Higher Order Freeness: A Survey. Roland Speicher Queen s University Kingston, Canada
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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 (2006), Mingo + Sniady + Speicher: II. Unitary random matrices Adv. Math. 209 (2007), Collins + Mingo + Sniady + Speicher: III. Higher order freeness and free cumulants Documenta Math. 12 (2007), 1-70 Kusalik + Mingo + Speicher: CRELLES 604 (2007), 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 N. in the limit 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 Wigner s semicircle law N =
6 Convergence of µ XN towards µ X is governed by large deviation principle: Prob(µ XN ν) e N 2 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 N 2 } tr(p(x N )) Note: log E { e N 2 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
7 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 2r 2 ( k k r tr(a 1 N 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. 10
8 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
9 Contrast this with following situation: D N = η 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 N 2 I(ν) 12
10 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
11 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
12 Consider α 2,3,1 17
13 Consider α 2,3,1 does not count! 18
14 Consider α 2,3,1 counts! 19
15 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. 20
16 Example: Gaussian random matrix A (N = 40, trials=50.000). Var(Tr(A)) = 1 Var(Tr(A 2 )) = 2 Var(Tr(A 4 )) = 36 21
17 Now consider two random matrix ensembles A N, B N Relevant quantities are all joint correlation moments lim N 2r 2 ( k r tr(p1 (A N, B N )),..., tr(p r (A N, B N )) ) N for all r N and all polynomials p 1,..., p r asymptotic joint distribution of all orders of A N, B N 22
18 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
19 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
20 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
21 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
22 Examples: k 1 (a 79 ) =? 27
23 Examples: k 1 (a 79 ) = 0 28
24 Examples: k 1 (a 79 ) = 0 k 1 (a 77 ) =?????? 29
25 Examples: k 1 (a 79 ) = 0 k 1 (a 77 ) = κ((1)) 30
26 Examples: k 1 (a 79 ) = 0 k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) =????????? 31
27 Examples: k 1 (a 79 ) = 0 k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1, 2, )) 32
28 Examples: k 1 (a 79 ) = 0 k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1, 2, 3)) 33
29 Examples: k 1 (a 79 ) = 0 k 1 (a 77 ) = κ((1)) k 3 (a 79, a 95, a 57 ) = κ((1, 2, 3)) 34
30 Examples: k 1 (a 79 ) = 0 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
31 Examples: k 1 (a 79 ) = 0 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
32 Examples: k 1 (a 79 ) = 0 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
33 Examples: k 1 (a 79 ) = 0 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
34 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
35 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)) 40
36 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
37 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 r 2+#π κ (N) (π) N 42
38 Consider α 2,1 = lim N 2 ( k 2 tr(a 2 ), tr(a) ) N 43
39 Consider α 2,1 = lim N 2 ( k 2 tr(a 2 ), tr(a) ) N = lim N 2 1 ( ) N N 2 k 2 aij a ji, a kk i,j,k 44
40 Consider α 2,1 = lim N 2 ( k 2 tr(a 2 ), tr(a) ) N = lim N 2 1 ( ) N 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
41 Consider α 2,1 = lim N 2 ( k 2 tr(a 2 ), tr(a) ) N = lim N 2 1 ( ) N 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
42 Consider α 2,1 = lim N 2 ( k 2 tr(a 2 ), tr(a) ) N = lim N 2 1 ( ) N 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
43 Consider α 2,1 = lim N 2 ( k 2 tr(a 2 ), tr(a) ) N = lim N 2 1 ( ) N 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
44 Consider α 2,1 = lim N 2 ( k 2 tr(a 2 ), tr(a) ) N = lim N 2 1 ( ) N 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
45 Consider α 2,1 = lim N 2 ( k 2 tr(a 2 ), tr(a) ) N = lim N 2 1 ( ) N 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)) 50
46 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
47 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
48 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
49 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
50 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
51 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
52 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 N 6 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 N 9 k ( (1) 5 c N 12, c(3) 23, c(4) 31, c(6) 44, ) c(7) 55 57
53 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 N 6 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 N 9 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
54 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 N 6 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 N 9 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
55 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) 0, otherwise 60
56 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) (0,γπ 1 )=(U,γ) κ(v, π)[c 1,..., C n ] 61
57 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 ] = 0, 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
58 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
59 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
60 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
61 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
62 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
63 Second order R-transform formula and G(x, y) := R(x, y) = are related by the equation m,n 1 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
64 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 (2004) for the fluctuations of general Wishart matrices. 69
65 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 70
66 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
67 Then, in first order, A N = X N T N X N converges to A = CT C where C is circular T has the limit distribution of the T N C and T are -free 72
68 And A = CT C is a free compound Poisson element, determined by the fact that κ A n = α T 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
69 In second order, the situation is exactly the same: The limit A = CT C 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 = α T m,n for all n for all m, n 74
70 κ A n = α T n, κ A m,n = α T 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 = 0)), then this reduces to the formula of Bai-Silverstein 75
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