Operator-Valued Free Probability Theory and Block Random Matrices. Roland Speicher Queen s University Kingston

Transcription

1 Operator-Valued Free Probability Theory and Block Random Matrices Roland Speicher Queen s University Kingston

2 I. Operator-valued semicircular elements and block random matrices II. General operator-valued free probability theory Voiculescu: Asterisque 232, 1995 Speicher: Memoir of the AMS 627 (1998) Rashidi Far, Oraby, Bryc, Speicher: IEEE Transactions on Information Theory 54,

3 I. Operator-valued semicircular elements and block random matrices 2

4 Consider Gaussian N N-random matrices A N = ( a ij ) N i,j=1 i.e., A N is N N-matrix, where a ij are random variables whose distribution is determined as follows: 3

5 Consider Gaussian N N-random matrices A N = ( a ij ) N i,j=1 i.e., A N is N N-matrix, where a ij are random variables whose distribution is determined as follows: A N is selfadjoint, i.e., a ji = ā ij 4

6 Consider Gaussian N N-random matrices A N = ( a ij ) N i,j=1 i.e., A N is N N-matrix, where a ij are random variables whose distribution is determined as follows: A N is selfadjoint, i.e., a ji = ā ij otherwise, all entries are independent and identically distributed with centered normal distribution of variance 1/N 5

7 Convergence of typical eigenvalue distribution of Gaussian N N random matrices to Wigner s semicircle N = one realization another realization yet another one... 6

8 Consider the empirical eigenvalue distribution of A N µ AN (ω) = 1 N N δ λi (ω) i=1 λ i (ω) are the N eigenvalues (counted with multiplicity) of A N (ω) 7

9 Consider the empirical eigenvalue distribution of A N µ AN (ω) = 1 N N δ λi (ω) i=1 λ i (ω) are the N eigenvalues (counted with multiplicity) of A N (ω) Then Wigner s semicircle law says that µ AN = µ W almost surely, i.e., for all continuous and bounded f lim f(t)dµ A N R N (t) = R f(t)dµ W(t) = 1 2π 2 2 f(t) 4 t 2 dt 8

10 Show lim µ A N N (f) = µ W (f) almost surely in two steps: lim N E[µ AN (f)] = µ W (f) N Var[µ AN (f)] < 9

11 Convergence of averaged eigenvalue distribution of Gaussian N N random matrices to Wigner s semicircle N= N= N=50 number of realizations =

12 For lim E[µ A N N (f)] = µ W (f) it suffices to treat convergence of all averaged moments, i.e., lim E[ N t n dµ AN (t)] = t n dµ W (t) n N 11

13 Note: E[ t n dµ AN (t)] = E[ 1 N N i=1 λ n i ] = E[tr(An N )] 12

14 Note: E[ t n dµ AN (t)] = E[ 1 N N i=1 λ n i ] = E[tr(An N )] but E[tr(A n N )] = 1 N N i 1,...,i n =1 E[a i1 i 2 a i2 i 3 a in i 1 ] 13

15 Note: E[ t n dµ AN (t)] = E[ 1 N N i=1 λ n i ] = E[tr(An N )] but E[tr(A n N )] = 1 N N i 1,...,i n =1 E[a i1 i 2 a i2 i 3 a in i 1 ] }{{} expressed in terms of pairings Wick formula 14

16 Asymptotically, for N, only non-crossing pairings survive: lim N E[tr(An N )] = #NC 2(n) 15

17 Asymptotically, for N, only non-crossing pairings survive: lim N E[tr(An N )] = #NC 2(n) Define limiting semicircle element s by ϕ(s n ) := #NC 2 (n). (s A, where A is some unital algebra, ϕ : A C) 16

18 Asymptotically, for N, only non-crossing pairings survive: lim N E[tr(An N )] = #NC 2(n) Define limiting semicircle element s by ϕ(s n ) := #NC 2 (n). (s A, where A is some unital algebra, ϕ : A C) Then we say that our Gaussian random matrices A N converge in distribution to the semicircle element s, A N distr s 17

19 What is distribution of s? Claim: ϕ(s n ) = t n dµ W (t) 18

20 What is distribution of s? Claim: ϕ(s n ) = t n dµ W (t) more concretely: #NC 2 (n) = 1 2π +2 2 tn 4 t 2 dt 19

21 What is distribution of s? n = 2: ϕ(s 2 ) = n = 4: ϕ(s 4 ) = n = 6: ϕ(s 6 ) = 20

22 What is distribution of s? n = 2: ϕ(s 2 ) = 1 n = 4: ϕ(s 4 ) = n = 6: ϕ(s 6 ) = 21

23 What is distribution of s? n = 2: ϕ(s 2 ) = 1 n = 4: ϕ(s 4 ) = 2 n = 6: ϕ(s 6 ) = 22

24 What is distribution of s? n = 2: ϕ(s 2 ) = 1 n = 4: ϕ(s 4 ) = 2 n = 6: ϕ(s 6 ) = 5 23

25 What is distribution of s? Claim: ϕ(s 2k ) = C k k-th Catalan number 24

26 What is distribution of s? Claim: ϕ(s 2k ) = C k k-th Catalan number C k = 1 k+1 ( ) 2k k 25

27 What is distribution of s? Claim: ϕ(s 2k ) = C k k-th Catalan number C k = 1 k+1 ( ) 2k k C k is determined by C 0 = C 1 = 1 and the recurrence relation C k = k l=1 C l 1 C k l. 26

28 ϕ(s 2k ) = k l=1 ϕ(s 2l 2 )ϕ(s 2k 2l ) 27

29 Put M(z) := ϕ(s 2k ) = n=0 k l=1 ϕ(s 2l 2 )ϕ(s 2k 2l ). ϕ(s n )z n = 1 + k=1 ϕ(s 2k )z 2k 28

30 Put M(z) := ϕ(s 2k ) = n=0 k l=1 ϕ(s 2l 2 )ϕ(s 2k 2l ). ϕ(s n )z n = 1 + k=1 ϕ(s 2k )z 2k Then M(z) = 1 + z 2 k k=1 l=1 ϕ(s 2l 2 )z 2l 2 ϕ(s 2k 2l )z 2k 2l 29

31 Put M(z) := ϕ(s 2k ) = n=0 k l=1 ϕ(s 2l 2 )ϕ(s 2k 2l ). ϕ(s n )z n = 1 + k=1 ϕ(s 2k )z 2k Then M(z) = 1 + z 2 k k=1 l=1 ϕ(s 2l 2 )z 2l 2 ϕ(s 2k 2l )z 2k 2l = 1 + z 2 M(z) M(z) 30

32 M(z) = 1 + z 2 M(z) M(z) Instead of moment generating series M(z) consider Cauchy transform G(z) := ϕ( 1 z s ) Note G(z) = n=0 ϕ(s n ) z n+1 = 1 z n=0 ϕ(s n ) ( 1) n 1 = z z M(1/z), 31

33 M(z) = 1 + z 2 M(z) M(z) Instead of moment generating series M(z) consider Cauchy transform G(z) := ϕ( 1 z s ) Note G(z) = n=0 ϕ(s n ) z n+1 = 1 z n=0 ϕ(s n ) ( 1) n 1 = z z M(1/z), thus zg(z) = 1 + G(z) 2 32

34 For any probability measure µ on R corresponding Cauchy transform 1 G(z) := R z t dµ(t) is analytic function on complex upper half plane C + and allows to recover µ via Stieltjes inversion formula dµ(t) = 1 π lim Im G(t + iε) ε 0 33

35 For semicircle s: implies zg(z) = 1 + G(z) 2 G(z) = z z

36 For semicircle s: implies zg(z) = 1 + G(z) 2 G(z) = z z and thus dµ(t) = 1 4 t 2 dt on [ 2,2] 2π

37 Consider now more general random matrices Keep the entries independent, but change distribution of entries, globally or depending on position of entry Arnold Bai und Silverstein Molchanov, Pastur, Khorunzhii (1996) Khorunzhy, Khoruzhenko, Pastur (1996) Shlyakhtenko (1996) Guionnet (2002) Anderson, Zeitouni (2006) 37

38 Keep the distributions normal, but allow correlations between entries for weak correlations one still gets semicircle Chatterjee (2006) Götze + Tikhomirov (2005) Schenker und Schulz-Baldes (2006) for stronger correlations other distributions occur Boutet de Monvel, Khorunzhy, Vasilchuck (1996) Girko (2001) Hachem, Loubaton, Najim (2005) Anderson, Zeitouni (2008) Rashidi Far, Oraby, Bryc, Speicher (2008)

39 Consider block matrix X N = A N B N C N B N A N B N C N B N A N where A N, B N, C N are independent Gaussian N N-random matrices., 38

40 Consider block matrix X N = A N B N C N B N A N B N C N B N A N where A N, B N, C N are independent Gaussian N N-random matrices., What is eigenvalue distribution of X N for N? 39

41 typical eigenvalue distribution for N = N= one realization......another realization... 40

42 averaged eigenvalue distributions N= N= N=150 41

43 This limiting distribution is not a semicircle, and it cannot be described nicely within usual free probability theory. 42

44 This limiting distribution is not a semicircle, and it cannot be described nicely within usual free probability theory. However, it fits well into the frame of operator-valued free probability theory! 43

45 What is an operator-valued probability space? scalars operator-valued scalars C B 44

46 What is an operator-valued probability space? scalars operator-valued scalars C B state conditional expectation ϕ : A C E : A B E[b 1 ab 2 ] = b 1 E[a]b 2 45

47 What is an operator-valued probability space? scalars operator-valued scalars C B state conditional expectation ϕ : A C E : A B E[b 1 ab 2 ] = b 1 E[a]b 2 moments operator-valued moments ϕ(a n ) E[ab 1 ab 2 a ab n 1 a] 46

48 What is an operator-valued semicircular element? Consider an operator-valued probability space E : A B s A is semicircular if 47

49 What is an operator-valued semicircular element? Consider an operator-valued probability space E : A B s A is semicircular if second moment is given by E[sbs] = η(b) for a completely positive map η : B B 48

50 What is an operator-valued semicircular element? Consider an operator-valued probability space E : A B s A is semicircular if second moment is given by E[sbs] = η(b) for a completely positive map η : B B higher moments of s are given in terms of second moments by summing over non-crossing pairings 49

51 sbs E[sbs] = η(b) 50

52 sbs E[sbs] = η(b) E[sb 1 sb 2 s sb n 1 s] = π NC 2 (n) ( ) iterated application of η according to π 51

53 sb 1 sb 2 sb 3 sb 4 sb 5 s sb 1 sb 2 sb 3 sb 4 sb 5 s sb 1 sb 2 sb 3 sb 4 sb 5 s η(b 1 ) b 2 η(b 3 ) b 4 η(b 5 ) η(b 1 ) b 2 η ( b 3 η(b 4 ) b 5 ) η (b ( ) 1 η b 2 η(b 3 ) b 4 ) b5 sb 1 sb 2 sb 3 sb 4 sb 5 s sb 1 sb 2 sb 3 sb 4 sb 5 s η ( b 1 η(b 2 ) b 3 ) b4 η(b 5 ) η ( b 1 η(b 2 ) b 3 η(b 4 ) b 5 ) 52

54 E[sb 1 sb 2 sb 3 sb 4 sb 5 s] =η(b 1 ) b 2 η(b 3 ) b 4 η(b 5 ) + η(b 1 ) b 2 η ( b 3 η(b 4 ) b 5 ) + η (b 1 η ( ) ) b 2 η(b 3 ) b 4 b5 + η ( b 1 η(b 2 ) b 3 ) b4 η(b 5 ) + η ( b 1 η(b 2 ) b 3 η(b 4 ) b 5 ) 53

55 E[ssssss] =η(1) η(1) η(1) + η(1) η ( η(1) ) + η (η ( η(1) )) + η ( η(1) ) η(1) + η ( η(1) η(1) ) 54

56 We have the recurrence relation E[s 2k ] = k l=1 η ( E[s 2l 2 ] ) E[s 2k 2l ]. 55

57 We have the recurrence relation Put thus E[s 2k ] = M(z) := n=0 k l=1 η ( E[s 2l 2 ] ) E[s 2k 2l ]. E[s n ]z n = 1 + k=1 E[s 2k ]z 2k, M(z) = 1 + zη ( M(z) ) M(z) 56

58 Consider the operator-valued Cauchy transform 1 G(z) := E[ z s ] for z C +. Note thus G(z) = E[ 1 z 1 1 sz 1] = 1 z M(sz 1 ), zg(z) = 1 + η ( G(z) ) G(z) 57

59 Thus: operator-valued Cauchy-transform of s satisfies G : C + B G analytic G solution of zg(z) = 1 + η ( G(z) ) G(z) G(z) 1 z 1 for z 58

60 back to random matrices special classes of random matrices are asymptotically described by operator-valued semicircular elements, e.g. band matrices (Shlyakhtenko 1996) block matrices (Rashidi Far, Oraby, Bryc, Speicher 2006) 59

61 Example: X N = A N B N C N B N A N B N C N B N A N where A N, B N, C N are independent Gaussian N N random matrices, For N, X N converges to s = s 1 s 2 s 3 s 2 s 1 s 2 s 3 s 2 s 1, where s 1, s 2, s 3 is free semicircular family. 60

62 s = s 1 s 2 s 3 s 2 s 1 s 2, s 1, s 2, s 3 (Ã, ϕ) s 3 s 2 s 1 s is an operator-valued semicircular element over M 3 (C) with respect to 61

63 s = s 1 s 2 s 3 s 2 s 1 s 2, s 1, s 2, s 3 (Ã, ϕ) s 3 s 2 s 1 s is an operator-valued semicircular element over M 3 (C) with respect to A = M 3 (Ã), B = M 3 (C) 62

64 s = s 1 s 2 s 3 s 2 s 1 s 2, s 1, s 2, s 3 (Ã, ϕ) s 3 s 2 s 1 s is an operator-valued semicircular element over M 3 (C) with respect to A = M 3 (Ã), B = M 3 (C) E = id ϕ : M 3 (Ã) M 3 (C), ( aij ) 3 i,j=1 ( ϕ(a ij ) ) 3 i,j=1 63

65 s = s 1 s 2 s 3 s 2 s 1 s 2, s 1, s 2, s 3 (Ã, ϕ) s 3 s 2 s 1 s is an operator-valued semicircular element over M 3 (C) with respect to A = M 3 (Ã), B = M 3 (C) E = id ϕ : M 3 (Ã) M 3 (C), ( aij ) 3 i,j=1 ( ϕ(a ij ) ) 3 i,j=1 η : M 3 (C) M 3 (C) given by η(d) = E[sDs] 64

66 s = s 1 s 2 s 3 s 2 s 1 s 2, s 1, s 2, s 3 (Ã, ϕ) s 3 s 2 s 1 Asymptotic eigenvalue distribution µ of X N is given by distribution of s with respect to tr 3 ϕ: H(z) = 1 z t dµ(t) = tr 3 ϕ ( 1 z s ) 65

67 s = s 1 s 2 s 3 s 2 s 1 s 2, s 1, s 2, s 3 (Ã, ϕ) s 3 s 2 s 1 Asymptotic eigenvalue distribution µ of X N is given by distribution of s with respect to tr 3 ϕ: H(z) = 1 z t dµ(t) = tr 3 ϕ ( 1 z s ) = tr { 1 3 E[ z s ]}, 66

68 s = s 1 s 2 s 3 s 2 s 1 s 2, s 1, s 2, s 3 (Ã, ϕ) s 3 s 2 s 1 Asymptotic eigenvalue distribution µ of X N is given by distribution of s with respect to tr 3 ϕ: H(z) = 1 z t dµ(t) = tr 3 ϕ ( 1 z s ) = tr { 1 3 E[ z s ]}, and G(z) = E[ z s 1 ] is solution of zg(z) = 1 + η ( G(z) ) G(z) 67

69 X = A B C B A B C B A : 68

70 X = A B C B A B : G(z) = C B A f(z) 0 h(z) 0 g(z) 0 h(z) 0 f(z) 69

71 X = A B C B A B : G(z) = C B A f(z) 0 h(z) 0 g(z) 0 h(z) 0 f(z) η ( G(z) ) = f(z) + g(z) 0 g(z) + 2 h(z) 0 2 f(z) + g(z) + 2 h(z) 0 g(z) + 2 h(z) 0 2 f(z) + g(z), 70

72 X = A B C B A B : G(z) = C B A f(z) 0 h(z) 0 g(z) 0 h(z) 0 f(z) η ( G(z) ) = f(z) + g(z) 0 g(z) + 2 h(z) 0 2 f(z) + g(z) + 2 h(z) 0 g(z) + 2 h(z) 0 2 f(z) + g(z), zg(z) = 1 + η ( G(z) ) G(z) 71

73 X = A B C B A B : G(z) = C B A f(z) 0 h(z) 0 g(z) 0 h(z) 0 f(z) η ( G(z) ) = f(z) + g(z) 0 g(z) + 2 h(z) 0 2 f(z) + g(z) + 2 h(z) 0 g(z) + 2 h(z) 0 2 f(z) + g(z), zg(z) = 1 + η ( G(z) ) G(z) H(z) = tr 3 (G(z)) = 1 (2f(z) + g(z)) 3 72

74 Comparison of this solution with simulations L 3 (n)

75 some more examples 0.35 L 3 (n) L 4 (n) L 5 (n) A B C B A B C B A A B C D B A B C C B A B D C B A A B C D E B A B C D C B A B C D C B A B E D C B A, 74

76 II. General operator-valued free probability theory 75

77 What can we say about the relation between two matrices, when we know that the entries of the matrices are free? X = (x ij ) N i,j=1 Y = (y kl ) N k,l=1 with {x ij } and {y kl } free w.r.t. ϕ X and Y are not free w.r.t. tr ϕ in general however: relation between X and Y is more complicated, but still treatable operator-valued freeness 76

78 What can we say about the relation between two matrices, when we know that the entries of the matrices are free? X = (x ij ) N i,j=1 Y = (y kl ) N k,l=1 with {x ij } and {y kl } free w.r.t. ϕ X and Y are not free w.r.t. tr ϕ in general however: relation between X and Y is more complicated, but still treatable in terms of operator-valued freeness 77

79 Let (C, ϕ) be non-commutative probability space. Consider N N matrices over C: M N (C) := {(a ij ) N i,j=1 a ij C} M N (C) = M N (C) C is a non-commutative probability space with respect to tr ϕ : M N (C) C but there is also an intermediate level 78

80 Instead of M N (C) tr ϕ C consider... 79

81 M N (C) = M N (C) C=: A id ϕ=: E M N (C)=: B tr ϕ tr C 80

82 M N (C) = M N (C) C=: A id ϕ=: E M N (C)=: B tr ϕ tr C 81

83 Let B A be a unital subalgebra. A linear map is a conditional expectation if E : A B E[b] = b b B and E[b 1 ab 2 ] = b 1 E[a]b 2 a A, b 1, b 2 B An operator-valued probability space consists of B A and a conditional expectation E : A B 82

84 Consider an operator-valued probability space (A, E : A B). The operator-valued distribution of a A is given by all operator-valued moments E[ab 1 ab 2 b n 1 a] B (n N, b 1,..., b n 1 B) 83

85 Consider an operator-valued probability space (A, E : A B). The operator-valued distribution of a A is given by all operator-valued moments E[ab 1 ab 2 b n 1 a] B (n N, b 1,..., b n 1 B) Random variables x, y A are free with respect to E (or free with amalgamation over B) if E[p 1 (x)q 1 (y)p 2 (x)q 2 (y) ] = 0 whenever p i, q j are polynomials with coefficients from B and E[p i (x)] = 0 i and E[q j (y)] = 0 j. 84

86 Note: polynomials in x with coefficients from B are of the form x 2 b 0 x 2 b s and x do not commute in general! 85

87 Note: polynomials in x with coefficients from B are of the form x 2 b 0 x 2 b 1 xb 2 xb 3 b s and x do not commute in general! 86

88 Note: polynomials in x with coefficients from B are of the form x 2 b 0 x 2 b 1 xb 2 xb 3 b 1 xb 2 xb 3 + b 4 xb 5 xb 6 + etc. b s and x do not commute in general! 87

89 Operator-valued freeness works mostly like ordinary freeness, one only has to take care of the order of the variables; in all expressions they have to appear in their original order! Example: If x and {y 1, y 2 } are free, then one has E[y 1 xy 2 ] = E [ y 1 E[x]y 2 ] ; and more general E[y 1 b 1 xb 2 y 2 ] = E [ y 1 b 1 E[x]b 2 y 2 ]. 88

90 Consider E : A B. Define free cumulants by κ B n : A n B E[a 1 a n ] = π N C(n) κ B π[a 1,..., a n ] arguments of κ B π are distributed according to blocks of π but now: cumulants are nested inside each other according to nesting of blocks of π 89

91 Example: π = { {1,10}, {2,5,9}, {3,4}, {6}, {7,8} } NC(10), κ B π[a 1,..., a 10 ] = κ B 2 ( a 1 κ B ( 3 a2 κ B 2 (a 3, a 4 ), a 5 κ B 1 (a 6) κ B 2 (a ) ) 7, a 8 ), a 9, a10 90

92 For a A define its operator-valued Cauchy transform 1 G a (b) := E[ b a ] = E[b 1 (ab 1 ) n ] n 0 and operator-valued R-transform R a (b) : = κ B n+1 (ab, ab,..., ab, a) n 0 = κ B 1 (a) + κb 2 (ab, a) + κb 3 (ab, ab, a) + Then bg(b) = 1 + R(G(b)) G(b) or G(b) = 1 b R(G(b)) 91

93 If x and y are free over B, then mixed B-valued cumulants in x and y vanish R x+y (b) = R x (b) + R y (b) G x+y (b) = G x [ b Ry ( Gx+y (b) )] subordination If s is a semicircle over B then R s (b) = η(b) where η : B B is a linear map given by η(b) = E[sbs]. 92

94 Back to our matrix example Proposition and Exercise: Assume {x ij } and {y kl } are free in the n.c.p.s. (C, ϕ). Then the matrices X = (x ij ) N i,j=1, Y = (y kl) N k,l=1 M N (C) are free with amalgamation over M N (C) M N (C), i.e., with respect to E = id ϕ. 93

95 A final remark The analytic theory of operator-valued free probability lacks at the moment some of the deeper statements of the scalar-valued theory (like Nevalinna-like characterizations of operator-valued Cauchy-transforms); developing a reasonable analogue of complex function theory on an operator-valued level is an active area in free probability at the moment, see Voiculescu: Free Analysis Questions, IMRN 2004 (see also recent work of Helton, Vinnikov, Belinschi, Popa) 94