Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random matrices depends only on the spectra of the components Øyvind Ryan October 2010 Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Main question Given A , B two n × n independent square Hermitian (or symmetric) random matrices 1. What can we say about the eigenvalue distribution of A , once we know those of A + B and B ? 2. What can we say about the eigenvalue distribution of A , once we know those of AB and B ? Such questions can also be asked starting with any functional of A and B . When we can infer on the mentioned eigenvalue distributions, the corresponding operation is called deconvolution . Two main techniques used in the literature: ◮ The Stieltjes transform method, ◮ The method of moments. We will focus on the latter. Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Moments and mixed moments Many probability distributions are uniquely determined by their � t n d µ ( t ) (Carlemans theorem), and can thus be used to moments characterize the spectrum of a random matrix. ◮ Let tr be the normalized trace, and E [ · ] the expectation. ◮ The quantities A k = E [ tr ( A k )] are the moments (or individual moments) of A . ◮ More generally, if A i are random matrices, E [ tr ( A i 1 A i 2 · · · A i k )] is called a mixed moment in the A i , when i 1 � = i 2 , i 2 � = i 3 , . . . . ◮ More generally, we can define a mixed moment in terms of algebras: if A i are algebras, A i ∈ A k i with k i � = k i + 1 for all i . Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Freeness: a computational rule for mixed moments Definition A family of unital ∗ -subalgebras { A i } i ∈ I is called a free family if a j ∈ A i j i 1 � = i 2 , i 2 � = i 3 , · · · , i n − 1 � = i n ⇒ φ ( a 1 · · · a n ) = 0 . (1) φ ( a 1 ) = φ ( a 2 ) = · · · = φ ( a n ) = 0 ◮ Defined at the algebraic level. Can be thought of as “spectral separation”. ◮ A concrete rule for computing mixed moments in terms of individual moments ( E [ tr ( · )] replaced with general φ ). ◮ Defining σ as the partition where k ∼ l if and only if i k = i l , the same formula for the mixed moment applies for any a 1 · · · a n giving rise to σ . Is in this way a particularly nice type of spectral separation. Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Instead of free algebras, assume that we have subalgebras A i of random matrices, where any random matrix from one algebra is independent from those in the other algebras. ◮ For which collection of algebras do mixed moments E [ tr ( A i 1 A i 2 · · · A i k )] , (2) depend only on individual moments? In other words: when do we have spectral separation? ◮ The question is often more easily answered in the large n -limit: n →∞ E [ tr ( A ( n ) i 1 A ( n ) i 2 · · · A ( n ) lim i k )] , where we now assume that we have ensembles of random matrices, their dimensions growing so that lim N →∞ N L = c . ◮ In the large n -limit, the problem is coupled with finding what modes of convergence apply. Almost sure convergence? ◮ When is the computational rule for computing (2) the same for any choice of matrices from the algebras, as for freeeness? If positive answers: good starting point for deconvolution. Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Gaussian matrices ◮ If the A i are Gaussian matrices, there exist results in the finite regime [1], on computational rules for mixed moments of Gaussian matrices and matrices independent from them. ◮ combinations of Gaussian matrices converge almost surely. ◮ Asymptotically free, so same convenient computational rule in the limit as for freeness. ◮ No need to expect that the same computational rule applies in the finite regime! Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Vandermonde matrices An N × L Vandermonde matrix with entries on the unit circle [2] is on the form 1 · · · 1 e − j ω 1 e − j ω L · · · 1 V = √ (3) . . ... . . N . . e − j ( N − 1 ) ω 1 e − j ( N − 1 ) ω L · · · ω 1 ,..., ω L , also called phases, are assumed i.i.d., taking values in [ 0 , 2 π ) . N and L go to infinity at the same rate, c = lim N →∞ L N (the aspect ratio). Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Algebraic result for Vandermonde matrices [3] Theorem Let { V i } i ∈ I , { V j } j ∈ J be independent Vandermonde matrices, with arbitrary phase distributions { ω i } i ∈ I and { ω j } j ∈ J , respectively, with continuous density. ◮ Let A I be the algebra generated by { ( V i 1 ) H V i 2 } i 1 , i 2 ∈ I . ◮ Let A J be the algebra generated by { ( V j 1 ) H V j 2 } j 1 , j 2 ∈ J . We have that any mixed moment N →∞ E [ tr ( a i 1 a j 1 a i 2 a j 2 · · · a i n a j n )] with a i k ∈ A I , a j k ∈ A J , lim (4) depends only on individual moments of the form N →∞ E [ tr ( a )] with a ∈ A I , lim N →∞ E [ tr ( a )] with a ∈ A J . lim (5) Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Sketch of proof We need to compute � � �� V H k 1 V k 2 · · · V H N →∞ E lim k 2 n − 1 V k 2 n . tr ◮ Define σ ∈ P ( 2 n ) defined by r ∼ σ s if and only if ω k r = ω k s , ◮ let σ j be the block of σ where ω k i = ω j for i ∈ σ j . ◮ For π ∈ P ( n ) , define ρ ( π ) ∈ P ( 2 n ) as the partition in P ( 2 n ) generated by the relations: � ⌊ k / 2 ⌋ + 1 ∼ π ⌊ l / 2 ⌋ + 1 and k ∼ ρ ( π ) l if k ∼ σ 1 l where σ 1 defined by r ∼ σ 1 s if and only if V k r = V k s . ◮ B ( n ) ⊂ P ( n ) be defined as in [3], ◮ write ρ ( π ) ∨ [ 0 , 1 ] n = { ρ 1 , ..., ρ r ( π ) } , with each ρ i ≥ [ 0 , 1 ] � ρ i � / 2 ( r ( π ) the number of blocks). Can be written so in a unique way. Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random By carefully collecting terms we obtain in the limit π ∈B ( n ) K ρ, u ( 2 π ) | ρ |− 1 � r ( π ) � � j p ω j ( x ) | ρ i ∩ σ j | dx , � (6) i = 1 ◮ Here p ω is the density of the phase distribution ω . ◮ The K ρ, u are called Vandermonde mixed moment expansion coefficients ◮ When each V H k 2 j − 1 V k 2 j is in either A I or A J , in each integral � � j p ω j ( x ) | ρ i ∩ σ j | dx , all ω j are either contained in { ω i } i ∈ I , or in { ω j } j ∈ J , ◮ Each such integral can be written in terms of moments from either A I or A J , showing that we have spectral separation. Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Due to (6), the moments of Vandermonde matrices are in the large n -limit essentially determined from � 2 π I k ,ω = ( 2 π ) k − 1 p ω ( x ) k dx . (7) 0 ◮ Reduces the dimensionality of the problem. ◮ In the finite regime, the moments are probably not uniquely determined from such simple quantities. Øyvind Ryan On general criteria for when the spectrum of a combination of
Workshop on Large Random Matrices, Paris, 2010 On general criteria for when the spectrum of a combination of random Vandermonde mixed moment expansion coefficients ◮ Write ρ ( π ) = { W 1 , ..., W | ρ ( π ) | } , ◮ write W j = W · j ∪ W H j , with W · j the even elements of W j (the V -terms), W H the odd elements of W j (the V H -terms). j ◮ Form the | ρ ( π ) | equations � � x ( k + 1 ) / 2 + 1 = x k / 2 + 1 (8) k ∈ W · k ∈ W H r r in n variables x 1 , . . . , x n . ◮ K ρ, u is the volume of the solution set to (8), when all x i are constrained to [ 0 , 1 ] . ◮ K ρ, u can be found with Fourier-Motzkin elmimination, and always computes to a rational number in [ 0 , 1 ] . ◮ Matrices such as Hankel and Toeplitz matrices also have asymptotic eigenvalue distributions which can be determined from such quantities. Øyvind Ryan On general criteria for when the spectrum of a combination of
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