Limit Laws for Random Matrices from Traffic Probability

Transcription

1 Limit Laws for Random Matrices from Traffic Probability arxiv: Slides available at math.berkeley.edu/ bensonau Benson Au UC Berkeley May 9th, 2016 Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

2 Wigner s Semicircle Law Definition (Wigner matrix) Let (W i,j ) 1 i<j< and (W i,i ) 1 i< be independent families of i.i.d. real-valued random variables such that EW 1,2 = 0, Var(W 1,2 ) = 1, and Var(W 1,1 ) <. We call the random real symmetric n n matrix W n defined by { Wi,j / n if i < j W n (i, j) = W i,i / n if i = j a Wigner matrix. Theorem (Wigner, 1955) The empirical spectral distributions (ESDs) µ(w n ) converge weakly almost surely to the standard semicircle distribution SC(0, 1), where SC(m, σ 2 )(dx) = 1 2πσ 2 (4σ2 (x m) 2 ) 1/2 + dx. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

3 Wigner s Semicircle Law Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

4 Non-commutative Probability Consider first the case of a usual measurable space (Ω, F). For a given probability measure P on (Ω, F), we may form the (commutative) -algebra L (Ω, F, P) of measurable complex-valued functions having finite moments of all orders, i.e., L (Ω, F, P) = L p (Ω, F, P). The expectation E : L (Ω, F, P) C recovers the probability measure P; thus, the passage from the probability space (Ω, F, P) = ((Ω, F), P) to the pair (L (Ω, F, P), E) involves no loss of information. p=1 Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

5 Non-commutative Probability Definition (Non-commutative probability space) A non-commutative probability space is a pair (A, ϕ) consisting of a unital algebra A over C equipped with a unital linear functional ϕ : A C. Examples (L (Ω, F, P), E) (M n (C), 1 n tr) (M n (L (Ω, F, P)), E 1 n tr) (C[G], τ G ) Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

6 Notions of Independence For a collection of random variables S A, we write S for the subset (possibly empty) of centered random variables in S. Definition (Classical independence) Subalgebras (A i ) i I are classically independent if the A i commute and ϕ is multiplicative across the A i in the following sense: for all k 1 and distinct indices i(1),..., i(k) I, ( k ) ϕ a i(j) = j=1 k ϕ(a i(j) ), a i(j) A i(j), (1) j=1 or, equivalently, ( k ) ϕ a i(j) j=1 = 0, a i(j) A i(j). (2) Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

7 Notions of Independence Voiculescu: What is free probability theory? It is not a euphemism for the advocacy of an unconstrained attitude in the practice of probability. It can rather be described by the exact formula free probability = non-commutative probability theory+free independence. Definition (Free independence) Subalgebras (A i ) i I are freely independent if for all k 1 and consecutively distinct indices i(1) i(2) i(k) I, ( k ) ϕ a i(j) j=1 = 0, a i(j) A i(j). (3) Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

8 Non-commutative Central Limit Theorems (CLTs) Theorem (de Moivre, 1733; Voiculescu, 1985) Let (a n ) be a sequence of identically distributed self-adjoint random variables in a -probability space (A, ϕ). Assume that the a n are centered with unit variance, i.e., ϕ(a n ) = 0 with ϕ(an) 2 = 1, and write s n = 1 n n j=1 a j. We consider two cases. (i) If the a n are classically independent, then (s n ) converges in distribution to a standard normal random variable, i.e., lim n ϕ(sm n ) = t m 1 e t2 /2 dt, m N. 2π R (ii) If the a n are freely independent, then (s n ) converges in distribution to a standard semicircular random variable, i.e., 2 lim n ϕ(sm n ) = t m 1 4 t 2 2π 2 dt, m N. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

9 Limit Laws for Random Matrices from Free Probability Theorem (Voiculescu, 1991; Dykema, 1993) With the appropriate moment assumptions, independent Wigner matrices are asymptotically freely independent. Corollary The ESDs µ(w n ) converge in expectation to the standard semicircle distribution SC(0, 1). Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

10 Random Markov Matrices Definition (Markov matrix) Let W n be a Wigner matrix, and let D n be the diagonal matrix of row sums of W n, i.e., D n (i, i) = n W n (i, j) = j=1 n W i,j / n. We call the random real symmetric n n matrix M n defined by j=1 a Markov matrix. M n = W n D n Theorem (Bryc, Dembo, and Jiang, 2006) The ESDs µ(m n ) converge weakly almost surely to the free convolution N (0, 1) SC(0, 1). Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

11 n W 1,j W 1,2 W 1,3 W 1,n j 1 n W 2,1 W 2,j W 2,3 W 2,n j 2. M n = 1 W 3,1 W.. 3,2. n.. n W k,1 W k,2 W k,j W k,n j k n W n,1 W n,2 j n W n,j Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

12 Generalized Notions of Independence Do there exist other notions of independence in the non-commutative probabilistic setting? Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

13 Generalized Notions of Independence Do there exist other notions of independence in the non-commutative probabilistic setting? Theorem (Speicher, 1997) No. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

14 -graph polynomials We write x = (x i ) i I for a set of indeterminates. We implicitly assume an associated set of indeterminates x = (x i ) i I such that (x, x ) = (x i, x i ) i I gives a set of pairwise distinct indeterminates satisfying the natural -relation. Definition ( -graph monomial) A -graph monomial in the indeterminates x is a finite, connected, bi-rooted -graph in x. We denote the set of -graph monomials in x by G x, x. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

15 -graph polynomials Example in x 2 x 1 x x3 x1 2 x 1 out Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

16 -graph polynomials Definition ( -graph polynomials) We write CG x, x for the complex vector space of finite linear combinations in G x, x, the elements of which we call the -graph polynomials. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

17 -graph polynomials Example = Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

18 -graph polynomials Example = Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

19 -graph polynomials Example ( x 1 ) x2 = x 3 x 1 x 2 x 3 Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

20 -graph polynomials Example ( x 1 ) x2 = x 3 x 3 x 2 x 1 Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

21 Traffic Probability Definition (Traffic space) A traffic space is a tracial -probability space (A, ϕ) such that A is an algebra over the symmetric operad of -graph polynomials. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

22 Traffic Probability Definition (Traffic space) A traffic space is a tracial -probability space (A, ϕ) such that A is an algebra over the symmetric operad of -graph polynomials. with the additional structure to evaluate -graph polynomials in the random variables a A. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

23 Traffic Probability Definition (Traffic space) A traffic space is a tracial -probability space (A, ϕ) such that A is an algebra over the symmetric operad of -graph polynomials. with the additional structure to evaluate -graph polynomials in the random variables a A. Example Let A = (A i ) i I be a family of random n n matrices. For a -graph monomial t = (T, v in, v out ) = (V, E, f, g, γ, ε, v in, v out ) in x = (x i ) i I, we define t(a) to be the random n n matrix with entries t(a)(i, j) = φ:v [n] φ(v in )=i, φ(v out)=j e E A ε(e) γ(e) (φ(f (e)), φ(g(e))) Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

24 Traffic Probability Example (A 1, A 2 ) x 1 x 2 Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

25 Traffic Probability Example (A 1, A 2 ) x 1 x 2 = A 1 A 2 Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

26 Traffic Probability Example x 1 (A 1, A 2 ) x 2 Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

27 Traffic Probability Example x 1 ) (A 1, A 2 = A 1 A 2 x 2 Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

28 Traffic Probability Example x (A) = row(a) Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

29 CLTs Revisited Theorem (de Moivre, 1733; Voiculescu, 1985) Let (a n ) be a sequence of identically distributed self-adjoint random variables in a -probability space (A, ϕ). Assume that the a n are centered with unit variance, i.e., ϕ(a n ) = 0 with ϕ(an) 2 = 1, and write s n = 1 n n j=1 a j. We consider two cases. (i) If the a n are classically independent, then (s n ) converges in distribution to a standard normal random variable, i.e., lim n ϕ(sm n ) = t m 1 e t2 /2 dt, m N. 2π R (ii) If the a n are freely independent, then (s n ) converges in distribution to a standard semicircular random variable, i.e., 2 lim n ϕ(sm n ) = t m 1 4 t 2 2π 2 dt, m N. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

30 CLTs Revisited Theorem (Male, 2012) (iii) Assume that A has the additional structure of a traffic space (A, ϕ, τ). We split the variance of a n as 1 = ϕ(a 2 n) = τ [ T 1 (a n ) ] = τ 0[ T 1 (a n ) ] + τ 0[ T 2 (a n ) ] = α + (1 α), where x x T 1 = and T 2 =. x x If the a n are traffic independent, then (s n ) converges in distribution to the free convolution µ α = SC(0, α) N (0, 1 α), i.e., lim n ϕ(sm n ) = t m µ α (dt), m N. R Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

31 Limit Laws for Random Matrices from Traffic Probability Definition ((p, q)-markov matrices) Let W n be a Wigner matrix and D n = row(w n ) the diagonal matrix of row sums of W n. For p, q R, we call the real symmetric n n matrix M n,p,q defined by M n,p,q = pw n + qd n a (p, q)-markov matrix. Theorem (Au, 2016) Let (W n (l) : 1 l < ) be a sequence of independent finite moments Wigner matrices. Then the families ((M (l) n,p,q) p,q R : 1 l < ) are asymptotically traffic independent with stable universal limiting traffic distribution. Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

32 Limit Laws for Random Matrices from Traffic Probability Corollary The ESDs µ(m n,p,q ) converge in expectation to the free convolution SC(0, p 2 ) N (0, q 2 ). Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

33 Limit Laws for Random Matrices from Traffic Probability Corollary The ESDs µ(m n,p,q ) converge in expectation to the free convolution SC(0, p 2 ) N (0, q 2 ). 100 p = 1, q = Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

34 Limit Laws for Random Matrices from Traffic Probability Corollary The ESDs µ(m n,p,q ) converge in expectation to the free convolution SC(0, p 2 ) N (0, q 2 ). 100 p =.8, q = ± Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

35 Limit Laws for Random Matrices from Traffic Probability Corollary The ESDs µ(m n,p,q ) converge in expectation to the free convolution SC(0, p 2 ) N (0, q 2 ). 150 p =.6, q = ± Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

36 Limit Laws for Random Matrices from Traffic Probability Corollary The ESDs µ(m n,p,q ) converge in expectation to the free convolution SC(0, p 2 ) N (0, q 2 ). 150 p =.4, q = ± Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

37 Limit Laws for Random Matrices from Traffic Probability Corollary The ESDs µ(m n,p,q ) converge in expectation to the free convolution SC(0, p 2 ) N (0, q 2 ). 200 p =.2, q = ± Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

38 Limit Laws for Random Matrices from Traffic Probability Corollary The ESDs µ(m n,p,q ) converge in expectation to the free convolution SC(0, p 2 ) N (0, q 2 ). 250 p = 0, q = Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

39 Limit Laws for Random Matrices from Traffic Probability Theorem (Au, 2016) Let W n be a Wigner matrix, and let (X n ) and (Y n ) be sequences of real-valued random variables converging almost surely to X and Y respectively. Then the ESDs µ(m n,xn,y n ) converge weakly almost surely to the random free convolution SC(0, X 2 ) N (0, Y 2 ). Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27

40 Limit Laws for Random Matrices from Traffic Probability Theorem (Au, 2016) Let W n be a Wigner matrix, and let (X n ) and (Y n ) be sequences of real-valued random variables converging almost surely to X and Y respectively. Then the ESDs µ(m n,xn,y n ) converge weakly almost surely to the random free convolution SC(0, X 2 ) N (0, Y 2 ). Thank you! Benson Au (UC Berkeley) Random Matrices from Traffic Probability May 9th, / 27