Combinatorial Aspects of Free Probability and Free Stochastic Calculus

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1 Combinatorial Aspects of Free Probability and Free Stochastic Calculus Roland Speicher Saarland University Saarbrücken, Germany supported by ERC Advanced Grant Non-Commutative Distributions in Free Probability

2 Part I: Free Probability and Non-Crossing Partitions

3 Some History 1985 Voiculescu introduces freeness in the context of isomorphism problem of free group factors 1991 Voiculescu discovers relation with random matrices (which leads, among others, to deep results on free group factors) 1994 Speicher develops combinatorial theory of freeness, based on free cumulants later... many new results on operator algebras, eigenvalue distribution of random matrices, and much more...

4 Definition of Freeness Let (A, ϕ) be non-commutative probability space, i.e., A is a unital algebra and ϕ : A C is unital linear functional (i.e., ϕ(1) = 1) Unital subalgebras A i (i I) are free or freely independent, if ϕ(a 1 a n ) = 0 whenever a i A j(i), j(i) I i, j(1) j(2) = j(n) ϕ(a i ) = 0 i Random variables x 1,..., x n A are free, if their generated unital subalgebras A i := algebra(1, x i ) are so.

5 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 ) )

6 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 )

7 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 )

8 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 )

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

10 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)

11 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)

12 Where Does Freeness Show Up? generators of the free group in the corresponding free group von Neumann algebras L(F n ) creation and annihilation operators on full Fock spaces for many classes of random matrices

13 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

14 Non-Crossing Partitions A partition of {1,..., n} is a decomposition π = {V 1,..., V r } with V i, V i V j = (i y), The V i are the blocks of π P(n). π is non-crossing if we do not have p 1 < q 1 < p 2 < q 2 i V i = {1,..., n} such that p 1, p 2 are in same block, q 1, q 2 are in same block, but those two blocks are different. NC(n) := {non-crossing partitions of {1,...,n}} N C(n) is actually a lattice with refinement order.

15 Moments and Cumulants For unital linear functional ϕ : A C we define cumulant functionals κ n (for all n 1) κ n : A n C as multi-linear functionals by moment-cumulant relation ϕ(a 1 A n ) = π N C(n) κ π [A 1,..., A n ] Note: classical cumulants are defined by a similar formula, where only NC(n) is replaced by P(n)

16 ϕ(a 1 ) =κ 1 (A 1 ) A 1 ϕ(a 1 A 2 ) = κ 2 (A 1, A 2 ) A 1 A 2 + κ 1 (A 1 )κ 1 (A 2 ) thus κ 2 (A 1, A 2 ) = ϕ(a 1 A 2 ) ϕ(a 1 )ϕ(a 2 )

17 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 )

18 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 )

19 Freeness ˆ= Vanishing of Mixed Cumulants Theorem [Speicher 1994]: The fact that A and B are free is equivalent to the fact that 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

20 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 1A n 2 ) is given by sum over planar diagrams with monochromatic (blue or red) blocks

21 Vanishing of Mixed Cumulants ϕ(abab) = κ 1 (A)κ 1 (A)κ 2 (B, B)+κ 2 (A, A)κ 1 (B)κ 1 (B)+κ 1 (A)κ 1 (B)κ 1 (A)κ 1 (B) A B A B A B A B A B A B

22 Factorization of Non-Crossing Moments The iteration of the rule ϕ(a 1 BA 2 ) = ϕ(a 1 A 2 )ϕ(b) if {A 1, A 2 } and B free leads to the factorization of all non-crossing moments in free variables x 1 x 2 x 3 x 3 x 2 x 4 x 5 x 5 x 2 x 1 ϕ(x 1 x 2 x 3 x 3 x 2 x 4 x 5 x 5 x 2 x 1 ) = ϕ(x 1 x 1 ) ϕ(x 2 x 2 x 2 ) ϕ(x 3 x 3 ) ϕ(x 4 ) ϕ(x 5 x 5 )

23 Sum of Free Variables Consider A, B free. Then, by freeness, the moments of A+B are uniquely determined by the moments of A and the moments of B. Notation: We say the distribution of A + B is the free convolution of the distribution of A and the distribution of B, µ A+B = µ A µ B.

24 Sum of Free Variables In principle, freeness determines this, but the concrete nature of this rule on the level of moments is not apriori clear. Example: ϕ ( (A + B) 1) = ϕ(a) + ϕ(b) ϕ ( (A + B) 2) = ϕ(a 2 ) + 2ϕ(A)ϕ(B) + ϕ(b 2 ) ϕ ( (A + B) 3) = ϕ(a 3 ) + 3ϕ(A 2 )ϕ(b) + 3ϕ(A)ϕ(B 2 ) + ϕ(b 3 ) ϕ ( (A + B) 4) = ϕ(a 4 ) + 4ϕ(A 3 )ϕ(b) + 4ϕ(A 2 )ϕ(b 2 ) + 2 ( ϕ(a 2 )ϕ(b)ϕ(b) + ϕ(a)ϕ(a)ϕ(b 2 ) ϕ(a)ϕ(b)ϕ(a)ϕ(b) ) + 4ϕ(A)ϕ(B 3 ) + ϕ(b 4 )

25 Sum of Free Variables Corresponding rule on level of free cumulants is easy: If A and B are free then κ n (A + B, A + B,..., A + B) =κ n (A, A,..., A) + κ n (B, B,..., B) +κ n (..., A, B,... ) +

26 Sum of Free Variables Corresponding rule on level of free cumulants is easy: If A and B are free then κ n (A + B, A + B,..., A + B) =κ n (A, A,..., A) + κ n (B, B,..., B) +κ n (..., A, B,... ) +

27 Sum of Free Variables Corresponding rule on level of free cumulants is easy: If A and B are free then κ n (A + B, A + B,..., A + B) =κ n (A, A,..., A) + κ n (B, B,..., B) i.e., we have additivity of cumulants for free variables κ A+B n = κ A n + κ B n Also: Combinatorial relation between moments and cumulants can be rewritten easily as a relation between corresponding formal power series.

28 Sum of Free Variables Corresponding rule on level of free cumulants is easy: If A and B are free then κ n (A + B, A + B,..., A + B) =κ n (A, A,..., A) + κ n (B, B,..., B) i.e., we have additivity of cumulants for free variables κ A+B n = κ A n + κ B n Also: Combinatorial relation between moments and cumulants can be rewritten easily as a relation between corresponding formal power series.

29 Relation between Moments and Free Cumulants We have m n := ϕ(a n ) moments and κ n := κ n (A, A,..., A) free cumulants Combinatorially, the relation between them is given by m n = ϕ(a n ) = κ π π N C(n) Example: m 1 = κ 1, m 2 = κ 2 + κ 2 1, m 3 = κ 3 + 3κ 2 κ 1 + κ 3 1

30 m 3 = κ + κ + κ + κ + κ = κ 3 + 3κ 2 κ 1 + κ 3 1 Theorem [Speicher 1994]: Consider formal power series M(z) = 1 + Then the relation k=1 m n z n, C(z) = 1 + m n = κ π π N C(n) k=1 between the coefficients is equivalent to the relation M(z) = C[zM(z)] κ n z n

31 Proof First we get the following recursive relation between cumulants and moments m n = κ π π N C(n) = n s=1 i 1,...,is 0 i 1 + +is+s=n π 1 NC(i 1 ) π s NC(i s ) κ s κ π1 κ πs = n s=1 i 1,...,is 0 i 1 + +is+s=n κ s m i1 m is

32 m n = n s=1 i 1,...,is 0 i 1 + +is+s=n κ s m i1 m is Plugging this into the formal power series M(z) gives M(z) = 1 + n m n z n = 1 + n n s=1 i 1,...,is 0 i 1 + +is+s=n k s z s m i1 z i 1 m is z i s = 1 + s=1 κ s z s( M(z) ) s = C[zM(z)]

33 Remark on Classical Cumulants Classical cumulants c k are combinatorially defined by m n = c π π P(n) In terms of generating power series M(z) = 1 + this is equivalent to n=1 m n n! zn, C(z) = n=1 c n n! zn C(z) = log M(z)

34 From Moment Series to Cauchy Transform Instead of M(z) we consider Cauchy transform 1 G(z) := ϕ( z A ) = 1 z t dµ A(t) = ϕ(a n ) z n+1 and instead of C(z) we consider R-transform = 1 z M(1/z) R(z) := n 0 Then M(z) = C[zM(z)] becomes R[G(z)] + 1 G(z) κ n+1 z n = C(z) 1 z = z or G[R(z) + 1/z] = z

35 Sum of Free Variables Consider a random variable A A and define its Cauchy transform G and its R-transform R by G(z) = 1 z + n=1 ϕ(a n ) z n+1, R(z) = n=1 κ n (A,..., A)z n 1 Theorem [Voiculescu 1986, Speicher 1994]: Then we have 1 G(z) + R(G(z)) = z R A+B (z) = R A (z) + R B (z) if A and B are free

36 What is Advantage of G(z) over M(z)? For any probability measure µ, its Cauchy transform G(z) := 1 z t dµ(t) is an analytic function G : C + C and we can recover µ from G by Stieltjes inversion formula dµ(t) = 1 π lim IG(t + iε)dt ε 0

37 Calculation of Free Convolution The relation between Cauchy transform and R-transform, and the Stieltjes inversion formula give an effective algorithm for calculating free convolutions; and thus also, e.g., the asymptotic eigenvalue distribution of sums of random matrices in generic position: A G A R A B G B R B R A +R B = R A+B G A+B A + B

38 What is the Free Binomial ( 1 2 δ δ ) 2 +1 µ := 1 2 δ δ +1, ν := µ µ Then G µ (z) = 1 z t dµ(t) = 1 2 ( 1 z ) z = z 1 z 2 1 and so z = G µ [R µ (z) + 1/z] = R µ (z) + 1/z (R µ (z) + 1/z) 2 1 thus R µ (z) = 1 + 4z 2 1 2z and so R ν (z) = 2R µ (z) = 1 + 4z 2 1 z

39 R ν (z) = and thus So 1 + 4z 2 1 z dν(t) = 1 π I 1 dt = t 2 4 gives G ν (z) = 1 π 4 t 2, t 2 0, otherwise ( 1 2 δ δ ) 2 +1 = ν = arcsine-distribution 1 z 2 4

40 1/(π (4 x 2 ) 1/2 ) x 2800 eigenvalues of A + UBU, where A and B are diagonal matrices with 1400 eigenvalues +1 and 1400 eigenvalues -1, and U is a randomly chosen unitary matrix

41 Some Literature on Free Probability Fields Institute Monographs Fields Institute Monographs 35 James A. Mingo Roland Speicher Free Probability and Random Matrices Th is volume opens the world of free probability to a wide variety of readers. From its roots in the theory of operator algebras, free probability has intertwined with non-crossing partitions, random matrices, applications in wireless communications, representation theory of large groups, quantum groups, the invariant subspace problem, large deviations, subfactors, and beyond. This book puts a special emphasis on the relation of free probability to random matrices, but also touches upon the operator algebraic, combinatorial, and analytic aspects of the theory. Th e book serves as a combination textbook/research monograph, with self-contained chapters, exercises scattered throughout the text, and coverage of important ongoing progress of the theory. It will appeal to graduate students and all mathematicians interested in random matrices and free probability from the point of view of operator algebras, combinatorics, analytic functions, or applications in engineering and statistical physics. 35 Mingo Speicher 1 Free Probability and Random Matrices The Fields Institute for Research in Mathematical Sciences James A. Mingo Roland Speicher Free Probability and Random Matrices ISBN

42 Part II: Free Brownian Motion and Free Stochastic Analysis

43 What is the free analogue of the normal distribution? The free anlogue of the normal distribution is what we get as limit in a free central limit theorem: Let x 1, x 2,... be free and identically distributed with ϕ(x i ) = 0 and ϕ(x 2 i ) = 1. To what does x x m m converge for m? Denote the limit by s.

44 We have κ n (s,..., s) = lim m κ n( x x m,..., x x m ) m m = lim m 1 m n/2 m i(1),...,i(n)=1 κ n (x i(1),..., x i(n) ) = lim m 1 m n/2 m i=1 κ n (x i,..., x i ) = lim m mn/2 1 κ n (x 1,..., x 1 ) = 0, n 2 1, n = 2

45 µ s has cumulants thus κ n = 0, n 2 1, n = 2 R(z) = n 0 κ n+1 z n = κ 2 z = z and hence z = R[G(z)] + 1 G(z) = G(z) + 1 G(z) or G(z) = zg(z)

46 G(z) = zg(z) thus G(z) = z ± z We have -, because G(z) 1/z for z ; then dµ s (t) = 1 π I( t t 2 4) dt = 1 2π 4 t 2 dt, if t [ 2, 2] 0, otherwise s has a semicircular distribution.

47 Wigner s semicircle law Consider selfadjoint Gaussian N N random matrix one realization... N=4000

48 A free Brownian motion is given by a family ( S(t) ) (A, ϕ) t 0 of random variables (A von Neumann algebra, ϕ faithful trace), such that S(0) = 0 each increment S(t) S(s) (s < t) is semicircular with mean = 0 and variance = t s, i.e., 1 dµ S(t) S(s) (x) = 4(t s) x 2 dx 2π(t s) disjoint increments are free: for 0 < t 1 < t 2 < < t n, S(t 1 ), S(t 2 ) S(t 1 ),..., S(t n ) S(t n 1 ) are free

49 A free Brownian motion is given abstractly, by a family ( S(t) ) t 0 of random variables with S(0) = 0 each S(t) S(s) (s < t) is (0, t s)-semicircular disjoint increments are free concretely, by the sum of creation and annihilation operators on the full Fock space asymptotically, as the limit of matrix-valued (Dyson) Brownian motions

50 Free Brownian motions as matrix limits Let (X N (t)) t 0 be a symmetric N N-matrix-valued Brownian motion, i.e., X N (t) = B 11 (t)... B 1N (t)..... B N1 (t)... B NN (t), where B ij are, for i j, independent classical Brownian motions B ij (t) = B ji (t). Then, ( X N (t) ) distr t 0 ( S(t) ) t 0, in the sense that almost surely lim N tr( X N (t 1 ) X N (t n ) ) = ϕ ( S(t 1 ) S(t n ) ) 0 t 1, t 2,..., t n

51 Intermezzo on realisations on Fock spaces Classical Brownian motion can be realized quite canonically by operators on the symmetric Fock space. Similarly, free Brownian motion can be realized quite canonically by operators on the full Fock space

52 First: symmetric Fock space... For real Hilbert space H R put H := H R + ih R and F s (H) := n 0 H symn, where H 0 = CΩ with inner product f 1 f n, g 1 g m sym = δ nm n π S n i=1 Define creation and annihilation operators (for f H) a (f)f 1 f n = f f 1 f n a(f)f 1 f n = a(f)ω = 0 n i=1 f i, g π(i). f, f i f 1 ˇf i f n

53 ... and classical Brownian motion Put ϕ( ) := Ω, Ω, x(f) := a(f) + a (f) then (x(f)) f HR is Gaussian family with covariance ϕ ( x(f)x(g) ) = f, g. In particular, choose H := L 2 (R + ), f t := 1 [0,t), then B t := x(f t ) = a(1 [0,t) ) + a (1 [0,t) ) is classical Brownian motion W t, meaning ϕ(b t1 B tn ) = E[W t1 W tn ] 0 t 1,..., t n

54 Now: full Fock space... For real Hilbert space H R put H := H R + ih R and F(H) := n 0 H n, where H 0 = CΩ with usual inner product f 1 f n, g 1 g m = δ nm f 1, g 1 f n, g n. Define creation and annihilation operators (for f H) b (f)f 1 f n = f f 1 f n b(f)f 1 f n = f, f 1 f 2 f n b(f)ω = 0

55 ... and free Brownian motion Put ϕ( ) := Ω, Ω, x(f) := b(f) + b (f) then (x(f)) f HR is semicircular family with covariance ϕ ( x(f)x(g) ) = f, g. In particular, choose H := L 2 (R + ), f t := 1 [0,t), then S t := x(f t ) = b(1 [0,t) ) + b (1 [0,t) ) is free Brownian motion.

56 Semicircle as real part of one-sided shift Consider case of one-dimensional H = Cv. Then this reduces to orthonormal basis for F(H): Ω = e 0, e 1, e 2, e 3,... and one-sided shift l = b (v) le n = e n+1, l e n = e n 1, n 1 0, n = 0 One-sided shift is canonical non-unitary isometry l l = 1, ll 1 (= 1 projection on Ω) With ϕ(a) := Ω, aω we claim: distribution of l+l is semicircle.

57 Moments of l + l In the calculation of Ω, (l + l ) n Ω only such products in creation and annihilation contribute, where we never annihilate the vacuum, and where we start and end at the vacuum. So odd moments are zero. Examples ϕ ( (l + l ) 2) : l l ϕ ( (l + l ) 4) : l l ll, l ll l ϕ ( (l + l ) 6) : l l l lll, l ll l ll, l ll ll l, l l ll ll, l l lll l Those contributing terms are in clear bijection with non-crossing pairings (or with Dyck paths).

58 (l, l, l, l, l, l) (l, l, l, l, l, l) (l, l, l, l, l, l) (l, l, l, l, l, l) (l, l, l, l, l, l)

59 Some History on Free Stochastic Analysis 1992 stochastic integration by Kümmerer and Speicher with respect to creation and annihilation operators on full Fock space (in analogy with theory of Hudson/Pathasarathy on symmetric Fock space) 1998 improved theory of stochastic integration (and free Malliavin calculus) for free Brownian motion by Biane and Speicher 2012 free version of the fourth moment theorem by Kemp, Nourdin, Peccati, Speicher 2013 rough-paths approach to non-commutative stochastic calculus, in particular to free stochastic calculus, by Deya and Schott

60 Stochastic Analysis on Wigner space we define mul- Starting from a free Brownian motion ( S(t) ) t 0 tiple Wigner integrals I(f) = f(t 1,..., t n )ds(t 1 )... ds(t n ) for scalar-valued functions f L 2 (R n + ), by avoiding the diagonals, i.e. we understand this as I(f) = all t i distinct f(t 1,..., t n )ds(t 1 )... ds(t n )

61 Definition of Wigner integrals More precisely: for f of form f = 1 [s1,t 1 ] [s n,t n ] for pairwisely disjoint intervals [s 1, t 1 ],..., [s n, t n ] we put I(f) := (S t1 S s1 ) (S tn S sn ) Extend I( ) linearly over set of all off-diagonal step functions (which is dense in L 2 (R n + ). Then observe Ito-isometry ϕ[i(g) I(f)] = f, g L 2 (R n + ) and extend I to closure of off-diagonal step functions, i.e., to L 2 (R n + ).

62 Note: free stochastic integrals are usually bounded operators Free Haagerup Inequality [Bozejko 1991; Biane, Speicher 1998]: f(t 1,..., t n )ds(t 1 )... ds(t n ) (n + 1) f L 2 (R n + )

63 Intermezzo: combinatorics and norms Consider free semicirculars s 1, s 2,... of variance 1. Since (s s n )/ n is again a semicircular element of variance 1 (and thus of norm 2), we have 1 n n k/2 i(1),...,i(k)=1 s i(1) s i(k) = ( s1 + + c n n ) k = 2k The free Haagerup inequality says that this is drastically reduced if we subtract the diagonals, i.e., lim n 1 n k/2 n i(1),...,i(k)=1 all i(.) different s i(1) s i(k) = k + 1

64 Intermezzo: combinatorics and norms Note: one can calculate norms from asymptotic knowledge of moments! If x is selfadjoint and ϕ faithful (as our ϕ for the free Brownian motion is) then one has x = lim p x p = lim p p ϕ( x p ) = So, if s is a semicircular element, then thus lim m ϕ(s 2m ) = c m = 1 ( 2m) 4 m, 1 + m m 2m ϕ(x 2m ) s = lim m 2m c m 2m 4 m = 2

65 Exercise: combinatorics and norms Consider free semicirculars s 1, s 2,... of variance 1. Prove by considering moments that 1 n n i,j=1 s i s j = 4, lim n 1 n n i,j=1,i j s i s j = 3 Realize that the involved NC pairings are in bijection with NC partitions and NC partitions without singletons, respectively.

66 2000 eigenvalues of the matrix i,j=1 X i X j, i,j=1,i j X i X j, for X 1,..., X 50 independent GUE.

67 Multiplication of Multiple Wigner Integrals The Ito formula shows up in multiplicaton of two multiple Wigner integrals f(t 1 )ds(t 1 ) g(t 2 )ds(t 2 ) = f(t 1 )g(t 2 )ds(t 1 )ds(t 2 ) + f(t)g(t) ds(t)ds(t) }{{} dt = f(t 1 )g(t 2 )ds(t 1 )ds(t 2 ) + f(t)g(t)dt

68 Multiplication of Multiple Wigner Integrals The Ito formula shows up in multiplicaton of two multiple Wigner integrals f(t 1 )ds(t 1 ) g(t 2 )ds(t 2 ) = f(t 1 )g(t 2 )ds(t 1 )ds(t 2 ) + f(t)g(t) ds(t)ds(t) }{{} dt = f(t 1 )g(t 2 )ds(t 1 )ds(t 2 ) + f(t)g(t)dt

69 Multiplication of Multiple Wigner Integrals The Ito formula shows up in multiplicaton of two multiple Wigner integrals f(t 1 )ds(t 1 ) g(t 2 )ds(t 2 ) = f(t 1 )g(t 2 )ds(t 1 )ds(t 2 ) + f(t)g(t) ds(t)ds(t) }{{} dt = f(t 1 )g(t 2 )ds(t 1 )ds(t 2 ) + f(t)g(t)dt

70 Multiplication of Multiple Wigner Integrals f(t 1, t 2 )ds(t 1 )ds(t 2 ) g(t 3 )ds(t 3 ) = f(t 1, t 2 )g(t 3 )ds(t 1 )ds(t 2 )ds(t 3 ) + + f(t 1, t)g(t)ds(t 1 ) ds(t)ds(t) }{{} dt f(t, t 2 )g(t) ds(t)ds(t 2 )ds(t) }{{} dtϕ[ds(t 2 )]=0

71 Multiplication of Multiple Wigner Integrals f(t 1, t 2 )ds(t 1 )ds(t 2 ) g(t 3 )ds(t 3 ) = f(t 1, t 2 )g(t 3 )ds(t 1 )ds(t 2 )ds(t 3 ) + + f(t 1, t)g(t)ds(t 1 ) ds(t)ds(t) }{{} dt f(t, t 2 )g(t) ds(t)ds(t 2 )ds(t) }{{} dtϕ[ds(t 2 )]=0

72 Multiplication of Multiple Wigner Integrals Free Ito Formula [Biane, Speicher 1998]: ds(t)ads(t) = ϕ(a)dt for A adapted

73 Multiplication of Multiple Wigner Integrals Consider f L 2 (R n + ), g L2 (R m + ) For 0 p min(n, m), define by f p g(t 1,..., t m+n 2p ) f p g L 2 (R n+m 2p + ) = f(t 1,..., t n p, s 1,..., s p )g(s p,..., s 1, t n p+1,..., t n+m 2p )ds 1 ds p Then we have I(f) I(g) = min(n,m) p=0 I(f p g)

74 Example: f L 2 (R 3 + ), g L2 (R 4 + ) I(f) I(g) = f(t 1, t 2, t 3 )ds(t 1 )ds(t 2 )ds(t 3 ) g(s 1, s 2, s 3, s 4 )ds(s 1 )... ds(s 4 ) = = = I(f 0 g) + I(f 1 g) + I(f 2 g) + I(f 3 g)

75 1 [0,1] (t)ds(t) = S(1) S(0) = S semicircular variable What is U n := 1 [0,1] n(t 1,..., t n )ds(t 1 ) ds(t n ) We have S U n = = + Thus S U n = U n+1 + U n 1 = U n+1 + U n 1 recursion for Chebycheff polynomials U 1 = S, U 2 = S 2 1, U 3 = S 3 2S,...

76 Note U n = S n + smaller degree polynomial In this special case, Haagerup inequality is saying that is reduced to For example, S n = 2 n U n = n + 1 S = 2, S 2 = 4, S 3 = 8 S = 2, S 2 1 = 3, S 3 2S = 4

77 Note U n = S n + smaller degree polynomial In this special case, Haagerup inequality is saying that is reduced to For example, S n = 2 n U n = n + 1 S = 2, S 2 = 4, S 3 = 8 S = 2, S 2 1 = 3, S 3 2S = 4 This follows here from U n = sup U n (t) U n (cos θ) = t 2 sin(n + 1)θ sin θ

78 Compare to classical analogue 1 [0,1] (t)db(t) = B(1) B(0) = N normal variable H n := 1 [0,1] n(t 1,..., t n )db(t 1 ) db(t n ) We have N H n = = = H n+1 + nh n 1 N H n = H n+1 + nh n 1 recursion for Hermite polynomials H 1 = N, H 2 = N 2 1, H 3 = N 3 3N,...

79 Note that for n 1: ϕ[i(f)] = f(t 1,..., t n )ϕ[ds(t 1 ) ds(t n )] = 0 Thus, for f i L 2 (R n i + ) ϕ[i(f 1 ) I(f r )] = only terms with total contractions = π NC 2 (n 1 n r ) π f 1 f r

80 Example: I(f 1 )I(f 2 )I(f 3 )I(f 4 )I(f 5 ) with (n 1, n 2, n 3, n 4, n 5 ) = (4, 3, 1, 2, 2) The first picture contributes in the classical case, but not in the free case. The other two pictures contribute here. More general: all π NC(n n r ) contribute with π {[n 1 ], [n 1, n 1 + n 2 ],... } = 0

81 In particular, we have for f, g L 2 (R n + ) ϕ[i(f)i(g) ] = = f(t 1,..., f n )ḡ(s 1,..., s n )ϕ[ds(t 1 )... ds(t n ) ds(s n )... ds(s 1 )] f(t 1,..., t n )ḡ(t 1,..., t n )dt 1... dt n = f, g L 2 (R n + ) and for f L 2 (R n + ) and g L2 (R m + ) with n m ϕ[i(g)i(g) ] = 0 or more general, for f, g n=0 L 2 (R n + ) ϕ[i(f)i(g) ] = f, g

82 Free Chaos Decomposition One has the canonical isomorphism via L 2 ({S(t) t 0}) ˆ= f = n=0 I(f n ) = n=0 n=0 L 2 (R n + ), f ˆ= n=0 f n, f n (t 1,..., t n )ds(t 1 )... ds(t n ). The set {I(f n ) f n L 2 (R n + )} is called n-th (Wigner) chaos.

83 Theorem [Kemp, Nourdin, Peccati, Speicher 2012]: Consider, for fixed n, a sequence f 1, f 2, L 2 (R n + ) with f k = f k and f k 2 = 1 for all k N. Then the following statements are equivalent. (i) We have lim k ϕ[i(f k ) 4 ] = 2. (ii) We have for all p = 1, 2,..., n 1 that lim k f k p f k = 0 in L 2 (R 2n 2p + ). (iii) The selfadjoint variable I(f k ) converges in distribution to a semicircular variable of variance 1.

84 Corollary: semicircular For n 2 and f L 2 (R n + ), the law of I(f) is not Thus for n 2 {distributions in first chaos} {distribution in n-th chaos} = The more general question for n m {distributions in m-th chaos} {distribution in n-th chaos} =??? is still open. [For the classical case one knows that all Wiener chaoses have disjoint distributions.]

85 Idea of proof of 4 th moment theorem I(f k ) s implies ϕ[i(f k ) 4 ] ϕ[s 4 ] = 2 is clear for the other direction first show: convergence of fourth moment implies vanishing of non-trivial contractions then show: if non-trivial contractions are zero, then all moments calculate as the moments for a semicircular

86 Vanishing of contractions I(f) = I(f ) where f (t 1,..., f n ) := f(t n,..., t 1 ) Then I(f)I(f ) = n p=0 I(f p f ) Note: f p f L 2 (R 2n 2p + ), thus terms for different p are orthogonal in L 2 and thus ϕ[ I(f) 4 ] = ϕ[(i(f)i(f )) 2 ] = n p=0 ϕ[i(f p f ) 2 ]

87 But for the two trivial contractions p = 0 and p = n we have I(f 0 f ) = I(f f ), thus ϕ[i(f 0 f ) 2 ] = f f 2 L 2 = 1 and and so f n f = f 2 = 1, thus ϕ[i(f n f ) 2 ] = 1 ϕ[ I(f) 4 ] = n p=0 ϕ[i(f p f ) 2 ] = 2 + n 1 p=1 ϕ[i(f p f ) 2 ] Thus: if ϕ[ I(f) 4 ] = 2, then f p f = 0 for all p = 1,..., n 1 [Note: all this cannot happen for one f if n > 1, but all arguments are also valid in the limit k for sequences f k ]

88 But for the two trivial contractions p = 0 and p = n we have I(f 0 f ) = I(f f ), thus ϕ[i(f 0 f ) 2 ] = f f 2 L 2 = 1 and and so f n f = f 2 = 1, thus ϕ[i(f n f ) 2 ] = 1 ϕ[ I(f) 4 ] = n p=0 ϕ[i(f p f ) 2 ] = 2 + n 1 p=1 ϕ[i(f p f ) 2 ] Thus: if ϕ[ I(f) 4 ] = 2, then f p f = 0 for all p = 1,..., n 1 [Note: all this cannot happen for one f if n > 1, but all arguments are also valid in the limit k for sequences f k ]

89 Calculation of higher moments The vanishing of non-trivial contractions implies that in the calculation of ϕ[i(f) r ] = π NC 2 (n r) π f r contractions between p arguments of two of the involved f must be trivial, i.e, either no contractions at all (p = 0), or a total contraction (p = n) But such contractions are in bijection with non-crossing pairings of r elements, i.e., we get ϕ[i(f) r ] = #NC 2 (r) = ϕ(s r )

90 Results on Regularity of Distributions Theorem [Shlyakhtenko, Skoufranis 2013; Mai, Speicher, Weber 2014]: Let p be a non-constant selfadjoint polynomial and s 1,..., s n free semicirculars. Then the distribution of p(s 1,..., s n ) does not have atoms. Theorem [Mai 2015]: The distribution of a non-constant finite selfadjoint Wigner integral n k=1 does not have atoms. f k (t 1,..., t k )ds(t 1, ) ds(t k )

91 Idea of Proofs The results of Mai, Speicher, Weber rely on having a calculus of non-commutative derivatives for our polynomials: having atoms for some polynomials implies by differentiation that one also has atoms for the derivative but then, by iteration, one should have atoms for linear polynomials which is not the case The result of Mai on stochastic integrals relies on a version of such a differential calculus in the setting of stochastic integrals... this is the free Malliavin calculus

92 Some literature on free stochastic analysis Biane, Speicher: Stochastic Calculus with Respect to Free Brownian Motion and Analysis on Wigner Space. Prob. Theory Rel. Fields, 1998 Kemp, Nourdin, Peccati, Speicher: Wigner Chaos and the Fourth Moment. Ann. Prob., 2012 Nourdin, Peccati, Speicher: Multidimensional Semicircular Limits on the Free Wigner Chaos. Seminar on Stochastic Analysis, Random Fields and Applications VII, 2013

93 Afterword on Quantitative Estimates... Given two self-adjoint random variables X, Y, define the distance where d C2 (X, Y ) := sup{ ϕ[h(x)] ϕ[h(y )] : I 2 (h) 1}; I 2 (h) ˆ= h and X n = n 1 k=0 X k X n 1 k Rigorously: If h is the Fourier transform of a complex measure ν on R, then we define h(x) = ν(x) = I 2 (h) = R eixξ ν(dξ) R ξ2 ν (dξ)

94 Afterword on Quantitative Estimates... Given two self-adjoint random variables X, Y, define the distance where d C2 (X, Y ) := sup{ ϕ[h(x)] ϕ[h(y )] : I 2 (h) 1}; I 2 (h) ˆ= h and X n = n 1 k=0 X k X n 1 k Rigorously: If h is the Fourier transform of a complex measure ν on R, then we define h(x) = ν(x) = I 2 (h) = R eixξ ν(dξ) R ξ2 ν (dξ)

95 ... in Terms of the Free Gradient Operator Define the free Malliavin gradient operator by ( ) t f(t 1,..., t n ) ds t1 ds tn := n k=1 f(t 1,..., t k 1, t, t k+1,..., t n ) ds t1 ds tk 1 ds tk+1 ds tn Theorem (Kemp, Nourdin, Peccati, Speicher): d C2 (F, S) 1 ( 2 ϕ ϕ s (N 1 F ) ( s F ) ds 1 1 ) But no estimate against Fourth Moment in this case!

96 ... in Terms of the Free Gradient Operator Define the free Malliavin gradient operator by ( ) t f(t 1,..., t n ) ds t1 ds tn := n k=1 f(t 1,..., t k 1, t, t k+1,..., t n ) ds t1 ds tk 1 ds tk+1 ds tn Theorem [Kemp, Nourdin, Peccati, Speicher 2012]: d C2 (F, S) 1 ( 2 ϕ ϕ s (N 1 F ) ( s F ) ds 1 1 ) But no estimate against Fourth Moment in this case!

97 ... in Terms of the Free Gradient Operator Define the free Malliavin gradient operator by ( ) t f(t 1,..., t n ) ds t1 ds tn := n k=1 f(t 1,..., t k 1, t, t k+1,..., t n ) ds t1 ds tk 1 ds tk+1 ds tn Theorem [Kemp, Nourdin, Peccati, Speicher 2012]: d C2 (F, S) 1 ( 2 ϕ ϕ s (N 1 F ) ( s F ) ds 1 1 ) But no estimate against Fourth Moment in this case!

98 In the classical case one can estimate the corresponding expression of the above gradient, for F living in some n-th chaos, in terms of the fourth moment of the considered variable, thus giving a quantitative estimate for the distance between the considered variable (from a fixed chaos) and a normal variable in terms of the difference between their fourth moments. In the free case such a general estimate does not seem to exist; at the moment we are only able to do this for elements F from the second chaos. Corollary: Let F = I(f) = I(f) (f L 2 (R 2 + )) be an element from the second chaos with variance 1, i.e., f 2 = 1, and let S be a semiciruclar variable with mean 0 and variance 1. Then we have d C2 (F, S) 1 3 ϕ(f 4 )

99 Some general literature D. Voiculescu, K. Dykema, A. Nica: Free Random Variables. CRM Monograph Series, Vol. 1, AMS 1992 F. Hiai, D. Petz: The Semicircle Law, Free Random Variables and Entropy. Math. Surveys and Monogr. 77, AMS 2000 A. Nica, R. Speicher: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, 2006 J. Mingo, R. Speicher: Free Probability and Random Matrices. Fields Institute Monographs, vol. 35, Springer, 2017