Free Meixner distributions and random matrices

Transcription

1 Free Meixner distributions and random matrices Michael Anshelevich July 13, 2006

2 Some common distributions first... 1

3 Gaussian Negative binomial Gamma Pascal chi-square geometric exponential 1 2πt e x2 /2t ( ) n=0 t+n 1 t 1 (1 µ) t µ n δ 1 n Γ(t) e x x t 1 1 [0, ) (x) x x x (, ) {0,1,2,...} [0, ) Poisson Binomial Hyperbolic tangent e t t n n=0 n! δ ( ) T T n n=0 n p n (1 p) T n δ 2n T C t,ζ e (π 2ζ)x Γ(t + ix) k k k x {0,1,2,...} { T, T + 2,..., T 2, T } (, ) 2

4 all shifted to have mean zero. Meixner family of probability distributions. What do these have in common? Look at their orthogonal polynomials. For the Gaussian distribution, Hermite polynomials are orthogonal: 1 2πt H n(x, t)h k (x, t)e x2 /2t dx = 0 if n k. H 1 = x, H 2 = x 2 t, H 3 = x 3 3xt,... 3

5 Distributions Gaussian Poisson Gamma (chi-square, exponential) Negative binomial (geometric) Binomial Hyperbolic tangent Polynomials Hermite Charlier Laguerre Meixner Krawtchouk Meixner-Pollaczek Generating functions for polynomials F(x, t, z) = n=0 1 n! P n(x, t)z n. 4

6 Polynomials Generating function Hermite exp(xz tz 2 /2) Charlier Laguerre Meixner Krawtchouk Meixner-Pollaczek (1 + z) x+t e tz (1 + z) t exp ( ) x+t 1+z (1 µ) (µ1/2 µ 1/2 ) 2 ((1 µ) + z) x (1 µ)t ((1 µ) + µz) x+1 µ µ t (1 + (1 p)z) pt+x (1 pz) (1 p)t x ( ) t+ix ( ) t ix 1 2sin zeiζ ζ 1 2sin ze iζ ζ All of the form e xu(z) tv(z). 5

7 Meixner distributions µ have orthogonal polynomials with generating functions e xu(z) tv(z). Theorem. (Meixner 1934) Up to re-scalings, translations, they are the only ones. Another way to see: recursion relations. Hermite xh n = H n+1 + nth n 1. Charlier xc n = C n+1 + nc n + ntc n 1. 6

8 Laguerre xl n = L n+1 + 2nL n + n(t + (n 1))L n 1. Meixner xm n = M n µ 1 µ nm n + n Krawtchouk ( t + µ (1 µ) 2(n 1) ) M n 1. xk n = K n+1 + (1 2p)nK n + n(t p(1 p)(n 1))K n 1. Meixner-Pollaczek xp n (x, t) = P n+1 (x, t) + 2(cos ζ)np n (x, t) + n(t + (n 1))P n 1 (x, t). Thus all of the Meixner form xm n (x, t) = M n+1 (x, t) + bnm n (x, t) + n(t + c(n 1))M n 1 (x, t). 7

9 Generating function H(x, z, t) = exp ( x dz 1 + bz + cz 2 t ) z dz 1 + bz + cz 2. More details: Wim Schoutens, STOCHASTIC PROCESSES AND ORTHOGONAL POLYNOMI- ALS, Springer P. Feinsilver, R. Schott, ALGEBRAIC STRUCTURES AND OPERATOR CALCULUS I. REPRESENTATIONS AND PROBABILITY THEORY, Kluwer

10 Other properties: Natural exponential families with quadratic variance function (Morris 83); generalized linear models. Quadratic regression property (Laha, Lukacs 60). Distributions of stochastic processes with linear conditional expectations and quadratic conditional variances (Wesołowski 93). Polynomial basis for representations of Ð2. Positive integer linearization coefficients (Kim, Zeng 2001). Combinatorics: Rota, Sheffer, Viennot, etc. 9

11 FREE ANALOGS. In e xu(z) tv(z), replace e z with 1 1 z. e z = 1 + z + 1 2! z ! z3 +..., 1 1 z = 1 + z + z2 + z Distributions with orthogonal polynomials {P n }, n=0 P n (x)z n = 1 1 xu(z) + tv(z). 10

12 Again can describe completely (M.A. 2003): up to re-scaling 1 2π t 4(t + c) (x b) 2 t 2 + tbx + cx 2 + zero, one, or two atoms. 11

13 b = c = 0 ( Gaussian ) no atoms. c = 0 ( Poisson ) at most one atom. c > 0, b 2 > 4c ( Negative binomial ) at most one atom. c > 0, b 2 = 4c ( Gamma ) no atoms. c > 0, b 2 < 4c ( Hyperbolic tangent ) no atoms. 1 c < 0 ( Binomial ) at most two atoms. 12

14 2 1 1 b

15 1 2π t 4(t + c) (x b) 2 t 2 + tbx + cx 2 + zero, one, or two atoms. APPEARANCE IN RANDOM MATRIX THEORY. 1. b = c = 0. Semicircular distribution. Limit of spectral distribution of GUE. 2. c = 0. Marchenko-Pastur distribution. Limit of Wishart matrices c < 0. Limit of Jacobi or Beta (Wachter 80; Capitaine, Casalis 2002). 4. c > 0, b 2 4c. Wireless communications (Gaussian matrices with correlated entries)? 14

16 Jacobi(n, α, β) has distribution Also, (Zn α,β ) 1 det(1 M) α det(m) β 1 0 M 1 dm J = (X + X ) 1/2 X(X + X ) 1/2 for some independent Wishart matrices. Limiting distribution 1 (x λ )(λ + x) 2π x x 2 + max(0,1 α)δ 0 + max(0,1 β)δ 1. for some λ ±. These are all free Meixner distributions. 15

17 FREE PROBABILITY. Random n n matrices A, B non-commuting objects x, y. Expectation 1 n TrE state ϕ. A, B diagonal, U random unitary matrix with uniform (Haar) distribution. lim n 1 n TrE[ A 2 (UBU 1 )A(UBU 1 ) 3 A 2 ] [ = ϕ x 2 yxy 3 x 2 ], etc. Then x, y freely independent. Explicit formulas to calculate all moments. 16

18 Free central limit theorem. Let x 1, x 2,... be identically distributed, mean zero, variance one, freely independent. Then ( ) x1 + x dist x k k Semicircular. k Free Poisson limit theorem. Let { x i,k : 1 i k, k = 1,2,... } have distributions ( dist(x i,k ) = 1 1 ) δ k k δ 1 and be freely independent. Then dist(x 1,k + x 2,k x k,k ) k Marchenko-Pastur. 17

19 APPEARANCE IN FREE PROBABILITY. 1. b = c = 0. Appears in the free central limit theorem, free analog of Gaussian. 2. c = 0. Appears in the free Poisson limit theorem, free analog of the Poisson distribution c < 0. Free binomial distributions. P α = n n diagonal matrix with αn 1 s, (1 α)n 0 s on the diagonal. Asymptotic distribution of P α is Bernoulli (1 α)δ 0 + αδ 1. U 1, U 2,..., U T = n n independent random unitary matrices with uniform (Haar) distribution. 18

20 lim n dist( U 1 P α U1 + U 2P α U U TP α UT ) = FreeBinomial(α, T) (up to a shift). In fact, can do this for any real T 1: for β = 1 T, dist ( P β UP α U P β ) shift of FreeBinomial(α, T) d = lim Jacobi True for finite matrices (Collins 2005). 4. c > 0, b 2 4c. S-transform is a rational function. 19

21 OTHER PROPERTIES. Polynomial recursion. almost constant coefficients xp 0 = P 1, xp 1 = P 2 + bp 1 + P 0, xp n = P n+1 + bp n + (t + c)p n 1. Stieltjes transform. Modified R-transform G µ (z) = G ( 1 + R(z) z dµ(x) z x. ) = z. 20

22 R(z) z 2 = 1 + br(z) z + c ( R(z) z ) 2. cf. (log F) = 1 + b(log F) + c[(log F) ] 2 Second order difference equation, with f (z) replaced by f(z) f(0) z. Rao, Edelman. 21

23 CONDITIONAL CHARACTERIZATION (Bożejko, Bryc 2006). Let, be freely independent, with conditional expectation + ] = α( + ) + α E[ 0 I and conditional variance Var[ + ] = C(I + b(( + ) + c( + ) 2 ). Then both and have free Meixner distributions. Matrix version? 22

24 GAMMA CHARACTERIZATIONS. Classical: Suppose X, Y are non-degenerate, independent non-negative random variables, and S = X + Y positive. Let Z = X/S. S and Z are independent if and only if X, Y have gamma-type distributions. (Lukacs 1955). Wishart: Suppose, are non-degenerate, independent positive random Ë matrices, and = + is strictly positive. Ë Let = 1/2 1/2. Ë and Ë are independent if and only if and have Wishart distributions. (Olkin, Rubin 1962). Symmetric cones: Casalis, Letac

25 Wishart: gamma-like but limit free Poisson. Suppose, are non-degenerate, freely independent non-commutative random variables, and Ë = + is strictly positive. Let = Ë 1/2 Ë 1/2. Ë and are freely independent if and only if and have free Poisson-type distributions. Suppose, are non-degenerate, freely independent, identically distributed, and strictly positive non-commutative random variables. Let = Ë 1 2 Ë 1. If and Ë are free, and are free gamma-type. Do not know if free gamma has this property. No matrix version (Letac) 24

26 Questions. Realize free Meixner distributions for general b, c by random matrices Realize free Meixner distributions for general b, c in free probability. Appearance in wireless communications? Meixner distributions for matrices of finite size? 25

27 Multivariate orthogonal polynomials in non-commuting variables math.co/ Trans. AMS 2008? Michael Anshelevich July 14, 2006

28 Polynomials in non-commuting variables. Ê x 1, x 2..., x n. Example. P(x 1, x 2, x 3 ) = 3x 2 1 x 2x 1 x 2 x 1 x 2 3 x 2. ORTHOGONAL POLYNOMIALS. state ϕ. Not with respect to a measure on Ê n, but a Linear, ϕ[1] = 1, and equals to zero only if P(x) = 0. ϕ [ P(x) P(x) ] 0 1

29 P(x), Q(x) orthogonal if P(x), Q(x) = ϕ [ P(x) Q(x) ] = 0. Commutative case: P(x), Q(x) = ϕ [ P(x) Q(x) ] = P(x) Q(x) dµ(x) = 0 A polynomial family {P u }, for all choices of multi-indices u. P u (x) = x u +... Example. P (1,2,1,1) (x) = x 1 x 2 x Monic, cf. polynomials in commuting variables. 2

30 Proposition. Monic polynomials are orthogonal with respect to some state if and only if they satisfy a recursion with x i P u = P (i, u) + w = u B i, w, u P w + v = u 1 C i, v, u P v (a) C i, s, u = 0 unless u = (i, s), and C i, s,(i, s) > 0, (b) Denoting s j = (s(j),..., s(k)), k B i, s, u j=1 C s(j), sj+1, s j = B i, u, s k j=1 C u(j), uj+1, u j. 3

31 In 2 variables: x 1 P 2,1,2 = P 1,2,1,2 + B w P w, w =3 x 2 P 2,1,2 = P 2,2,1,2 + w =3 B w P w + CP 1,2. 4

32 Generating functions of special form F(x,z) = P u (x)z u = H(z) 1 x V(z). Example. 1 1 z 2 1 +z2 2 ( x 1 z 1 z 2 1 +z2 2 + x 2 z 2 z 2 1 +z2 2 ) = 1 1 (x 1 z 1 + x 2 z 2 ) + (z z2 2 ). Free Meixner states = states whose orthogonal polynomials have such a generating function. 5

33 Theorem. Start with {P u } as above. Define a functional ϕ by Not always positive. ϕ[1] = 1, ϕ[p u (x)] = 0. (1) Necessarily F(x,z) = 1 1 x U(z) + R(U(z)). R(w) = R ϕ (w) = multivariate R-transform. Can be calculated using the implicit relation with the moment series R ( w 1 ( 1 + M(w) ),..., wn ( 1 + M(w) ) ) = M(w). Thus from {P u }, can calculate ϕ. 6

34 Now suppose that {P u } are orthogonal (so ϕ positive). (2) U is determined by the relation D i (R) ( U 1 (z), U 2 (z),..., U n (z) ) = z i. D i = left partial derivative D 1 (z 2 1 z 2z 1 ) = z 1 z 2 z 1, D 2 (z 2 1 z 2z 1 ) = 0. 7

35 (3) x i P (t, u) = P (i,t, u) + j B t ij P (j, u) + δ it(1 + C i,u(1) )P u, where (a) C ij > 1. (b) B t ij = Bj it. (c) For each j, t, either B t ij = 0 for all i, or C ju = C tu for all u. Equivalently, D i D j R = δ ij + t B t ij D tr + C ij D i R D j R, i.e. R-transform satisfies a second order PDE ( partial difference equation ) 8

36 In 2 variables: x 1 P 2,1,2 = P 1,2,1,2 + B 1 P 1,1,2 + B 2 P 2,1,2, x 2 P 2,1,2 = P 2,2,1,2 + B 1 P 1,1,2 + B 2 P 2,1,2 + CP 1,2. 9

37 Example. One-dimensional free Meixner states. R(z) z 2 = 1 + br(z) z + c ( R(z) z ) 2. One variable commutative states are measures. 1 4(1 + c) (x b) 2 2π 1 + bx + cx 2 + zero, one, or two atoms. Semicircular=free Gaussian Marchenko-Pastur=free Poisson Jacobi=free binomial. 10

38 Free product of n one-dimensional states = state on Ê x 1, x 2,..., x n. Example. { Two states ϕ 1 on Ê[x 1 ], ϕ 2 on Ê[x 2 ]. Corresponding polynomials P n (i) } (x i ), i = 1,2. ϕ [ 5 2P (2) 3 (x 2) + P (1) 1 (x 1)P (2) 4 (x 2) + 3P (1) 2 (x 1)P (2) 2 (x 2)P (1) ] 3 (x 1) = 5. Free products of free Meixner states are again free Meixner. 11

39 R(z) z 2 = 1 + br(z). z Semicircular, Marchenko-Pastur. Free products still satisfy a first-order equation. Rotation y = Ox, O orthogonal still OK. ϕ a tracial state if (note: not ϕ[bac]). ϕ[abc] = ϕ[cab]. 12

40 Proposition. If ϕ is a tracial free Meixner state with D i D j R(z) = δ ij + B t ij D tr(z), then ϕ is a rotation of a free product of semicircular and Marchenko-Pastur distributions. 13

41 Can construct a state ϕ so that R ϕ (z) = i z i M free product of shifted scaled semicircular (z)z i ϕ will be free Meixner, non-tracial, freely infinitely divisible. Non-freely-infinitely-divisible examples? Free multinomial? 14