Free Probability Theory and Non-crossing Partitions. Roland Speicher Queen s University Kingston, Canada

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1 Free Probability Theory and Non-crossing Partitions Roland Speicher Queen s University Kingston, Canada

2 Freeness Definition [Voiculescu 1985]: Let (A, ϕ) be a non-commutative probability space, i.e. A is unital algebra, and ϕ : A C is linear functional with ϕ(1) = 1. Unital subalgebras A 1,..., A r A are called free if we have whenever ϕ(a 1 a k ) = 0 a j A i(j) for all j = 1,..., k ϕ(a j ) = 0 for all j = 1,..., k i(1) i(2), i(2) i(3),..., i(k 1) i(k) 1

3 Freeness of random variables Elements ( random variables ) a and b in A are called free if their generated unital subalgebras are free, i.e. whenever ϕ ( p 1 (a)q 1 (b)p 2 (a)q 2 (b) ( ) ϕ q 1 (b)p 1 (a)q 2 (b)p 2 (a) ) = 0 = 0 p i, q j are polynomials ϕ ( p i (a) ) = 0 = ϕ ( q j (b) ) for all i, j 2

4 Examples Canonical examples for free random variables appear in the context of operator algebras: creation and annihilation operators on full Fock spaces von Neumann algebras of free groups random matrices 3

5 Operators on full Fock spaces For a Hilbert space H we define full Fock space F(H) = CΩ H n n=1 4

6 Operators on full Fock spaces For a Hilbert space H we define full Fock space F(H) = CΩ H n n=1 For each g H we have corresponding creation operator l(g) and annihilation operator l (g) and l(g)h 1 h n = g h 1 h n l (g)ω = 0 l (g)h 1 h n = h 1, g h 2 h n 5

7 Operators on full Fock spaces For a Hilbert space H we define full Fock space F(H) = CΩ H n n=1 For each g H we have corresponding creation operator and annihilation operator l (g) l(g) and l(g)h 1 h n = g h 1 h n l (g)ω = 0 l (g)h 1 h n = h 1, g h 2 h n We have vacuum expectation ϕ(a) = Ω, aω 6

8 Freeness for operators on full Fock spaces If H 1,..., H r are orthogonal sub-hilbert spaces in H and A := B(F(H)), ϕ( ) := Ω, Ω A i := unital -algebra generated by l(f) (f H i ) then A 1,..., A r are free in (A, ϕ). 7

9 Freeness for operators on full Fock spaces If H 1,..., H r are orthogonal sub-hilbert spaces in H and A := B(F(H)), ϕ( ) := Ω, Ω A i := unital -algebra generated by l(f) (f H i ) then A 1,..., A r are free in (A, ϕ). Reason: If a j A i(j) with i(1) i(2) = i(k) and ϕ(a j ) = 0 for all j then a 1 a 2 a k Ω = a 1 Ω a 2 Ω a k Ω 8

10 Gaussian random matrices A N = (a ij ) N i,j=1 : Ω M N(C) with 9

11 Gaussian random matrices A N = (a ij ) N i,j=1 : Ω M N(C) with a ij = ā ji (i.e. A N = A N ) 10

12 Gaussian random matrices A N = (a ij ) N i,j=1 : Ω M N(C) with a ij = ā ji (i.e. A N = A N ) {a ij } 1 i j N are independent Gaussian random variables with E[a ij ] = 0 E[ a ij 2 ] = 1 N 11

13 Freeness for Gaussian random matrices If A N and B N are independent Gaussian random matrices, i.e., A N is Gaussian random matrix, and B N is Gaussian random matrix entries of A N are independent from entries of B N Then A N and B N are asymptotically free 12

14 Asymptotic freeness asymptotic freeness ˆ= freeness relations hold in the large N-limit whenever ( ) lim ϕ p 1 (A N )q 1 (B N )p 2 (A N )q 2 (B N ) N = 0 p i, q j polynomials lim N ϕ ( p i (A N ) ) = 0 = lim N ϕ ( q j (A N ) ) for all i, j 13

15 What is state ϕ for random matrices ( ) lim ϕ p 1 (A N )q 1 (B N )p 2 (A N )q 2 (B N ) N = 0 Two possibilities: (averaged) asymptotic freeness: ϕ = E tr almost sure asymptotic freeness: ϕ = tr, and lim-equations hold almost surely 14

16 What is Freeness? Freeness between a and b is an infinite set of equations relating various moments in a and b: ϕ ( p 1 (a)q 1 (b)p 2 (a)q 2 (b) ) = 0 15

17 What is Freeness? Freeness between a and b is an infinite set of equations relating various moments in a and b: ϕ ( p 1 (a)q 1 (b)p 2 (a)q 2 (b) ) = 0 Basic observation: freeness between a and b is actually a rule for calculating mixed moments in a and b from the moments of a and the moments of b: ( ϕ a n 1b m 1a n 2b m 2 )= polynomial ( ϕ(a i ), ϕ(b j ) ) 16

18 Example: ϕ( (a n ϕ(a n )1 )( b m ϕ(b m )1 ) ) = 0, thus ϕ(a n b m ) ϕ(a n 1)ϕ(b m ) ϕ(a n )ϕ(1 b m )+ϕ(a n )ϕ(b m )ϕ(1 1) = 0, and hence ϕ(a n b m ) = ϕ(a n ) ϕ(b m ) 17

19 Freeness is a rule for calculating mixed moments, analogous to the concept of independence for random variables. Thus freeness is also called free independence 18

20 Freeness is a rule for calculating mixed moments, analogous to the concept of independence for random variables. Note: free independence is a different rule from classical independence; free independence occurs typically for non-commuting random variables, like operators on Hilbert spaces or (random) matrices Example: ϕ( (a ϕ(a)1 ) ( b ϕ(b)1 ) ( a ϕ(a)1 ) ( b ϕ(b)1 ) ) = 0, which results in ϕ(abab) = ϕ(aa) ϕ(b) ϕ(b) + ϕ(a) ϕ(a) ϕ(bb) ϕ(a) ϕ(b) ϕ(a) ϕ(b) 19

21 Consider independent Gaussian random matrices A N and B N. Then one has lim ϕ(a N) = 0, N lim N ϕ(a2 N ) = 1, lim N ϕ(a3 N ) = 0, lim ϕ(b N) = 0 N lim N ϕ(b2 N ) = 1 lim N ϕ(b3 N ) = 0 lim N ϕ(a4 N ) = 2, lim N ϕ(b4 N ) = 2 Asymptotic freeness between A N and B N implies then for example: lim ϕ(a NA N B N B N A N B N B N A N ) = 2 N 20

22 tr(a N A N B N B N A N B N B N A N ) one realization averaged over 1000 realizations 4 4 gemittelt uber 1000 Realisierungen tr(aabbabba) E[tr(AABBABBA)] N N 21

23 Understanding the freeness rule: the idea of cumulants write moments in terms of other quantities, which we call free cumulants freeness is much easier to describe on the level of free cumulants: vanishing of mixed cumulants relation between moments and cumulants is given by summing over non-crossing or planar partitions 22

24 Example: independent Gaussian random matrices Consider two independent Gaussian random matrices A and B Then, in the limit N, the moments are given by ϕ(a n 1B m 1A n 2B m 2 ) { # non-crossing/planar pairings of pattern A A A }{{} n 1 -times B B B }{{} m 1 -times A A A }{{} n 2 -times B } B {{ B}, m 2 -times which do not pair A with B } 23

25 Example: ϕ(aabbabba) = 2 A A B B A B B A A A B B A B B A 24

26 Example: ϕ(aabbabba) = 2 A A B B A B B A A A B B A B B A 25

27 Moments and cumulants For ϕ : A C we define cumulant functionals κ n (for all n 1) κ n : A n C by moment-cumulant relation ϕ(a 1 a n ) = π N C(n) κ π [a 1,..., a n ] 26

28 ϕ(a 1 ) =κ 1 (a 1 ) a 1 27

29 ϕ(a 1 a 2 ) = κ 2 (a 1, a 2 ) + κ 1 (a 1 )κ 1 (a 2 ) a 1 a 2

30 a 1 a 2 a 3 ϕ(a 1 a 2 a 3 ) = κ 3 (a 1, a 2, a 3 ) + κ 1 (a 1 )κ 2 (a 2, a 3 ) + κ 2 (a 1, a 2 )κ 1 (a 3 ) + κ 2 (a 1, a 3 )κ 1 (a 2 ) + κ 1 (a 1 )κ 1 (a 2 )κ 1 (a 3 ) 28

31 29

32 ϕ(a 1 a 2 a 3 a 4 ) = ϕ(a 1 a 2 a 3 a 4 ) = κ 4 (a 1, a 2, a 3, a 4 ) + κ 1 (a 1 )κ 3 (a 2, a 3, a 4 ) + κ 1 (a 2 )κ 3 (a 1, a 3, a 4 ) + κ 1 (a 3 )κ 3 (a 1, a 2, a 4 ) + κ 3 (a 1, a 2, a 3 )κ 1 (a 4 ) + κ 2 (a 1, a 2 )κ 2 (a 3, a 4 ) + κ 2 (a 1, a 4 )κ 2 (a 2, a 3 ) + κ 1 (a 1 )κ 1 (a 2 )κ 2 (a 3, a 4 ) + κ 1 (a 1 )κ 2 (a 2, a 3 )κ 1 (a 4 ) + κ 2 (a 1, a 2 )κ 1 (a 3 )κ 1 (a 4 ) + κ 1 (a 1 )κ 2 (a 2, a 4 )κ 1 (a 3 ) + κ 2 (a 1, a 4 )κ 1 (a 2 )κ 1 (a 3 ) + κ 2 (a 1, a 3 )κ 1 (a 2 )κ 1 (a 4 ) + κ 1 (a 1 )κ 1 (a 2 )κ 1 (a 3 )κ 1 (a 4 ) 30

33 Freeness ˆ= vanishing of mixed cumulants free product ˆ= direct sum of cumulants ϕ(a n ) given by sum over blue planar diagrams ϕ(b m ) given by sum over red planar diagrams then: for a and b free, ϕ(a n 1b m 1 ) is given by sum over planar diagrams with monochromatic (blue or red) blocks 31

34 Freeness ˆ= vanishing of mixed cumulants free product ˆ= direct sum of cumulants We have: a and b free is equivalent to whenever κ n (c 1,..., c n ) = 0 n 2 c i {a, b} for all i there are i, j such that c i = a, c j = b 32

35 ϕ(a 1 b 1 a 2 b 2 ) = κ 1 (a 1 )κ 1 (a 2 )κ 2 (b 1, b 2 ) + κ 2 (a 1, a 2 )κ 1 (b 1 )κ 1 (b 2 ) +κ 1 (a 1 )κ 1 (b 1 )κ 1 (a 2 )κ 1 (b 2 ) 33

36 One random variable and free convolution Consider one random variable a A and define their Cauchy transform G and their R-transform R by G(z) = 1 z + Then we have m=1 ϕ(a m ) z m+1, R(z) = m=1 κ m (a,..., a)z m 1 1 G(z) + R(G(z)) = z R a+b (z) = R a (z) + R b (z) if a and b are free 34

37 Random matrices: a route from classical to free probability Consider unitarily invariant random matrices A = (a ij ) N i,j=1 entries matrix independence between {a ij } and {b kl } freeness between A and B 35

38 Random matrices: a route from classical to free probability Consider unitarily invariant random matrices A = (a ij ) N i,j=1 entries matrix independence between {a ij } and {b kl } classical cumulants c n of a ij freeness between A and B free cumulants κ n of A 36

39 Free cumulants as classical cumulants of cycles Let A = (a ij ) N i,j=1 be a unitarily invariant random matrix. Then we have κ n (A,..., A) = lim N Nn 1 c n (a 12, a 23, a 34,..., a n1 ) 37