Fluctuations of Random Matrices and Second Order Freeness

Transcription

1 Fluctuations of Random Matrices and Second Order Freeness james mingo with b. collins p. śniady r. speicher SEA 06 Workshop Massachusetts Institute of Technology July 9-14,

2 random matrix free probability 2

3 references J. Mingo and A. Nica: Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices, Int. Math. Res. Not., (28): , 2004 J. Mingo and R. Speicher: Second Order Freeness and Fluctuations of Random Matrices: I. Gaussian and Wishart matrices and Cyclic Fock spaces, J. Funct. Anal., 235, 2006, pp J. Mingo, P. Śniady, and R. Speicher. Second order freeness and fluctuations of random matrices: II. 3

4 Unitary random matrices. Adv. in Math. to appear, ariv:math.oa/ B. Collins, J. Mingo, P. Śniady, and R. Speicher. Second order freeness and fluctuations of random matrices: III. Higher order freeness and free cumulants, ariv:math.oa/

5 second order freeness & fluctuations Voiculescu s freeness property gives a universal rule for calculating mixed moments of a and b if a and b are free and moments of a and b are known; second order freeness will do for fluctuations what first order freeness (i.e. Voiculescu s freeness) did for moments; n a N N random matrix 1 α k = lim N N E(Tr( k N )) = moments of limit distribution in many examples we also have a second order limit 5

6 distribution: Y N,k = Tr( k N α k I N ) converges in distribution: α k,l = lim N cov(y N,k, Y N,l ) k r (Tr(Y n 1 N,i i ),..., Tr(Y n r N,i r )) 0 for r > 2 6

7 examples GUE: N = ( f i, j ) an N N random matrix, f i, j a complex Gaussian r.v.; E( f i, j ) = 0, E( f i, j 2 ) = 1/N 1 lim N N E(Tr( k N )) = 2 4 t t k 2 dt 2 2π (Wigner s semi-circle law) Y N,k = Tr( k N α k I N ) is asymptotically Gaussian α m,n = lim N cov(y N,m, Y N,n ) ρ(x, y) = = x m y m x n y n ρ(x, y) dx dy [ 2,2] 2 x y x y 4 xy 4π 2 4 x 2 4 y 2 7

8 combinatorial interpretation Wigner s semi-circle law 4 t 2 dt has moments 2π 2 given by n o non-crossing pairings: covariance given by fluctuation moments α m,n = lim cov(y N,m, Y N,n ) N = x m y m x n y n ρ(x, y) dx dy [ 2,2] 2 x y x y 4 xy ρ(x, y) = 4π 2 4 x 2 4 y 2 8

9 fluctuation moments are given by n o non-crossing annular pairings 9

10 second order freeness 1 N E(Tr( k N )) = α k + O(N 2 ) E(Tr( m N α m I N )Tr( n N α n I N )) = α m,n + O(N 2 ) general situation: given {A N } N with A N a i.e. lim N 1 N E(Tr(Ak N )) = ϕ(ak ) lim n E(Tr(A m N ϕ(am )I N ) Tr(A n N ϕ(an )I N )) = ϕ 2 (a m, a n ) and {B N } N with B N b i.e. lim N 1 N E(Tr(Bk N )) = ϕ(bk ) lim n E(Tr(B m N ϕ(bm )I N ) Tr(B n N ϕ(bn )I N )) = ϕ 2 (b m, b n ) 10

11 first order freeness tells us how to calculate ϕ(a r 1 b r2 a r k 1b r k) from {ϕ(a k )} k and {ϕ(b k )} k second order freeness tells us how to calculate ϕ 2 (a r 1 b r2 a r k 1 b r k, a s 1 b s2 a s l 1 b s l ) from the fluctuation moments of a and b suppose we want to calculate ϕ 2 (a r 1 b r2 a r k 1 b r k, a s 1 b s2 a s l 1 b s l ) then it suffices to calculate ϕ 2 ((a r 1 α r1 ) (b r 2 β r2 ) (a r k 1 α rk 1 ) (b r k β rk )), (a s 1 α s1 ) (b s 2 β s2 ) (a s l 1 α sl 1 ) (b s l β sl )) and apply induction (α k = ϕ(a k ), β k = ϕ(b k )) 11

12 the definition so let p 1, p 2,... p k and q 1, q 2,..., q l be centred polynomials alternating from a to b, i.e. ϕ(p i ) = ϕ(q j ) = 0 for all i and j and if p i is a polynomial in a then p i+1 (setting p k+1 = p 1 ) is a polynomial in b, likewise for q j if a and b are free of second order then ϕ 2 (p 1 p 2 p k, q l q l 1 q 1 ) = 0 if k l ϕ 2 (p 1 p 2 p k, q k q k 1 q 1 ) = ϕ(p 1 q 1 ) ϕ(p k q k ) + ϕ(p 1 q 2 ) ϕ(p k q 1 ) + ϕ(p 1 q 3 ) ϕ(p k q 2 ) + + ϕ(p 1 q k ) ϕ(p k q k 1 ) 12

13 example ϕ 2 (p 1 p 2 p 3, q 3 q 2 q 1 ) = ϕ(p 1 q 1 )ϕ(p 2 q 2 )ϕ(p 3 q 3 ) + ϕ(p 1 q 2 )ϕ(p 2 q 3 )ϕ(p 3 q 1 ) + ϕ(p 1 q 2 )ϕ(p 2 q 3 )ϕ(p 3 q 1 ) p 1 p 1 q 3 q 1 q 2 q 3 q 1 q 2 p 3 p 2 p 3 p 2 ϕ(p 1 q 1 )ϕ(p 2 q 2 )ϕ(p 3 q 3 ) ϕ(p 1 q 2 )ϕ(p 2 q 3 )ϕ(p 3 q 1 ) 13

14 p 1 q 1 ϕ(p 1 q 3 )ϕ(p 2 q 1 )ϕ(p 3 q 2 ) q 3 q 2 p 3 p 2 ϕ 2 (a r, b s ) = 0 ϕ 2 (a r 1 b r 2, a s ) = ϕ 2 (a r 1, a s )ϕ(b r 2 ) ϕ 2 (a r 1 b r 2, a s 1 b s 2 ) = (ϕ(a r 1+s 1 ) ϕ(a r 1 )ϕ(a s 1 ))(ϕ(b r 2+s 2 ) ϕ(b r 2 )ϕ(b s 2 )) + ϕ 2 (a r 1, a r 2 )ϕ(b r 2 )ϕ(b s 2 ) + ϕ(a r 1 )ϕ(a s 1 )ϕ 2 (b s 1, b s 2 ) 14

15 second order limit distributions N = GUE; lim N 1 N E(Tr( k N )) = α k = n o non-crossing pairings of [k] lim N E(Tr( m N α m )Tr( n N α n)) = α m,n = n o non-crossing (m, n)-annular pairings 15

16 complex Wishart G = G M,N = M N cx. i.i.d. Gaussian, N = G M,N G M,N, cx. Wishart with = I M, M/N c 1 lim M,N N E(Tr( k N )) = α k = c #(π) π NC(k) lim E(Tr( m M,N N α m )Tr( n N α n)) = α m,n = c #(π) π S NC (m,n) 16

17 more Wishart matrices suppose {D M,1,..., D M,s } M are independent from G M,N and have a second order limit distribution, i.e. lim M 1 M E(Tr(D i 1 D ik ) = ϕ(d i1 d ik ) lim M k 2 (Tr(D i1 D im ), Tr(D im+1 D im+n )) = ϕ 2 (d i1 d im, d im+1 d im+n ) (and a vanishing condition on higher cumulants) 1 lim M,N N E(Tr(G D i 1 G G D i2 G G D ik G ) = c #(π) ϕ π 1 γ (d i1,..., d ik ) π NC(k) if π = ( j 1,..., j l ) is a cycle, ϕ π (d 1,..., d n ) := ϕ(d j1 d j2 d jl ); γ = (1, 2, 3,..., k) 17

18 second order limit distributions lim k 2 (Tr(G D i1 G G D i2 G G D im G ), M,N Tr(G D im+1 G G D im+2 G G D im+n G )) = π S NC (m,n) + c #(π) ϕ π 1 γ m,n (d i1,..., d im+n ) π 1 π 2 NC(m) NC(n) π 1 =s 1 s p,π 2 =t 1 t q c #(π1)+#(π2) i, j ϕ si,t j ϕ s1 ϕ si ϕ sp ϕ t1 ϕ t j ϕ tq the argument of each ϕ, (d i1,..., d im+n ) has been suppressed; γ m,n = (1, 2, 3,..., m)(m + 1,..., m + n) 18

19 asymptotic second order freeness Suppose {A N } N and {B N } N are two families of random matrices such that: each of {A N } N and {B N } N have second order limit distributions A N and B N are independent the joint distribution of the entries of B N is invariant under conjugation by a unitary then {A N } N and {B N } N are asymptotically free of second order 19

20 epilogue P.S. freeness of third and higher order 20

21 Fluctuations of Random Matrices and Second Order Freeness Part 2: R-transform formulas Roland Speicher Queen s University Kingston, Canada

22 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 math (to appear in Adv. Math.) Collins + Mingo + Sniady + Speicher: III. Higher order freeness and free cumulants math

23 Warning In the following all our random matrices are complex. On the level of eigenvalue distributions the theory is robust against changing complex to real or quaternionic. For fluctuations, however, the phenomena are more sensitive and there are typically extra factors for the real case (still to worked out rigorously!). 2

24 Definition [Mingo, Speicher]: Consider N N random matrices (A N ) N N. They have 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. In many cases (Gaussian, Wishart, Haar unitary random matrices) such second order limit distribution exists. 3

25 Example: Gaussian random matrix A (N = 40, trials=50.000) 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 4

26 Example: Wishart matrix B (N = M = 40, trials=50.000) Var(Tr(B)) = 1 Var(Tr(B 2 )) = 18 Var(Tr(B 3 )) = 300 Normal Probability Plot Normal Probability Plot Normal Probability Plot Probability Probability Probability cov= cov= cov= Data Data Data 5

27 Problem: What can we say about the fluctuations of C := A + B in a generic situation. What we need to understand are the mixed fluctuations in A and B, i.e. limits of cov ( Tr(A m 1B n 1A m 2 ),Tr(A m 1Bñ1A m 2 ) ). 6

28 Theorem [Mingo, Sniady, Speicher]: Consider random matrices A N and B N such that A N has second order limit distribution. B N has second order limit distribution. A N and B N are independent B N is invariant under unitary conjugation Then A N and B N are asymptotically free of second order. 7

29 Definition [Mingo, Speicher]: A and B are free of second order if special mixed covariances cov ( Tr(cyclically alternating centered), Tr(cyclically alternating centered) calculate in a specific way (by summing over spoke diagrams in corresponding annular representation of the two traces) from expectations of A and of B. ) All mixed covariances can be reduced to above situation, thus second order freeness is a rule for calculating mixed correlations in A and B from the expectations and covariances of A and of B. 8

30 In particular: 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. 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) ) 9

31 To treat these formulas in general, linearize the problem by going over to free cumulants κ or R-transforms R. Recall first order case: Put and define by the relation G(x) = 1 x + R(x) = n=1 n=1 1 α n x n+1 κ n x n 1 Cauchy transform R-transform G(x) + R(G(x)) = x. 10

32 There is a combinatorial structure behind this, the relation between the α s and the κ s is given by summing over non-crossing partitions: α 1 = α 2 = + = κ 1 = 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

33 Theorem [Voiculescu 1986, Speicher 1994]: Let A and B be free. Then one has R A+B (z) = R A (z) + R B (z), or equivalently κ A+B m = κ A m + κ B m m. 12

34 Second order R-transform formula From the expectations α m and the covariances α m,n we define second order free cumulants κ m,n as follows. Put and define by the equation G(x, y) := m,n 1 R(x, y) = m,n 1 α m,n 1 x m+1 1 y n+1 κ m,n x m 1 y n 1 13

35 G(x, y) = G (x) G (y) R ( G(x), G(y) ) + G (x)g (y) 1 ( ) 2 G(x) G(y) (x y) 2. or G(x, y) = G (x) G (y) R ( G(x), G(y) ) + 2 x y [ log ( G(x) G(y) x y )]

36 These equations encode relations between the α m,n and the κ m,n (containing also the κ m ), e.g.: 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 There is again a rich combinatorial theory behind this, the above sums are running over annular non-crossing permutations. 14

37 Note also the following special case: If the second order free cumulants are zero (as it happens for Gaussian and Wishart random matrices), the above formula reduces to G(x, y) = G (x)g (y) ( G(x) G(y) ) 2 1 (x y) 2 or G(x, y) = 2 x y [ log ( G(x) G(y) x y )], which says that 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. 15

38 Theorem [Collins,Mingo,Sniady,Speicher]: Let A and B be free of second order. Then one has κ A+B m,n = κ A m,n + κ B m,n m, n or equivalently R A+B (x, y) = R A (x, y) + R B (x, y). This allows to calculate the fluctuations of A + B from the moments and fluctuations of A and the moments and fluctuations of B if A and B are free of second order (for example, for independent random matrices, where one of them is unitarily invariant). 16

39 Examples: Take independent Gaussian random matrix A and Wishart random matrix B (N = M) and consider C = A + B and D = A 2 + B Note: C has vanishing second order free cumulants, thus the Bai-Silverstein formula can be applied to get its fluctuations; for D this is not the case! 17

40 Example: C = A + B (N = 40, trials=50.000) Var(Tr(C)) = 2 Var(Tr(C 2 )) = 28 Var(Tr(C 3 )) = 600 Normal Probability Plot Normal Probability Plot Normal Probability Plot Probability Probability Probability cov= cov= cov Data Data Data 18

41 Example: D = A 2 + B (N = 40, trials=50.000) x Var(Tr(D)) = 3 Var(Tr(D 2 )) = 112 Var(Tr(D 3 )) = 4090 Normal Probability Plot Normal Probability Plot Normal Probability Plot Probability Probability Probability cov= cov= cov= Data Data Data 19