Properties of Free Multiplicative Convolution Hong Chang Ji Korea - PowerPoint PPT Presentation

Properties of Free Multiplicative Convolution Hong Chang Ji Korea Advanced Institute of Science and Technology (KAIST) Random Matrices and Related Topics May 6 - 10, 2019 Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 1 / 34

Introduction Wishart ensemble Suppose that we have sample N independent random vectors { x 1 , · · · , x N } from N -dimensional complex standard Gaussian distribution. Then their sample covariance matrix is defined by XX ∗ , where X = ( x ij ) 1 ≤ i , j ≤ N = ( x 1 , · · · , x N ) . X and XX ∗ are known as (complex) Ginibre and Wishart ensemble . Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 2 / 34

Introduction Marˇ cenko-Pastur law The empirical distribution of eigenvalues of N − 1 XX ∗ converges to the Marˇ cenko-Pastur distribution µ MP : 2.0 1.5 1.0 0.5 0.0 0 1 2 3 4 5 Figure: Histogram of eigenvalues of N − 1 XX ∗ with N = 5000 and density � (4 − x ) 1 of Marˇ cenko-Pastur distribution. 2 π x Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 3 / 34

Introduction Sample covariance matrix with general population In some occasions, we wish to consider the case in which the law of x is non-standard Gaussian, so that the variables are dependent. Thus we take y i := D x i where D is another ( N × N ) matrix, called population matrix . In this case, the sample covariance matrix becomes YY ∗ = ( y 1 , · · · , y N )( y 1 , · · · , y N ) T = DXX ∗ D ∗ , referred as non-white Wishart ensemble . Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 4 / 34

Introduction deformed Marˇ cenko-Pastur law If the e.s.d. of D ∗ D converges to a probability measure ν , then that of DXX ∗ D ∗ also converges(Marˇ cenko and Pastur, 1967 [8]). The limiting measure was characterized by an integral equation satisfied by its Stieltjes transform. 2.5 2.0 Figure: Eigenvalues of DXX ∗ D ∗ where e.s.d. of DD ∗ converges 1.5 to the arcsine distribution 1.0 µ AS ( d x ) := 1 √ 1 x (4 − x ) d x π 0.5 0.0 0 2 4 6 8 10 The limit is “free multiplicative convolution” of ν and µ MP . Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 5 / 34

Free convolutions Stieltjes transform and M -function Definition 1 For a probability measure µ on R + := [0 , ∞ ), we define its Stieltjes transform and M-function by, for z ∈ C \ R + , �� � − 1 � 1 x m µ ( z ) := x − z d µ ( x ) , and M µ ( z ) = 1 − x − z d µ ( x ) . Remark M µ ( z ) = 1 − ( zm µ ( z ) + 1) − 1 m µ , M µ : C \ R + → C \ R + are analytic. M µ : ( −∞ , 0) ∼ − → ( −∞ , − µ (0) / (1 − µ (0))) is increasing. M − 1 is analytic in a neighborhood of ( −∞ , − µ (0) / (1 − µ (0))). µ Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 6 / 34

Free convolutions Free multiplicative convolution Definition 2 For two probability measures µ and ν on [0 , ∞ ), both not δ 0 , µ ⊠ ν is the unique probability measure satisfying µ ⊠ ν ( z ) = 1 M − 1 z M − 1 µ ( z ) M − 1 ν ( z ) in a neighborhood of ( −∞ , − C ). Remark If X and Y are free random variables with distributions µ and ν , then √ √ √ √ µ ⊠ ν is the distribution of XY X (or Y X Y ). Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 7 / 34

Free convolutions Free additive convolution Definition 3 For two probability measures µ and ν on R , µ ⊞ ν is the unique probability measure satisfying F − 1 µ ⊞ ν ( z ) = F − 1 µ ( z ) + F − 1 ν ( z ) − z in a neighborhood of ( i M , i ∞ ), where F µ ( z ) := − 1 / m µ ( z ). Remark If X and Y are free random variables with distributions µ and ν , then µ ⊞ ν is the distribution of X + Y . As X and Y are noncommutative , log( XY ) = log X + log Y and e X + Y = e X e Y are no longer true. Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 8 / 34

Free convolutions Connection to random matrices U N : ( N × N )- Haar distributed random unitary matrix. C N = diag ( c ( N ) , · · · , c ( N ) N ) , D N = diag ( d ( N ) , · · · , d ( N ) ) such that 1 1 N � N � N 1 1 δ c ( N ) → µ and δ d ( N ) → ν, as N → ∞ . N N i i i =1 i =1 ( λ ( N ) , · · · , λ ( N ) N ): eigenvalues of C N + U N D N U ∗ N , 1 N ): those of √ C N U N D N U ∗ √ C N (for C N , D N ≥ 0). ( γ ( N ) , · · · , γ ( N ) 1 N Theorem (Voiculescu, 1998 [9]) N N � � 1 1 δ λ ( N ) → µ ⊞ ν and δ γ ( N ) → µ ⊠ ν. N N i i i =1 i =1 We may replace U N D N U ∗ N with Wishart ensemble and ν with µ MP . Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 9 / 34

Free convolutions Marˇ cenko-Pastur distribution revisited n →∞ (( n − 1) δ 0 / n + δ 1 / n ) ⊞ n is also known as free Poisson law . µ MP = lim In general for any a ≥ 1, (( n − a ) δ 0 / n + a δ 1 / n ) ⊞ n converges to � 1 µ ( a ) 4 a − ( x − ( a + 1)) 2 d x . MP ( d x ) := 2 π x The measure µ ( a ) MP ’s are also the limiting e.s.d. of the general sample covariance N − 1 XX ∗ , where X is ( N × M ) random matrix whose entries are i.i.d. and M / N → a . In fact, µ ( a ) MP are also ⊠ -infinitely divisible, so that the common properties of µ ( a ) MP are “desirable” in terms of the operation ⊠ . Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 10 / 34

Free convolutions Marˇ cenko-Pastur distribution revisited Examples of “desirable” properties are.. Having density, which is analytic in the bulk of spectrum. The density being bounded by 1 / x . The density decaying as square root at the edges. 0.25 0.20 a = 1 0.15 Figure: Densities of µ ( a ) a = 2  ]= MP a = 3 0.10 0.05 1 2 3 4 5 6 7 These properties of µ ( a ) MP hold even for convolution of two measures, under proper assumptions. Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 11 / 34

Main results Regularity(Lebesgue decomposition) of free convolution Let µ = 1 2 δ 0 + 1 2 δ 2 . Then µ ⊠ µ can be explicitly calculated as 1 1 2 δ 0 + � ✶ (0 , 4) ( x ) d x . 2 π x (4 − x ) 0.7 0.6 Figure: Nonzero eigenvalues of 0.5 √ CUDU ∗ √ (5 · 10 3 ) matrix C , 0.4 where { c i , d i } are i.i.d. with law 0.3 µ and U is independent of C 0.2 and D . 0.1 0 1 2 3 4 Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 12 / 34

Main results Known results for free additive convolution: Lebesgue decomposition Theorem (Belinschi, 2008 [4]) Let µ and ν be Borel probability measures on R , both not a point mass. (i) ( µ ⊞ ν )( { a } ) > 0 if and only if there exist b , c ∈ R with a = b + c and µ ( { b } ) + ν ( { c } ) > 1 . In this case, ( µ ⊞ ν )( { a } ) = µ ( { b } ) + ν ( { c } ) − 1 . (ii) ( µ ⊞ ν ) sc ≡ 0 . d x ( µ ⊞ ν ) ac ( x ) is analytic whenever positive and finite. d (iii) Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 13 / 34

Main results Lebesgue decomposition Theorem 1 (J., 2019 [7]) Let µ and ν be Borel probability measures on R + , both not a point mass. (i*) For c > 0 , ( µ ⊠ ν )( { c } ) > 0 if and only if there exist u , v ∈ (0 , ∞ ) with uv = c and µ ( { u } ) + ν ( { v } ) > 1 . In this case, ( µ ⊠ ν )( { c } ) = µ ( { u } ) + ν ( { v } ) − 1 . (ii*) ( µ ⊠ ν )( { 0 } ) = max( µ ( { 0 } ) , ν ( { 0 } )) . (iii) ( µ ⊠ ν ) sc ≡ 0 . d ( µ ⊠ ν ) ac ( x ) (iv) is analytic whenever positive and finite. d x *First two statements were proved in Belinschi, 2003 [3]. Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 14 / 34

Main results Boundedness of the density Letting µ = (1 − p ) δ 0 + p δ 1 / p , we find that µ ⊠ µ almost have an atom at p − 2 if p = 1 / 2. Figures below show what happens if p < 1 / 2. 1.0 0.7 0.6 0.8 0.5 0.6 0.4 0.3 0.4 0.2 0.2 0.1 0.0 0.0 0 1 2 3 4 0 1 2 3 4 5 Figure: Density of ( µ ⊠ µ ) ac Figure: Density of ( µ ⊠ µ ) ac where µ = 0 . 51 δ 0 + 0 . 49 δ 1 / 0 . 49 . where µ = 0 . 55 δ 0 + 0 . 45 δ 1 / 0 . 45 . Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 15 / 34

Main results Known results for free additive convolution: Boundedness of the density Theorem (Belinschi, 2013 [5]) Let µ and ν be Borel probability measures on R , both not a point mass. If F µ and F ν are continuous at infinity and µ ( { b } ) + ν ( { c } ) < 1 for all b , c ∈ R , then µ ⊞ ν = ( µ ⊞ ν ) ac and the density is bounded and continuous. Remark The density of µ MP diverges as x − 1 / 2 around x , thus we need different statement to cover µ MP . Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 16 / 34

Main results Boundedness of the density Theorem 2 (J., 2019 [7]) Let µ and ν be probability measures on R + such that M µ and M ν are continuous at 0 and ∞ . Further assume that µ ( { a } ) + ν ( { b } ) < 1 for all a , b ∈ (0 , ∞ ) . Then the density of ( µ ⊠ ν ) ac is continuous and uniformly O ( x − 1 ) on (0 , ∞ ) . Remark By Theorem 1 (i), µ ⊠ ν can have point mass at 0 under the assumptions of Theorem 2. Hong Chang Ji (KAIST) Free multiplicative convolution May 10, 2019 17 / 34