From random matrices to free groups, through non-crossing partitions. Michael Anshelevich

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1 From random matrices to free groups, through non-crossing partitions Michael Anshelevich March 4, 22

2 RANDOM MATRICES For each N, A (N), B (N) = independent N N symmetric Gaussian random matrices, i.e. A (N) ij = A (N) ji, otherwise independent Gaussian N (, N ). Arise in: nuclear physics, quantum chaos, communication theory, string theory. Want joint moments N Tr[p (A (N) )q (B (N) )... p k (A (N) )q k (B (N) )]. Example. M (N) = N Tr[(A(N) ) 2 B (N) A (N) B (N) ]. Note: this is a random variable.

3 Theorem. (Wigner 5) N Tr[(A(N) ) k ] N x k dσ(x) the moments of the semicircular distribution dσ(x) = 4 x 2 2π [ 2,2] (x)dx. Theorem. (Voiculescu 9) If p i (x)dσ(x) =, qi (x)dσ(x) =, then N Tr[p (A (N) )q (B (N) )... p k (A (N) )q k (B (N) )] N. Note this is enough to find all moments, e.g. M (N) N Tr[((A(N) ) 2 )B (N) A (N) B (N) ] + N Tr[] N Tr[B(N) A (N) B (N) ]. 2

4 FREE GROUPS F 2 = free group on 2 generators {a, b} = all words in a, b, a, b with cancellations. L 2 (F 2 ) = { f : x F 2 f(x) 2 < }, the Hilbert space of all square-integrable functions on F 2. Each x F 2 acts on L 2 (F 2 ) by (S x (f))(y) = f(xy). L(F 2 ) = von Neumann algebra generated by all S x. It has a state ϕ[s] = Sδ e, δ e ( = f(e) ). If ϕ[p i (a)] =, ϕ[q i (b)] =, then ϕ[p (a)q (b)... p k (a)q k (b)] =. 3

5 FREE PROBABILITY THEORY Voiculescu ( 8): this says a, b are freely independent. Large independent random matrices are asymptotically freely independent. Free probability: a non-commutative probability theory, with independence replaced by free independence. Lives in the large N limit, but not only there. Many statements from probability theory have free analogs. Theorem (Free central limit theorem). Let X, X 2,..., X n be freely independent with respect to a state ϕ, and have mean and variance. Then X X n n d σ. 4

6 Why semicircle appears in both contexts: +...+A n (N) n A (N)... A (N) n A(N) X... X n X +...+X n n Have analogs of Infinitely divisible distributions Convolution and harmonic analysis Entropy Connections to: combinatorics, representation theory, orthogonal polynomials, Yang-Mills theory, etc. 5

7 Another way to look at the CLT: as a fixed point theorem. Corresponding to X +X 2 2, have the operator C : µ (µ µ) S / 2. σ is an attracting fixed point for it. Behavior in the neighborhood of the fixed point (M.A. 99): C is a non-linear operator. Its derivative is compact, with eigenfunctions T n, eigenvalues 2 n/2. Here T n = Chebyshev polynomials of the st kind. Have similar results for other free convolution semigroups (M.A. 2). 6

8 Most importantly, applications to the theory of von Neumann algebras. Example: a von Neumann algebra is prime is A = B C for any infinite-dimensional B, C. A definition in search of an example. Why rare: A = A A common. For A = lim M n, M n M n = Mn 2 A A = A II -factors: von Neumann algebras with a trace and a trivial center ( simple). First known example of a prime II -factor: L(F n ). Still know little about L(F n ), for example do not know if L(F 2 ) = L(F 3 ) (Kadison 6). Know that either all L(F r ) = L(F s ) for all r, s R +, or they are all non-isomorphic. 7

9 NON-CROSSING PARTITIONS Free independence hard to use for calculations, e.g. ϕ[ab 2 aba] =? Speicher 9: ϕ[ ] = π NC (n) NC (n) = non-crossing partitions. R π. R = free cumulant functional,r π = B π R B. For example, R (,5)(2,3)(4) (X X 2 X 3 X 4 X 5 ) = R(X X 5 )R(X 2 X 3 )R(X 4 ). 8

10 Here r k = R(a k ) = free cumulants of a. Why simplifies calculations: R(free variables) =. This implies the free independence property. ϕ[a b a 2 b 2... a n b n ] = π NC (2n) R π (a b a 2 b 2... a n b n ). π connect only a s to a s, b s to b s π contains a singleton. Each a, b centered each term is. Example. ϕ[ab 2 aba] = R(a) 3 ϕ[b 3 ] + R(a 2 )R(a)ϕ[b 2 ]ϕ[b] + R(a 3 )ϕ[b 2 ]ϕ[b] 9

11 Non-crossing partitions and random matrices: N E[Tr[A4 ]] = N i,j,k,l E[A ij A jk A kl A li ] Gaussian matrices: all moments expressed through the 2nd order moments. E[ i,j,k + i,j,k A ij A ji A ik A ki A ij A jk A kj A ji + i,j A ii A ii A ij A ji + more terms]

12 Another appearance of non-crossing partitions. Let {X(t)} be a process. Let π be any partition, for example π = (, 3, 5)(2, 4, 6) The corresponding stochastic measure is St π (t) = [,t) 2 dx(s )dx(s 2 )dx(s )dx(s 2 )dx(s )dx(s 2 ). Defined and investigated by Rota and Wallstrom 97 for {X(t)} a (classical) Lévy process.

13 For scalar-valued measures, dµ(s)dν(t) [,) 2 = µ([, t))dν(t) + ν([, s))dµ(s) (+ dµ(s)dν(s)). If X(t) are operators, dx dx. For example, ( ) ( ) (( ) ( ) ( )) = + + (( ) ( ) ( )) + + ( ) ( ) ) ( ) = +( ( ) ( ) ) ( ) +( +( ( ) ( ) ) ( ) ) ( ) +( +( +( ( X(s)dX(s)) dx(s)x(s)) dx(s)dx(s)) 2

14 Instead, let {X(t)} be a bounded free Lévy process, i.e. a stationary (operator-valued) process with freely independent increments. Theorem. (M.A. ) St π are well-defined. Moreover, St π = unless π is non-crossing. Why want St π : can write products of multiple integrals as sums of integrals with respect to stochastic measures (Itô formulas). Example. ( dx(s)dx(t)) ( dx(u)dx(v)) = St ()(2)(3)(4) + St (,3)(2)(4) + St ()(2,4)(3). 3

15 Relation to free cumulants: R π = ϕ[st π ]. In fact (M.A. ) St π can be expressed through simple multiple integrals and the free cumulants dependent on the inner classes of the non-crossing partition π. Example. St ()(2,6)(3,4)(5)(7,8) (t) = [,t) 5 dx(s )d 2 (s 2 )r 2 ds 3 r ds 4 d 2 (s 5 ), where k (t) = t (dx(s)) k are the higher diagonal measures (higher variations). 4

16 Other projects: Relate the linearization around the fixed point results to fluctuation results for random matrices. Develop stochastic integration with respect to free processes. Partially done (M.A. 2), need more machinery, martingale inequalities etc. What algebras do these processes generate? (Free n-dimensional Brownian motion generates L(F n )). q-interpolations between the free and the classical world. Free corresponds to q =, while q = is symmetric and q = is anti-symmetric. Partially done: q-lévy processes (M.A. ). Have an interesting relation to the theory of orthogonal polynomials in the free case (M.A. 2). Would like such a relation for other q. Relate the free Lévy process results to the representation theory of S (Biane). 5