Fluctuations from the Semicircle Law Lecture 4

Transcription

1 Fluctuations from the Semicircle Law Lecture 4 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 23, 2014 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

2 1 An Extension to Smooth Functions Approximation Theory 2 Other results; other models Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

3 An Extension to Smooth Functions A Simple Approximation Lemma (Weierstrass) Let f be a continuous function on [a, b]. There exists a sequence of polynomials {p m } which converge to f uniformly. (I.e., ɛ > 0, N N such that m N, x [a, b], f (x) p m (x) < ɛ.) We know that we have a polynomial CLT. Can we use the above to get continuous, compactly supported functions? Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

4 An Extension to Smooth Functions A Simple Approximation Examine: want to argue that ( ) f (λ i ) E f (λ i ) ( ) p m (λ i ) E p m (λ i ) for some polynomials p m. Anything troublesome? Yes. We are adding n values of f, thus possibly the error is on the order of nɛ... which does NOT go to 0. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

5 An Extension to Smooth Functions A Simple Approximation Examine: want to argue that ( ) f (λ i ) E f (λ i ) ( ) p m (λ i ) E p m (λ i ) for some polynomials p m. Anything troublesome? Yes. We are adding n values of f, thus possibly the error is on the order of nɛ... which does NOT go to 0. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

6 A Better Idea An Extension to Smooth Functions So we must find something better; perhaps a better approximating sequence of polynomials, for which f p m = sup f (x) p m (x) < 1. m 1+ɛ x [a,b] Unfortunately, this is NOT possible for merely continuous functions... but it is possible for, e.g., C 2 c functions. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

7 A Better Idea An Extension to Smooth Functions So we must find something better; perhaps a better approximating sequence of polynomials, for which f p m = sup f (x) p m (x) < 1. m 1+ɛ x [a,b] Unfortunately, this is NOT possible for merely continuous functions... but it is possible for, e.g., C 2 c functions. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

8 Chebyshev Series An Extension to Smooth Functions Approximation Theory Wlog assume that f C 2 c[ 1, 1] for some large 1. Theorem (Chebyshev Series) f has a unique representation on [ 1, 1] as an absolutely and uniformly convergent series f (x) = a k T k (x), k=0 with coefficients given by the formulae a k = 2 π 1 1 f (x)t k (x) 1 x 2 dx, for all k 1; for a 0 the factor 2/π changes to 1/π. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

9 An Extension to Smooth Functions Bounds and coefficients Approximation Theory More is known and useful. Let f n be the n-th truncation of the series, i.e., f n (x) = n k=0 a kt k (x). If f C 2, then for any n 3, f f n = sup f (x) f n (x) C f x [ 1,1] n 2, where C f is a constant depending on f. And finally, a k A f k 3, where once again A f is a constant. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

10 Great candidates An Extension to Smooth Functions Approximation Theory So f n, the truncation of the Chebyshev Series, satisfies the requirement. In fact, it will follow that for any λ 1,..., λ n, X n,f X n,fn := ( ) (f (λ i ) f n(λ i )) E (f (λ i ) f n(λ i )) (f (λ i ) f n(λ i )) E ( ) (f (λ i ) f n(λ i )) 1 = O. n λ i 1 λ i 1 The variance of X n,fn is n k=0 ka2 k, and thus converges to k=0 ka2 k (convergent due to bound a k A f /k 3.) Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

11 Great candidates An Extension to Smooth Functions Approximation Theory It seems that it should immediately follow that X n,f / k=0 ka2 k converges in distribution to N(0, 1)... Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

12 An Extension to Smooth Functions Approximation Theory An apparent wrench in the spokes But in the second line the sum is only over λ i 1! As f C 2 c([ 1, 1]), this is not a problem for f : X n,f = f (λ i ) E f (λ i ) = λ i 1 λ i 1 ( ) f (λ i ) E f (λ i )! The same is not true for f n. So what to do?... Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

13 An Extension to Smooth Functions Approximation Theory A Fudge Factor and an Exponential Salvation Let ɛ > 0. We split the sum as follows: (f n(λ i ) Ef n(λ i ))+ (f n(λ i ) Ef n(λ i ))+ (f n(λ i ) Ef n(λ i )) =: S 1 +S 2 +S 3. λ i 1 1< λ i 1+ɛ λ i >1+ɛ As it turns out, S 3 is a sum which is nonzero with exponentially small probability (since the chance that the largest eigenvalue is larger, asymptotically, than 1 + ɛ, is minuscule). S 1 is the approximation we want. Showing that S 2 is small is technical and we skip it. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

14 Other results; other models Other CLTs for Wigner We have assummed that the entries of W n have all moments, and shown that one can obtain a CLT for C 2 c functions. The following are stronger versions. Assume the entries distribution µ satisfies a Poincaré-type inequality, i.e., for all C 1 functions, Var µ (f ) ce µ ( f 2 ), then one can prove a CLT for C 1, Lipschitz functions f (Anderson-Zeitouni 08). Assumming symmetry of the distributions, a Lindeberg-condition for the 4th moment of the distributions, one can show a CLT for f C 5 (Pastur-Lytova 08). Another CLT proved for Wigner type matrices with a moment condition, for f analytic in a domain including the support of the limiting ESD (Bai-Yao 06). Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

15 Other results; other models Other CLTs for Wigner We have assummed that the entries of W n have all moments, and shown that one can obtain a CLT for C 2 b functions. The following are stronger versions. Assume the entries distribution µ satisfies a Poincaré-type inequality, i.e., for all C 1 functions, Var µ (f ) ce µ ( f 2 ), then one can prove a CLT for C 1, Lipschitz functions f (Anderson-Zeitouni 08). Assumming symmetry of the distributions, a Lindeberg-condition for the 4th moment of the distributions, one can show a CLT for f C 5 (Pastur-Lytova 08). Another CLT proved for Wigner type matrices with a moment condition, for f analytic in a domain including the support of the limiting ESD (Bai-Yao 06). Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

16 Other results; other models Other CLTs for Wigner We have assummed that the entries of W n have all moments, and shown that one can obtain a CLT for C 2 b functions. The following are stronger versions. Assume the entries distribution µ satisfies a Poincaré-type inequality, i.e., for all C 1 functions, Var µ (f ) ce µ ( f 2 ), then one can prove a CLT for C 1, Lipschitz functions f (Anderson-Zeitouni 08). Assumming symmetry of the distributions, a Lindeberg-condition for the 4th moment of the distributions, one can show a CLT for f C 5 (Pastur-Lytova 08). Another CLT proved for Wigner type matrices with a moment condition, for f analytic in a domain including the support of the limiting ESD (Bai-Yao 06). Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

17 Other results; other models Other CLTs for Wigner Consider Wigner centered matrices with moment existence conditions (4th moment). Real or complex. Eigenvalues on [ 1, 1]. Sinai-Soshnikov 98 and Soshnikov 99: CLTs for large powers; universality for top eigenvalue fluctuations (Tracy-Widom). if k n << n 2/3 ; tr((w n ) kn ) E(tr((W n ) kn )) σ kn N(0, 1) tr((w n ) kn ) E(tr((W n ) kn )) σ kn F 1 if k n = n 2/3, for real W n and F 1 the Tracy-Widom distribution. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

18 Other results; other models Other models: general β-hermite ensembles Seminal paper (Johansson 98) for a very large variety of potential-defined matrix ensembles. Recall GOE/GUE/GSE have joint eigenvalue distribution f β (λ 1,..., λ n ) i j λ i λ j β e n λ2 i /2 ; where β = 1, 2, 4 for R, C, H. Let β > 0 and replace n λ2 i /2 with n V(λ i) for V a polynomial of even degree. Johansson proved a CLT for all such ensembles, for f sufficiently smooth (about 8.5 derivatives). Note these are NOT Wigner matrices, except for GOE/GUE/GSE. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

19 Other results; other models Other models: Wishart matrices, β-laguerre, β-jacobi Let m n and X an m n iid matrix of variables with mean 0 and variance 1; let W = X T X. Assume m/n c as m, n. Marčenko-Pastur 67: found ESD limit. Bai-Silverstein 04: moment conditions, CLT for analytic functions. Large body of work by Bai, Silverstein, others on more complicated models (with non-trivial covariance). Variances, covariances have complicated expressions. Free probability methods (Cabanal-Duvillard 01, Capitaine-Casalis 04) showed the Chebyshev (shifted/scaled) polynomials diagonalize covariance matrix if Σ = I n. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

20 Other results; other models Other models: Wishart matrices, β-laguerre, β-jacobi Let m n and X an m n iid matrix of variables with mean 0 and variance 1; let W = X T X. Assume m/n c as m, n. D.-Edelman 06: β-laguerre models generalizing Wishart real, complex for normal entries (Σ = 1): monomial CLT. D.-Paquette 13: β-jacobi models generalizing MANOVA ensembles with trivial covariance: C 1, Lipschitz CLT. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

21 Other results; other models Other models: regular graphs A regular graph has the same degree d for all vertices. The matrix considered is the adjacency matrix, where A ij = δ i j. Ben Arous-Dang 11: d = 2, random permutation matrices (directed), bounded variation functions (less actually needed); fluctuation, not CLT! Fluctuation converges to infinitely divisible distribution. D.-Johnson-Pal-Paquette 12: analytic+ conditions for a permutation-model random regular graph; d fixed implies fluctuations converge to infinitely divisible distribution; d slowly with n yields CLT. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

22 Other models:? Other results; other models The stage is yours. Thank you for your attention. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23

23 Other models:? Other results; other models The stage is yours. Thank you for your attention. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, / 23