Higher Order Freeness: A Survey Roland Speicher Queens University - PowerPoint PPT Presentation

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

Second order freeness and fluctuations of random matrices: Mingo + Speicher: I. Gaussian and Wishart matrices and cyclic Fock spaces JFA 235 (2006), 226-270 Mingo + Sniady + Speicher: II. Unitary random matrices Adv. Math. 209 (2007), 212-240 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:0708.0586 (to appear in TAMS) 1

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

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

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 eine Realisierung 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 zweite Realisierung 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 dritte Realisierung 4

1 2 0.9 1.8 0.8 1.6 0.7 1.4 0.6 1.2 0.5 1 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 eine Realisierung N=10 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 zweite Realisierung N=10 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 dritte Realisierung N=10 5

1 2 3 0.9 1.8 2.5 0.8 1.6 0.7 1.4 2 0.6 1.2 0.5 1 1.5 0.4 0.8 1 0.3 0.6 0.2 0.4 0.5 0.1 0.2 0 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 eine Realisierung N=10 N=100 1 1 3 0.9 0.9 2.5 0.8 0.8 0.7 0.7 2 0.6 0.6 0.5 0.5 1.5 0.4 0.4 1 0.3 0.3 0.2 0.2 0.5 0.1 0.1 0 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 zweite Realisierung N=10 N=100 1 1 4 0.9 0.9 3.5 0.8 0.8 3 0.7 0.7 2.5 0.6 0.6 0.5 0.5 2 0.4 0.4 1.5 0.3 0.3 1 0.2 0.2 0.5 0.1 0.1 0 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 dritte Realisierung N=10 N=100 6

30 1 2 3 0.9 1.8 25 2.5 0.8 1.6 0.7 1.4 2 20 0.6 1.2 0.5 1 1.5 15 0.4 0.8 1 10 0.3 0.6 0.2 0.4 0.5 5 0.1 0.2 0 0 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 eine Realisierung N=10 N=100 N=1000 30 1 1 3 0.9 0.9 25 2.5 0.8 0.8 0.7 0.7 20 2 0.6 0.6 0.5 0.5 1.5 15 0.4 0.4 1 10 0.3 0.3 0.2 0.2 0.5 5 0.1 0.1 0 0 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 zweite Realisierung N=10 N=100 N=1000 1 1 4 30 0.9 0.9 3.5 25 0.8 0.8 3 0.7 0.7 20 2.5 0.6 0.6 0.5 0.5 2 15 0.4 0.4 1.5 10 0.3 0.3 1 0.2 0.2 5 0.5 0.1 0.1 0 0 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 dritte Realisierung N=10 N=100 N=1000 7

Wigner’s semicircle law N = 4000 0.35 0.35 0.35 0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 ... one realization ... ... another realization ... ... yet another one ... 8

Convergence of µ X N towards µ X is governed by large deviation principle : Prob( µ X N ≈ ν ) ∼ e − N 2 I ( ν ) , where rate function ν �→ I ( ν ) is given as Legendre transform of 1 � e − N 2 tr( p ( X N )) � C � X � ∋ p �→ lim N 2 log E N →∞ Note: ( − 1) r � e − N 2 tr( p ( X N )) � � � � N 2 r · k r log E = tr( p ( X N )) , . . . , tr( p ( X N )) r ! r where k r are classical cumulants 9

This motivates our general assumption on the considered ran- dom matrices A N : For all r ∈ N and all k 1 , . . . , k r ∈ N the following limits exists � � N →∞ N 2 r − 2 k r tr( A k 1 N ) , . . . , tr( A k r =: α A lim N ) k 1 ,...,k r � �� � classical cumulants of traces of powers The α ’s are the asymptotic correlation moments of our random matrix ensemble A N and constitute its limiting distribution of all orders . 10

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 eigen- values have a correlation as for Gaussian random matrices: N ) = λ k 1 + · · · + λ k tr( A k N N Eigenvalues λ 1 , . . . , λ N of A N are not independent , but feel some interaction 11

Contrast this with following situation:   0 0 η 1 · · · 0 0  η 2 · · ·    D N =  , . . . ... . . .   . . .  0 0 · · · η N where η 1 , η 2 , . . . are independent and identically distributed ac- cording to η . Then N ) = η k 1 + · · · + η k tr( D k → E [ η k ] N N with large deviation principle ∼ e − N H ( ν ) ; not ∼ e − N 2 I ( ν ) 12

In this case: � � tr( D k 1 N ) , . . . , tr( D k r = N 1 − r k r ( η k 1 , . . . , η k r ) , k r N ) 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 eigen- value distribution, but a quite different type of convergence to- wards the semicircle 13

500 independent eigenvalues with semicircular distribution 1 0.5 0 −0.5 −1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 eigenvalues of a 500 x 500 Gaussian random matrix 1 0.5 0 −0.5 −1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 14

500 independent eigenvalues with semicircular distribution 20 15 10 5 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 eigenvalues of a 500 x 500 Gaussian random matrix 15 10 5 15 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

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

Consider α 2 , 3 , 1 17

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

Consider α 2 , 3 , 1 counts! 19

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 N →∞ E [tr( A n α n := lim N )] and � � Tr( A m N ) , Tr( A n α m,n := N →∞ cov lim 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

Example: Gaussian random matrix A ( N = 40, trials=50.000) 0.4 0.07 0.35 0.06 0.25 0.3 0.05 0.2 0.25 0.04 0.2 0.15 0.03 0.15 0.1 0.02 0.1 0.05 0.01 0.05 0 0 0 −4 −3 −2 −1 0 1 2 3 4 34 36 38 40 42 44 46 50 60 70 80 90 100 110 Var(Tr( A 2 )) = 2 Var(Tr( A 4 )) = 36 . Var(Tr( A )) = 1 Normal Probability Plot Normal Probability Plot Normal Probability Plot 0.999 0.999 0.999 0.997 0.997 0.997 0.99 0.99 0.99 0.98 0.98 0.98 0.95 0.95 0.95 0.90 0.90 0.90 0.75 0.75 0.75 Probability Probability Probability 0.50 0.50 0.50 0.25 0.25 0.25 0.10 0.10 0.10 0.05 0.05 0.05 0.02 0.02 0.02 0.01 0.01 0.01 0.003 0.003 0.003 cov=35.85 cov=0.99 0.001 0.001 0.001 cov=2.02 −4 −3 −2 −1 0 1 2 3 4 34 36 38 40 42 44 46 60 65 70 75 80 85 90 95 100 105 110 Data Data Data 21