d ( x, y ) = h d ( x − y ) + (2 d ) − 1 / 2 W 1 ( x, y ) , d � � h d ( x ) = d − 1 / 2 h 1 ( x j ) δ ( x k ) , h (0) = 0 , x = ( x 1 , . . . , x d ) , j =1 k � = i δ is the Kronecker symbol, h 1 : Z → C is as in H R G for d = 1 (e.g. the discrete Laplacian) and   W ( x, y ) , | x − y | = 1 , W 1 ( x, y ) =  0 , | x − y | � = 1 , and W ( x, y ) are as in H R G . 20