Higher Order Freeness: A Survey. Roland Speicher Queen s University Kingston, Canada

Transcription

1 Higher Order Freeness: A Survey Roland Speicher Queen s University Kingston, Canada

2 Second order freeness and fluctuations of random matrices: Mingo + Speicher: I. Gaussian and Wishart matrices and cyclic Fock spaces JFA 235 (26), Mingo + Sniady + Speicher: II. Unitary random matrices Adv. Math. 29 (27), Collins + Mingo + Sniady + Speicher: III. Higher order freeness and free cumulants Documenta Math. 12 (27), 1-7 Kusalik + Mingo + Speicher: CRELLES 64 (27), 1-46 Mingo + Speicher + Tan: arxiv: (to appear in TAMS) 1

3 Warning We deal only with complex random matrices. Higher order freeness for the real case still has to be worked out! 2

4 We want to consider N N random matrices A N in the limit N. Which kind of information about the random matrices do we want to keep in the limit N =? Consider selfadjoint Gaussian N N random matrices X N. One knows: empirical eigenvalue distribution of X N converges almost surely to deterministic limit distribution µ X one has a large deviation principle for convergence towards µ X 3

5 eine Realisierung zweite Realisierung dritte Realisierung 4

6 eine Realisierung N= zweite Realisierung N= dritte Realisierung N=1 5

7 eine Realisierung N= N= zweite Realisierung N= N= dritte Realisierung N= N=1 6

8 eine Realisierung N= N= N= zweite Realisierung N= N= N= dritte Realisierung N= N= N=1 7

9 Wigner s semicircle law N = one realization another realization yet another one... 8

10 Convergence of µ XN towards µ X is governed by large deviation principle: Prob(µ XN ν) e N2 I(ν), where rate function ν I(ν) is given as Legendre transform of C X p lim N 1 N 2 log } E{ e N2 tr(p(x N )) Note: log E { e N2 tr(p(x N )) } = r ( 1) r N 2r k ( r tr(p(xn )),...,tr(p(x N )) ) r! where k r are classical cumulants 9

11 This motivates our general assumption on the considered random matrices A N : For all r N and all k 1,..., k r N the following limits exists ( lim N N2r 2 k k r tr(a 1 N ),...,tr(ak r N }{{ )) } classical cumulants of traces of powers =: α A k 1,...,k r The α s are the asymptotic correlation moments of our random matrix ensemble A N and constitute its limiting distribution of all orders. 1

12 Typical examples for random matrices where limiting distribution of all orders exists: Gaussian random matrices, Wishart random matrices, Haar unitary random matrices, and combinations of independent copies of such ensembles Note: We are looking on random matrix ensembles whose eigenvalues have a correlation as for Gaussian random matrices: tr(a k N ) = λk λk N N Eigenvalues λ 1,..., λ N of A N are not independent, but feel some interaction 11

13 Contrast this with following situation: D N = η 1 η η N, where η 1, η 2,... are independent and identically distributed according to η. Then tr(d k N ) = ηk ηk N N E[η k ] with large deviation principle e NH(ν) ; not e N2 I(ν) 12

14 In this case: k r ( tr(d k 1 N ),...,tr(dk r N )) = N 1 r k r (η k 1,..., η kr ), and thus: D N has no limiting distribution of all orders in our sense. The Gaussian random matrices A N and the above ensemble with a semicircle distribution for η have the same asymptotic eigenvalue distribution, but a quite different type of convergence towards the semicircle 13

15 1 5 independent eigenvalues with semicircular distribution eigenvalues of a 5 x 5 Gaussian random matrix

16 2 5 independent eigenvalues with semicircular distribution eigenvalues of a 5 x 5 Gaussian random matrix

17 Remarks: 1) For Gaussian (and also for Wishart) random matrices there are nice combinatorial descriptions of the higher order limit distributions in terms of planar pictures α Gaussian k 1,...,k r = #NC-pairings of r circles, with k 1 points on first circle, k 2 points on second circle, etc. such that all circles are connected by pairing 16

18 Consider α 2,3,1 17

19 Consider α 2,3,1 does not count! 18

20 Consider α 2,3,1 counts! 19

21 2) Specialize general theory to second order: An N N random matrix ensemble (A N ) N N has a second order limit distribution if for all m, n 1 the limits and α n := lim N E[tr(An N )] α m,n := lim N cov( Tr(A m N ),Tr(An N )) exist and if all higher classical cumulants of Tr(A m N ) go to zero. This means that the family ( Tr(A m N ) E[Tr(A m N )]) m N converges to a Gaussian family. 2

22 Example: Gaussian random matrix A (N = 4, trials=5.) Var(Tr(A)) = 1 Var(Tr(A 2 )) = 2 Var(Tr(A 4 )) = 36 Normal Probability Plot Normal Probability Plot Normal Probability Plot Probability Probability Probability cov= cov= cov= Data Data Data 21

23 Now consider two random matrix ensembles A N, B N Relevant quantities are all joint correlation moments ( lim N N2r 2 k r tr(p1 (A N, B N )),...,tr(p r (A N, B N )) ) for all r N and all polynomials p 1,..., p r asymptotic joint distribution of all orders of A N, B N 22

24 Theorem: If A N and B N are in generic position, i.e., A N and B N are independent at least one of them is unitarily invariant and if A N as well as B N have asymptotic distributions of all orders then also the asymptotic joint distribution of all orders of A N, B N exists and it is, furthermore, determined uniquely and in a universal way by the joint distribution of A and the joint distribution of B. This universal calculation rule is the essence of freeness (of all orders) 23

25 lim N cov( Tr(A N B N ),Tr(A N B N ) ) = lim {E [ tr(a N A N ) ] E [ tr(b N B N ) ] N E [ tr(a N A N ) ] E [ tr(b N ) ] E [ tr(b N ) ] E [ tr(a N ) ] E [ tr(a N ) ] E [ tr(b N B N ) ] + E [ tr(a N ) ] E [ tr(a N ) ] E [ tr(b N ) ] E [ tr(b N ) ] + cov ( tr(a N ),tr(a N ) ) E [ tr(b N ) ] E [ tr(b N ) ] + E [ tr(a N ) ] E [ tr(a N ) ] cov ( tr(b N ),tr(b N ) )} 24

26 In order to understand this universal calculation rule use the idea of cumulants! Write our correlation moments k r ( tr(a k 1 ),...,tr(a kr ) ) in terms of cumulants of entries of our matrix, k r (a i(1)j(1),..., a i(r)j(r) ). Asymptotically, the later will give the cumulants in our theory. 25

27 To make this connection explicit, consider unitarily invariant A N = (a ij ), i.e., the joint distribution of the entries of A N is the same as the joint distribution of UA N U, for any unitary N N matrix U. Then, the only contributing cumulants of the entries are those with cycle structure in their indices! We have a Wick type formula: k r (a i(1)j(1),..., a i(r)j(r) ) = π S n δ i,j π κ(π) 26

28 Examples: k 1 (a 79 ) =? 27

29 Examples: k 1 (a 79 ) = 28

30 Examples: k 1 (a 79 ) = k 1 (a 77 ) =?????? 29

31 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) 3

32 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) =????????? 31

33 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1,2, )) 32

34 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1,2,3)) 33

35 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1,2,3)) 34

36 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1,2,3)) k 3 (a 79, a 97, a 77 ) =??????? 35

37 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1,2,3)) k 3 (a 79, a 97, a 77 ) = κ((1,2,3))+?????? 36

38 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1,2,3)) k 3 (a 79, a 97, a 77 ) = κ((1,2,3)) + κ((1,2) ) 37

39 Examples: k 1 (a 79 ) = k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1,2,3)) k 3 (a 79, a 97, a 77 ) = κ((1,2,3)) + κ((1,2)(3)) 38

40 Note: k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1,2,3)) k 3 (a 79, a 97, a 77 ) = κ((1,2,3)) + κ((1,2)(3)) 39

41 Note: κ depends actually on N k 1 (a 77 ) = κ (N) ((1)) k 3 (a 79, a 95, a 57 ) = κ (N) ((1,2,3)) k 3 (a 79, a 97, a 77 ) = κ (N) ((1,2,3)) + κ (N) ((1,2)(3)) 4

42 Note: κ depends actually on N k 1 (a 77 ) = κ (N) ((1)) k 3 (a 79, a 95, a 57 ) = κ (N) ((1,2,3)) k 3 (a 79, a 97, a 77 ) = κ (N) ((1,2,3)) + κ (N) ((1,2)(3)) π S r : κ (N) (π) N r+2 #π 41

43 Note: κ depends actually on N k 1 (a 77 ) = κ (N) ((1)) k 3 (a 79, a 95, a 57 ) = κ (N) ((1,2,3)) k 3 (a 79, a 97, a 77 ) = κ (N) ((1,2,3)) + κ (N) ((1,2)(3)) π S r : κ (N) (π) N r+2 #π κ(π) := lim N Nr 2+#π κ (N) (π) 42

44 Consider α 2,1 = lim N N2 k 2 ( tr(a 2 ),tr(a) ) 43

45 Consider α 2,1 = lim N N2 k 2 ( tr(a 2 ),tr(a) ) = lim 1 ( ) N N2 N 2 k 2 aij a ji, a kk i,j,k 44

46 Consider α 2,1 = lim N N2 k 2 ( tr(a 2 ),tr(a) ) = lim 1 ( ) N N2 N 2 k 2 aij a ji, a kk i,j,k }{{} k 3 (a ij, a ji, a kk ) + k 2 (a ij, a kk )k 1 (a ji ) + k 2 (a ji, a kk )k 1 (a ij ) 45

47 Consider α 2,1 = lim N N2 k 2 ( tr(a 2 ),tr(a) ) = lim 1 ( ) N N2 N 2 k 2 aij a ji, a kk i,j,k }{{} k 3 (a ij, a ji, a kk ) + k 2 (a ij, a kk )k 1 (a ji ) + k 2 (a ji, a kk )k 1 (a ij ) = κ((1,2)(3))+ 46

48 Consider α 2,1 = lim N N2 k 2 ( tr(a 2 ),tr(a) ) = lim 1 ( ) N N2 N 2 k 2 aij a ji, a kk i,j,k }{{} k 3 (a ij, a ji, a kk ) + k 2 (a ij, a kk )k 1 (a ji ) + k 2 (a ji, a kk )k 1 (a ij ) = κ((1,2)(3)) + κ((1,2,3))+ 47

49 Consider α 2,1 = lim N N2 k 2 ( tr(a 2 ),tr(a) ) = lim 1 ( ) N N2 N 2 k 2 aij a ji, a kk i,j,k }{{} k 3 (a ij, a ji, a kk ) + k 2 (a ij, a kk )k 1 (a ji ) + k 2 (a ji, a kk )k 1 (a ij ) = κ((1,2)(3)) + κ((1,2,3)) + κ((1,3,2))+ 48

50 Consider α 2,1 = lim N N2 k 2 ( tr(a 2 ),tr(a) ) = lim 1 ( ) N N2 N 2 k 2 aij a ji, a kk i,j,k }{{} k 3 (a ij, a ji, a kk ) + k 2 (a ij, a kk )k 1 (a ji ) + k 2 (a ji, a kk )k 1 (a ij ) = κ((1,2)(3)) + κ((1,2,3)) + κ((1,3,2)) + κ((1)(3))κ((2))+ 49

51 Consider α 2,1 = lim N N2 k 2 ( tr(a 2 ),tr(a) ) = lim 1 ( ) N N2 N 2 k 2 aij a ji, a kk i,j,k }{{} k 3 (a ij, a ji, a kk ) + k 2 (a ij, a kk )k 1 (a ji ) + k 2 (a ji, a kk )k 1 (a ij ) = κ((1,2)(3)) + κ((1,2,3)) + κ((1,3,2)) + κ((1)(3))κ((2)) + κ((1,3))κ((2)) + κ((2)(3))κ((1)) + κ((2,3))κ((1)) 5

52 Thus α 2,1 = κ((1,2)(3)) + κ((1,2,3)) + κ((1,3,2)) + κ((1)(3))κ((2)) + κ((1,3))κ((2)) + κ((2)(3))κ((1)) + κ((2,3))κ((1)) 51

53 Thus α 2,1 = κ((1,2)(3)) κ 2,1 + κ((1,2,3)) κ 3 + κ((1,3,2)) κ 3 + κ((1)(3))κ((2)) κ 1,2 κ 1 + κ((1,3))κ((2)) κ 2 κ 1 + κ((2)(3))κ((1)) κ 1,1 κ 1 + κ((2,3))κ((1)) κ 2 κ 1 52

54 Thus α 2,1 = κ((1,2)(3)) κ 2,1 κ ( {1,2,3},(1,2)(3) ) + κ((1,2,3)) κ 3 κ ( {1,2,3},(1,2,3) ) + κ((1,3,2)) κ 3 κ ( {1,2,3},(1,3,2) ) + κ((1)(3))κ((2)) κ 1,2 κ 1 + κ((1,3))κ((2)) κ 2 κ 1 + κ((2)(3))κ((1)) κ 1,1 κ 1 + κ((2,3))κ((1)) κ 2 κ 1 53

55 Thus α 2,1 = κ((1,2)(3)) κ 2,1 κ ( {1,2,3},(1,2)(3) ) + κ((1,2,3)) κ 3 κ ( {1,2,3},(1,2,3) ) + κ((1,3,2)) κ 3 κ ( {1,2,3},(1,3,2) ) + κ((1)(3))κ((2)) κ 1,2 κ 1 κ( ( {1,3}{2},(1)(3)(2) ) + κ((1,3))κ((2)) κ 2 κ 1 κ( ( {1,3}{2},(1,3)(2) ) + κ((2)(3))κ((1)) κ 1,1 κ 1 κ( ( {1}{2,3},(1)(2)(3) ) + κ((2,3))κ((1)) κ 2 κ 1 κ( ( {1}{2,3},(1)(2,3) ) 54

56 general combinatorial object partitioned permutation (V, π) PS n π S n, V P n, with V π Index both correlation moments ϕ(v, π) and cumulants κ(v, π) with (V, π): product of moments/cumulants according to blocks of V, distribution into slots for arguments according to cycles of π: 55

57 Let C 1,..., C 9 {A, B}, with C k = (c (k) ij )N i,j=1. ϕ ( {1,3,4,6,7}{2,5,8}{9},(1,3,4)(2,8)(5)(6)(7)(9) ) [C 1,..., C 9 ] κ ( {1,3,4,6,7}{2,5,8}{9},(1,3,4)(2,8)(5)(6)(7)(9) ) [C 1,..., C 9 ] 56

58 Let C 1,..., C 9 {A, B}, with C k = (c (k) ij )N i,j=1. ϕ ( {1,3,4,6,7}{2,5,8}{9},(1,3,4)(2,8)(5)(6)(7)(9) ) [C 1,..., C 9 ] = lim N N6 k 3 ( tr(c1 C 3 C 4 ),tr(c 6 ),tr(c 7 ) ) κ ( {1,3,4,6,7}{2,5,8}{9},(1,3,4)(2,8)(5)(6)(7)(9) ) [C 1,..., C 9 ] = lim N9 k ( (1) 5 c N 12, c(3) 23, c(4) 31, c(6) 44, ) c(7) 55 57

59 Let C 1,..., C 9 {A, B}, with C k = (c (k) ij )N i,j=1. ϕ ( {1,3,4,6,7}{2,5,8}{9},(1,3,4)(2,8)(5)(6)(7)(9) ) [C 1,..., C 9 ] = lim N N6 k 3 ( tr(c1 C 3 C 4 ),tr(c 6 ),tr(c 7 ) ) k 2 ( tr(c2 C 8 ),tr(c 5 ) ) κ ( {1,3,4,6,7}{2,5,8}{9},(1,3,4)(2,8)(5)(6)(7)(9) ) [C 1,..., C 9 ] = lim N9 k ( (1) 5 c N 12, c(3) 23, c(4) 31, c(6) 44, ( c(7) (2) 55 ) k3 c 12, c(8) 21, ) c(5) 33 58

60 Let C 1,..., C 9 {A, B}, with C k = (c (k) ij )N i,j=1. ϕ ( {1,3,4,6,7}{2,5,8}{9},(1,3,4)(2,8)(5)(6)(7)(9) ) [C 1,..., C 9 ] = lim N N6 k 3 ( tr(c1 C 3 C 4 ),tr(c 6 ),tr(c 7 ) ) k 2 ( tr(c2 C 8 ),tr(c 5 ) ) k 1 ( tr(c9 ) ) κ ( {1,3,4,6,7}{2,5,8}{9},(1,3,4)(2,8)(5)(6)(7)(9) ) [C 1,..., C 9 ] = lim N9 k ( (1) 5 c N 12, c(3) 23, c(4) 31, c(6) 44, ( c(7) (2) 55 ) k3 c 12, c(8) 21, ( c(5) (9) 33 ) k1 c 33 ) 59

61 Define length function (V, π) := n (2#V #π) We have triangle inequality ((V, π),(w, σ) PS n ) (V W, πσ) (V, π) + (W, σ) Define product (V, π) (W, σ) = (V W, πσ), (V W, πσ) = (V, π) + (W, σ), otherwise 6

62 Asymptotically, for N, only the geodesic terms corresponding to equality in the triangle inequality contribute. In particular, the relation between correlation moments and cumulants is given by the moment-cumulant formula for all orders ϕ(u, γ)[c 1,..., C n ] = (V,π) PSn (V,π) (,γπ 1 )=(U,γ) κ(v, π)[c 1,..., C n ] 61

63 If A N and B N are in generic position (i.e., asymptotically free of all orders), then we have for their asymptotic distribution the vanishing of mixed cumulants κ(1 n, π)[c 1,..., C n ] =, whenever C 1,..., C n contain A as well as B convolution formula for cumulants of products κ(u, γ)[ab, AB,..., AB] = (V,π) (W,σ)=(U,γ) κ(v, π)[a, A,..., A] κ(w, σ)[b, B,..., B] 62

64 Restrict now to special situation Consider only first and second order, and restrict to problem of the sum of A and B If A and B are free, then the second order distribution (covariances) of A+B depends only on the expectations and covariances of A and of B. 63

65 Example: We have α A+B 1,2 = α A 1,2 + αb 1,2 + 2αA 1 αb 1,1 + 2αB 1 αa 1,1, i.e., cov (Tr(A + B),Tr ( (A + B) 2)) = cov ( Tr(A),Tr(A 2 ) ) + cov ( Tr(B),Tr(B 2 ) ) + 2E[tr(A)] cov ( Tr(B),Tr(B) ) + 2E[tr(B)] cov ( Tr(A),Tr(A) ) 64

66 Moment-cumulant formulas for first and second order say α 1 = κ 1 α 2 = κ 2 + κ 1 κ 1 α 3 = κ 3 + κ 1 κ 2 + κ 2 κ 1 + κ 2 κ 1 + κ 1 κ 1 κ 1 α 4 = κ 4 + 4κ 1 κ 3 + 2κ κ2 1 κ 2 + κ 4 1. α 1,1 = κ 1,1 + κ 2 α 1,2 = κ 1,2 + 2κ 1 κ 1 + 2κ 3 + 2κ 1 κ 2 α 2,2 = κ 2,2 + 4κ 1 κ 1,2 + 4κ 2 1 κ 1,1 + 4κ κ 1 κ 3 + 2κ κ2 1 κ 2 65

67 Vanishing of mixed cumulants gives additivity of free cumulants for free A, B κ A+B m = κ A m + κ B m m and κ A+B m,n = κa m,n + κb m,n m, n 66

68 Combinatorial relation between moments and cumulants can be rewritten in terms of generating power series Recall: first order case (Voiculescu) and G(x) = 1 x + R(x) = n=1 n=1 are related by the relation 1 α n x n+1 κ n x n 1 Cauchy transform R-transform G(x) + R(G(x)) = x. 67

69 and Second order R-transform formula G(x, y) := m,n 1 R(x, y) = are related by the equation m,n 1 α m,n 1 x m+1 1 y n+1 κ m,n x m 1 y n 1 G(x, y) = G (x) G (y) R ( G(x), G(y) ) + 2 x y [ log ( G(x) G(y) x y )] 68

70 If second order free cumulants are zero, then formula reduces to G(x, y) = 2 x y [ log ( G(x) G(y) x y )], i.e. the fluctuations in such a case are determined by the eigenvalue distribution. This is the formula of Bai and Silverstein (24) for the fluctuations of general Wishart matrices. 69

71 G(x, y) = 2 x y [ log ( G(x) G(y) x y )], Second order free cumulants are zero for example for Gaussian random matrices Wishart matrices independent sums of Gaussian and Wishart 7

72 Consider How do Wishart matrices fit in this theory? A N = X N T N X N where X N are N N non-selfadjoint Gaussian random matrices T N are random matrix ensemble such that second order limit distribution exists X N and T N are independent (for example, T N are deterministic) 71

73 Then, in first order, A N = X N T N X N converges to A = CTC where C is circular T has the limit distribution of the T N C and T are -free 72

74 And A = CTC is a free compound Poisson element, determined by the fact that κ A n = αt n for all n In terms of transforms this gives the fixed point equation of Marchenko-Pastur for the Cauchy transform of A in terms of the Cauchy transform of T. 73

75 In second order, the situation is exactly the same: The limit A = CTC of is a A N = X N T N X N free compound Poisson element of second order, determined by the fact that and κ A n = α T n κ A m,n = αt m,n for all n for all m, n 74

76 κ A n = αt n, κa m,n = αt m,n for all m, n In terms of transforms this gives: G A (x, y) = G (x) G (y) G(x) 2 G(y) 2 ( ) GT 1/G(x),1/G(y) + 2 x y [ log ( G(x) G(y) x y )] If T N are deterministic (i.e., κ A m,n = α T m,n = )), then this reduces to the formula of Bai-Silverstein 75