Random matrix theory and log-correlated Gaussian fields

Transcription

1 Random matrix theory and log-correlated Gaussian fields Nick Simm Collaborators : Yan Fyodorov and Boris Khoruzhenko (Queen Mary, London) Mathematics Institute, Warwick University, Coventry, UK XI Brunel-Bielefeld RMT Workshop December 2015 Research supported by Leverhulme fellowship ECF

2 Outline

3 Contents Introduction Regularization 1 Regularization 2 Distribution of the maximum value

4 Log-correlated Gaussian fields: what and why?

5 Log-correlated Gaussian fields: what and why? A log-correlated Gaussian field is a random distribution V(x) with covariance kernel of the form E(V(x)V(x )) log x x, x x and Gaussian finite-dimensional distributions.

6 Log-correlated Gaussian fields: what and why? A log-correlated Gaussian field is a random distribution V(x) with covariance kernel of the form E(V(x)V(x )) log x x, x x and Gaussian finite-dimensional distributions. Why: Central object of Kahane s theory of multiplicative chaos: e V(x)

7 Log-correlated Gaussian fields: what and why? A log-correlated Gaussian field is a random distribution V(x) with covariance kernel of the form E(V(x)V(x )) log x x, x x and Gaussian finite-dimensional distributions. Why: Central object of Kahane s theory of multiplicative chaos: e V(x) Liouville quantum gravity. e.g. Rhodes, Vargas, et al. 2015

8 Log-correlated Gaussian fields: what and why? A log-correlated Gaussian field is a random distribution V(x) with covariance kernel of the form E(V(x)V(x )) log x x, x x and Gaussian finite-dimensional distributions. Why: Central object of Kahane s theory of multiplicative chaos: e V(x) Liouville quantum gravity. e.g. Rhodes, Vargas, et al Non-trivial extreme value statistics relating to branching Brownian motions, Gaussian free fields. Ding, Roy, Zeitouni 2015.

9 Log-correlated Gaussian fields: what and why? A log-correlated Gaussian field is a random distribution V(x) with covariance kernel of the form E(V(x)V(x )) log x x, x x and Gaussian finite-dimensional distributions. Why: Central object of Kahane s theory of multiplicative chaos: e V(x) Liouville quantum gravity. e.g. Rhodes, Vargas, et al Non-trivial extreme value statistics relating to branching Brownian motions, Gaussian free fields. Ding, Roy, Zeitouni Connections to random matrix theory, number theory Fyodorov, Keating, Le Doussal, Arguin, Belius, Harper, Bourgade, Webb,

10 Random matrices

11 Random matrices A random Hermitian matrix H, of size N N taken from, e.g. the Gaussian Unitary Ensemble: P(H) = 1 C e 2NTr(H2 )

12 Random matrices A random Hermitian matrix H, of size N N taken from, e.g. the Gaussian Unitary Ensemble: P(H) = 1 C e 2NTr(H2 ) Wigner s semi-circle law (1955): lim N 1 N R 1(x) = { (2/π) 1 x 2, x 1 0, x > 1

13 Random matrices A random Hermitian matrix H, of size N N taken from, e.g. the Gaussian Unitary Ensemble: P(H) = 1 C e 2NTr(H2 ) Wigner s semi-circle law (1955): lim N 1 N R 1(x) = { (2/π) 1 x 2, x 1 0, x > 1 In this talk we study the characteristic polynomial of H: p N (x) = det(xi H)

14 Random matrices A random Hermitian matrix H, of size N N taken from, e.g. the Gaussian Unitary Ensemble: P(H) = 1 C e 2NTr(H2 ) Wigner s semi-circle law (1955): lim N 1 N R 1(x) = { (2/π) 1 x 2, x 1 0, x > 1 In this talk we study the characteristic polynomial of H: p N (x) = det(xi H) Viewpoint: p N (x) is a sequence (N = 1,2,3,...) of stochastic processes in x.

15 Peaks and troughs of the characteristic polynomial Figure : A plot of a single realization of p N (x) e Elog p N(x) for N = 50. Above: a random curve (stochastic process) depending on N. Is there a limit as N?

16 Larger N: logarithmic scale Figure : A plot of a single realization of log p N (x) Elog p N (x) for N = x

17 Colours of noise

18 Colours of noise A time signal V(t) can be described by a spectral density 1 S(ω) = lim T T E T V(t)e iωt 2 dt. T

19 Colours of noise A time signal V(t) can be described by a spectral density 1 S(ω) = lim T T E T V(t)e iωt 2 dt. For example: T V(t) V(t) log(s(f)) z1 P(f) P(f) S(f)= 1 f 0 Whitenoise

20 Colours of noise A time signal V(t) can be described by a spectral density 1 S(ω) = lim T T E T V(t)e iωt 2 dt. For example: T V(t) V(t) log(s(f)) z1 P(f) P(f) S(f)= 1 f 0 Whitenoise V(t) V(t) log(s(f)) V(t) z1 P(f) P(f) S(f)= 1 f 2 z Red noise

21 Colours of noise A time signal V(t) can be described by a spectral density 1 S(ω) = lim T T E T V(t)e iωt 2 dt. For example: T V(t) V(t) log(s(f)) z1 P(f) P(f) S(f)= 1 f 0 Whitenoise V(t) V(t) log(s(f)) V(t) z1 P(f) P(f) A well-known example is the random Fourier series: V(t) = k=1 S(f)= 1 f 2 z Red noise 2sin(kπt) X k, X k i.i.d. standard Gaussians. kπ which represents a Brownian bridge on [0, 1].

22 Colours of noise A time signal V(t) can be described by a spectral density 1 S(ω) = lim T T E T V(t)e iωt 2 dt. For example: T V(t) V(t) log(s(f)) z1 P(f) P(f) S(f)= 1 f 0 Whitenoise V(t) V(t) log(s(f)) z P(f) P(f) S(f)= 1 )= f 1 Pink noise V(t) V(t) log(s(f)) V(t) z1 P(f) P(f) A well-known example is the random Fourier series: V(t) = k=1 S(f)= 1 f 2 z Red noise 2sin(kπt) X k, X k i.i.d. standard Gaussians. kπ which represents a Brownian bridge on [0, 1].

23 Characteristic function of log p N (x)

24 Characteristic function of log p N (x) Study the statistics in Fourier space: φ N ( x, α) = E (e ) K k=1 α k log p N (x k )

25 Characteristic function of log p N (x) Study the statistics in Fourier space: φ N ( x, α) = E (e ) K k=1 α k log p N (x k ) Diagonalize H = UDU : the above is equal to N c 1... w K (λ j ) (λ i λ j ) 2 dλ 1...dλ N R R j=1 1 i<j N

26 Characteristic function of log p N (x) Study the statistics in Fourier space: φ N ( x, α) = E (e ) K k=1 α k log p N (x k ) Diagonalize H = UDU : the above is equal to N c 1... w K (λ j ) (λ i λ j ) 2 dλ 1...dλ N R R j=1 1 i<j N Weight (symbol) has Fisher-Hartwig type singularities K w K (λ) = e 2Nλ2 λ x k α k, Re(α k ) > 1/2 k=1

27 Characteristic function of log p N (x) Study the statistics in Fourier space: φ N ( x, α) = E (e ) K k=1 α k log p N (x k ) Diagonalize H = UDU : the above is equal to N c 1... w K (λ j ) (λ i λ j ) 2 dλ 1...dλ N R R j=1 1 i<j N Weight (symbol) has Fisher-Hartwig type singularities K w K (λ) = e 2Nλ2 λ x k α k, Re(α k ) > 1/2 Hankel determinant: k=1 φ N ( x, α) det N N { λ i+j w K (λ)dλ R } N 1 i,j=0

28 Asymptotics of the Hankel determinant

29 Asymptotics of the Hankel determinant Krasovsky (2007) computed the asymptotics of this determinant: E (e ) K k=1 2α k log p N (x k ) = K C(α k )(1 xk 2 )α2 k /2 (N/2) α2 k e (2xk 2 1)α kn k=1 1 i<j K ( )] (2 x i x j ) 2α iα j logn [1+O N

30 Asymptotics of the Hankel determinant Krasovsky (2007) computed the asymptotics of this determinant: E (e ) K k=1 2α k log p N (x k ) = K C(α k )(1 xk 2 )α2 k /2 (N/2) α2 k e (2xk 2 1)α kn k=1 1 i<j K ( )] (2 x i x j ) 2α iα j logn [1+O N What does this tell us about the statistics of log p N (x)?

31 Asymptotics of the Hankel determinant Krasovsky (2007) computed the asymptotics of this determinant: E (e ) K k=1 2α k log p N (x k ) = K C(α k )(1 xk 2 )α2 k /2 (N/2) α2 k e (2xk 2 1)α kn k=1 1 i<j K ( )] (2 x i x j ) 2α iα j logn [1+O N What does this tell us about the statistics of log p N (x)? Quadratic α 2 k in the exponential: Gaussianity

32 Asymptotics of the Hankel determinant Krasovsky (2007) computed the asymptotics of this determinant: E (e ) K k=1 2α k log p N (x k ) = K C(α k )(1 xk 2 )α2 k /2 (N/2) α2 k e (2xk 2 1)α kn k=1 1 i<j K ( )] (2 x i x j ) 2α iα j logn [1+O N What does this tell us about the statistics of log p N (x)? Quadratic α 2 k in the exponential: Gaussianity x i x j factor: O(1) logarithmic correlations

33 Asymptotics of the Hankel determinant Krasovsky (2007) computed the asymptotics of this determinant: E (e ) K k=1 2α k log p N (x k ) = K C(α k )(1 xk 2 )α2 k /2 (N/2) α2 k e (2xk 2 1)α kn k=1 1 i<j K ( )] (2 x i x j ) 2α iα j logn [1+O N What does this tell us about the statistics of log p N (x)? Quadratic α 2 k in the exponential: Gaussianity x i x j factor: O(1) logarithmic correlations Factor (N/2) α2 k: variance diverges like log(n).

34 Asymptotics of the Hankel determinant Krasovsky (2007) computed the asymptotics of this determinant: E (e ) K k=1 2α k log p N (x k ) = K C(α k )(1 xk 2 )α2 k /2 (N/2) α2 k e (2xk 2 1)α kn k=1 1 i<j K ( )] (2 x i x j ) 2α iα j logn [1+O N What does this tell us about the statistics of log p N (x)? Quadratic α 2 k in the exponential: Gaussianity x i x j factor: O(1) logarithmic correlations Factor (N/2) α2 k: variance diverges like log(n). So limit does not exist! Unless one rescales by log(n)... But this kills the subtle correlations log x i x j. How can we regularize?

35 Regularizing the logarithmic divergence

36 Regularizing the logarithmic divergence Regularization 1: We embed the characteristic polynomial in a space of distributions, obtaining a limiting object which is a random generalized function. The latter will be a random Gaussian field on a function space, with logarithmic correlations. Here the convergence takes place on global scales (all eigenvalues are involved).

37 Regularizing the logarithmic divergence Regularization 1: We embed the characteristic polynomial in a space of distributions, obtaining a limiting object which is a random generalized function. The latter will be a random Gaussian field on a function space, with logarithmic correlations. Here the convergence takes place on global scales (all eigenvalues are involved). Regularization 2: Instead of putting x [ 1,1] we will go slightly complex: x z = x + t +iη d N for some large parameter d N and η > 0. This will actually lead to a well-defined Gaussian process in the limit (no normalization necessary). Here the limit takes place on mesoscopic scales (only a fraction of the eigenvalues are involved).

38 Contents Introduction Regularization 1 Regularization 2 Distribution of the maximum value

39 Theorem: Weak convergence in a Sobolev space

40 Theorem: Weak convergence in a Sobolev space Let V N (x) = log det(x H) and x ( 1,1). Act on test functions: V N [f] := c k (V N )c k (f), k=0 f W (a)

41 Theorem: Weak convergence in a Sobolev space Let V N (x) = log det(x H) and x ( 1,1). Act on test functions: V N [f] := c k (V N )c k (f), k=0 f W (a) where c k (f) are Chebyshev coefficients, given by c k (f) := 2 π 1 1 dx T k(x)f(x) 1 x 2, T k(x) = cos(k cos 1 (x)) and W (a) := {f L 2 [ 1,1] : k c k(f) 2 (1+k 2 ) a < }.

42 Theorem: Weak convergence in a Sobolev space Let V N (x) = log det(x H) and x ( 1,1). Act on test functions: V N [f] := c k (V N )c k (f), k=0 f W (a) where c k (f) are Chebyshev coefficients, given by c k (f) := 2 π 1 1 dx T k(x)f(x) 1 x 2, T k(x) = cos(k cos 1 (x)) and W (a) := {f L 2 [ 1,1] : k c k(f) 2 (1+k 2 ) a < }. Theorem (Fyodorov, Khoruzhenko and Simm 13) Let X 1,X 2,... be i.i.d. standard Gaussians. Then as N log det(x H) E[...] weakly in W (a) with a < 1/2. k=1 X k k T k (x)

43 Properties of the limiting object

44 Properties of the limiting object The limiting object V W ( a) is log-correlated: E(V(x)V(x T k (x)t k (x ) )) = 2 = 1 k 2 log(2 x x ) k=1

45 Properties of the limiting object The limiting object V W ( a) is log-correlated: E(V(x)V(x T k (x)t k (x ) )) = 2 = 1 k 2 log(2 x x ) k=1 It acts on test functions via X k V[f] = c k (f), f W (a). k k=1

46 Properties of the limiting object The limiting object V W ( a) is log-correlated: E(V(x)V(x T k (x)t k (x ) )) = 2 = 1 k 2 log(2 x x ) k=1 It acts on test functions via X k V[f] = c k (f), f W (a). k k=1 Thus, V is the centered Gaussian field on W (a) with covariance operator specified by E(V[f]V[g]) = log(2 x y )f(x)g(y)µ(dx)µ(dy) where µ(dx) = (1 x 2 ) 1/2 dx, i.e. V has logarithmic correlations.

47 Idea of proof:

48 Idea of proof: (Step 1) Convergence of finite-dimensional distributions:

49 Idea of proof: (Step 1) Convergence of finite-dimensional distributions: We expand the logarithm log det(x H) = k=1 T k (x)x k (H) k where (with λ 1,...,λ N denoting the eigenvalues of H), X k (H) := N j=1 2T k (λ j ) k + 2 k N j=1 ( (1 χ [ 1,1] (λ j )) T k (λ j ) ( ) ) k λ j λ 2 j 1.

50 Idea of proof: (Step 1) Convergence of finite-dimensional distributions: We expand the logarithm log det(x H) = k=1 T k (x)x k (H) k where (with λ 1,...,λ N denoting the eigenvalues of H), X k (H) := N j=1 2T k (λ j ) k + 2 k N j=1 ( (1 χ [ 1,1] (λ j )) T k (λ j ) ( ) ) k λ j λ 2 j 1. The first term converges as N to i.i.d. standard Gaussians X k (Johansson 98). We prove that the remainder 0 in probability.

51 Idea of proof: (Step 1) Convergence of finite-dimensional distributions: We expand the logarithm log det(x H) = k=1 T k (x)x k (H) k where (with λ 1,...,λ N denoting the eigenvalues of H), X k (H) := N j=1 2T k (λ j ) k + 2 k N j=1 ( (1 χ [ 1,1] (λ j )) T k (λ j ) ( ) ) k λ j λ 2 j 1. The first term converges as N to i.i.d. standard Gaussians X k (Johansson 98). We prove that the remainder 0 in probability. (Step 2) Tightness: We prove the following estimate lim N k=0 (1+k 2 ) a Var(X k (H)) <, a < 1/2.

52 Contents Introduction Regularization 1 Regularization 2 Distribution of the maximum value

53 Fractional Brownian motion

54 Fractional Brownian motion Let B(ds) be the Gaussian white noise measure with E(B(ds)) = 0 and E(B(ds 1 )B(ds 2 )) = δ(s 1 s 2 )ds 1 ds 2. For 0 < H < 1, we define fbm as (see Taqqu et al. 03) B H (t) = 1 C where d B(ω) = db 1 (ω)+idb 2 (ω). 0 [e iωt 1 1] d B(ω)+c.c. ωh+1/2

55 Fractional Brownian motion Let B(ds) be the Gaussian white noise measure with E(B(ds)) = 0 and E(B(ds 1 )B(ds 2 )) = δ(s 1 s 2 )ds 1 ds 2. For 0 < H < 1, we define fbm as (see Taqqu et al. 03) B H (t) = 1 C where d B(ω) = db 1 (ω)+idb 2 (ω). From the definition one sees B H (t) is a zero-mean Gaussian field. 0 [e iωt 1 1] d B(ω)+c.c. ωh+1/2 E[B H (t)b H (s)] = (σ 2 /2)( t 2H + s 2H t s 2H ) Self-similarity: B H (at) d = a H B H (t)

56 Fractional Brownian motion Let B(ds) be the Gaussian white noise measure with E(B(ds)) = 0 and E(B(ds 1 )B(ds 2 )) = δ(s 1 s 2 )ds 1 ds 2. For 0 < H < 1, we define fbm as (see Taqqu et al. 03) B H (t) = 1 C where d B(ω) = db 1 (ω)+idb 2 (ω). From the definition one sees B H (t) is a zero-mean Gaussian field. 0 [e iωt 1 1] d B(ω)+c.c. ωh+1/2 E[B H (t)b H (s)] = (σ 2 /2)( t 2H + s 2H t s 2H ) Self-similarity: B H (at) d = a H B H (t) The parameter H - Hurst index.

57 Fractional Brownian motion Let B(ds) be the Gaussian white noise measure with E(B(ds)) = 0 and E(B(ds 1 )B(ds 2 )) = δ(s 1 s 2 )ds 1 ds 2. For 0 < H < 1, we define fbm as (see Taqqu et al. 03) B H (t) = 1 C where d B(ω) = db 1 (ω)+idb 2 (ω). From the definition one sees B H (t) is a zero-mean Gaussian field. 0 [e iωt 1 1] d B(ω)+c.c. ωh+1/2 E[B H (t)b H (s)] = (σ 2 /2)( t 2H + s 2H t s 2H ) Self-similarity: B H (at) d = a H B H (t) The parameter H - Hurst index. Loosely speaking, one may say that fbm has spectral density S(ω) ω 2H 1.

58 Fractional Brownian motion Let B(ds) be the Gaussian white noise measure with E(B(ds)) = 0 and E(B(ds 1 )B(ds 2 )) = δ(s 1 s 2 )ds 1 ds 2. For 0 < H < 1, we define fbm as (see Taqqu et al. 03) B H (t) = 1 C where d B(ω) = db 1 (ω)+idb 2 (ω). From the definition one sees B H (t) is a zero-mean Gaussian field. 0 [e iωt 1 1] d B(ω)+c.c. ωh+1/2 E[B H (t)b H (s)] = (σ 2 /2)( t 2H + s 2H t s 2H ) Self-similarity: B H (at) d = a H B H (t) The parameter H - Hurst index. Loosely speaking, one may say that fbm has spectral density S(ω) ω 2H 1. This motivates us to consider the limit H 0. Undefined!

59 The regularization of fbm for H 0

60 The regularization of fbm for H 0 Solution: For η > 0, introduce the 1-parameter extension B (η) H (t) = e ηω ω H+1/2[e iωt 1]d B(ω)+c.c. 0 For fixed η > 0, the limit H 0 is well defined and yields a particular regularization of 1/f-noise. We denote it B (η) 0 (t).

61 The regularization of fbm for H 0 Solution: For η > 0, introduce the 1-parameter extension B (η) H (t) = e ηω ω H+1/2[e iωt 1]d B(ω)+c.c. 0 For fixed η > 0, the limit H 0 is well defined and yields a particular regularization of 1/f-noise. We denote it B (η) 0 (t). Properties: B (η) 0 (t) is a zero-mean Gaussian process. B (η) 0 (t) has stationary increments. B (η) 0 (t) is self-similar : B(Tη) 0 (Tt) = d B (η) 0 (t).

62 The regularization of fbm for H 0 Solution: For η > 0, introduce the 1-parameter extension B (η) H (t) = e ηω ω H+1/2[e iωt 1]d B(ω)+c.c. 0 For fixed η > 0, the limit H 0 is well defined and yields a particular regularization of 1/f-noise. We denote it B (η) 0 (t). Properties: B (η) 0 (t) is a zero-mean Gaussian process. B (η) 0 (t) has stationary increments. B (η) 0 (t) is self-similar : B(Tη) 0 (Tt) = d B (η) 0 (t). Covariance structure: ( E((B (η) 0 (t 1) B (η) 0 (t 2)) 2 (t1 t 2 ) 2 ) ) = log η 2 +1 Regularized log correlations on scale η > 0.

63 Scaling regimes in Random Matrix Theory

64 Scaling regimes in Random Matrix Theory Let λ 1,...,λ N be the eigenvalues of H. The results of the previous few slides describe the limiting law of X N := N f(d N (λ j x )) E[...] j=1 for specific f with d N = 1, x = 0.

65 Scaling regimes in Random Matrix Theory Let λ 1,...,λ N be the eigenvalues of H. The results of the previous few slides describe the limiting law of X N := N f(d N (λ j x )) E[...] j=1 for specific f with d N = 1, x = 0. Three distinct scales: Global regime: d N = 1. Many results available, one expects Gaussian fluctuations of X N as N.

66 Scaling regimes in Random Matrix Theory Let λ 1,...,λ N be the eigenvalues of H. The results of the previous few slides describe the limiting law of X N := N f(d N (λ j x )) E[...] j=1 for specific f with d N = 1, x = 0. Three distinct scales: Global regime: d N = 1. Many results available, one expects Gaussian fluctuations of X N as N. Mesoscopic regime: 1 d N N. e.g. d N = N α with 0 < α < 1. See Erdos and Knowles 13, Duits and Johansson 13, Bourgade et al. 14, Breuer and Duits 14, Lodhia and Simm 15, Johansson and Lambert 15, Lambert 15 (See his poster)

67 Scaling regimes in Random Matrix Theory Let λ 1,...,λ N be the eigenvalues of H. The results of the previous few slides describe the limiting law of X N := N f(d N (λ j x )) E[...] j=1 for specific f with d N = 1, x = 0. Three distinct scales: Global regime: d N = 1. Many results available, one expects Gaussian fluctuations of X N as N. Mesoscopic regime: 1 d N N. e.g. d N = N α with 0 < α < 1. See Erdos and Knowles 13, Duits and Johansson 13, Bourgade et al. 14, Breuer and Duits 14, Lodhia and Simm 15, Johansson and Lambert 15, Lambert 15 (See his poster) Local regime: d N = N. CLT fails, not enough eigenvalues.

68 Theorem: Mesoscopic regime and fbm Regularization 2: Shift the argument of p N (x): x z t := x + t +iη d N with x,t,η > 0 fixed. Define W (η) N (t) := log p N(z t ) +log p N (z 0 )

69 Theorem: Mesoscopic regime and fbm Regularization 2: Shift the argument of p N (x): x z t := x + t +iη d N with x,t,η > 0 fixed. Define W (η) N (t) := log p N(z t ) +log p N (z 0 ) Theorem (Fyodorov, Khoruzhenko and Simm 13) Let d N with d N = o(n/log(n)). After centering W := W E(W), we obtain the convergence in law ( W (η) N (t 1),..., W (η) N (t k)) d (B (η) 0 (t 1),...,B (η) 0 (t k)), N, where B (η) 0 (t) is our regularization of fbm with H = 0: B (η) 0 (t) = 0 e ηω ω [e itf 1]d B(ω)+c.c.

70 Theorem: Mesoscopic regime and fbm Regularization 2: Shift the argument of p N (x): x z t := x + t +iη d N with x,t,η > 0 fixed. Define W (η) N (t) := log p N(z t ) +log p N (z 0 ) Theorem (Fyodorov, Khoruzhenko and Simm 13) Let d N with d N = o(n/log(n)). After centering W := W E(W), we obtain the convergence in law ( W (η) N (t 1),..., W (η) N (t k)) d (B (η) 0 (t 1),...,B (η) 0 (t k)), N, where B (η) 0 (t) is our regularization of fbm with H = 0: B (η) 0 (t) = 0 e ηω ω [e itf 1]d B(ω)+c.c.

71 Theorem: Mesoscopic regime and fbm Regularization 2: Shift the argument of p N (x): x z t := x + t +iη d N with x,t,η > 0 fixed. Define W (η) N (t) := log p N(z t ) +log p N (z 0 ) Theorem (Fyodorov, Khoruzhenko and Simm 13) Let d N with d N = o(n/log(n)). After centering W := W E(W), we obtain the convergence in law ( W (η) N (t 1),..., W (η) N (t k)) d (B (η) 0 (t 1),...,B (η) 0 (t k)), N, where B (η) 0 (t) is our regularization of fbm with H = 0: B (η) 0 (t) = 0 e ηω ω [e itf 1]d B(ω)+c.c.

72 The Fourier decomposition

73 The Fourier decomposition An exact identity rewrites W (η) N (t) in the form of a Fourier integral: ( W (η) N (t) = log det H E t +iη ) ( +log det H E iη d N) = 0 d N e ηω ω [e itω 1]b N (ω)dω +c.c.

74 The Fourier decomposition An exact identity rewrites W (η) N (t) in the form of a Fourier integral: ( W (η) N (t) = log det H E t +iη ) ( +log det H E iη d N) = 0 d N e ηω ω [e itω 1]b N (ω)dω +c.c. where b N (ω) are Fourier coefficients b N (ω) = 1 Tr(e iωdn(h E) ) = 1 N e iωd N(λ j x ). ω ω j=1

75 The Fourier decomposition An exact identity rewrites W (η) N (t) in the form of a Fourier integral: ( W (η) N (t) = log det H E t +iη ) ( +log det H E iη d N) = 0 d N e ηω ω [e itω 1]b N (ω)dω +c.c. where b N (ω) are Fourier coefficients b N (ω) = 1 Tr(e iωdn(h E) ) = 1 N e iωd N(λ j x ). ω ω Our definition of the limit object was just B (η) 0 (t) = 0 j=1 e ηω ω [e itω 1]d B(ω)+c.c.

76 The Fourier decomposition An exact identity rewrites W (η) N (t) in the form of a Fourier integral: ( W (η) N (t) = log det H E t +iη ) ( +log det H E iη d N) = 0 d N e ηω ω [e itω 1]b N (ω)dω +c.c. where b N (ω) are Fourier coefficients b N (ω) = 1 Tr(e iωdn(h E) ) = 1 N e iωd N(λ j x ). ω ω Our definition of the limit object was just B (η) 0 (t) = 0 j=1 e ηω ω [e itω 1]d B(ω)+c.c. We are motivated to prove the following result.

77 White noise limit for Fourier coefficients

78 White noise limit for Fourier coefficients Let U U(N) with Haar measure. Recall Theorem (Diaconis and Shahshahani 94) Let b N (k) = 1 k Tr(U k ). Then for any fixed k 1,...,k m, we have (b N (k 1 ),...,b N (k m )) Z := (Z 1,...,Z m ), N where Z is a discrete white noise.

79 White noise limit for Fourier coefficients Let U U(N) with Haar measure. Recall Theorem (Diaconis and Shahshahani 94) Let b N (k) = 1 k Tr(U k ). Then for any fixed k 1,...,k m, we have (b N (k 1 ),...,b N (k m )) Z := (Z 1,...,Z m ), N where Z is a discrete white noise. Theorem (Fyodorov, Khoruzhenko, Simm 13) We defined: b N (ω) = 1 ω Tr(e id N(H x ) ) Let x ( 1,1) and set d N = N α with 0 < α < 1. Then we have the convergence in distribution (b N (ω 1 ),...,b N (ω m )) {d B(ω 1 ),...,d B(ω m )}, N

80 White noise limit for Fourier coefficients Let U U(N) with Haar measure. Recall Theorem (Diaconis and Shahshahani 94) Let b N (k) = 1 k Tr(U k ). Then for any fixed k 1,...,k m, we have (b N (k 1 ),...,b N (k m )) Z := (Z 1,...,Z m ), N where Z is a discrete white noise. Theorem (Fyodorov, Khoruzhenko, Simm 13) We defined: b N (ω) = 1 ω Tr(e id N(H x ) ) Let x ( 1,1) and set d N = N α with 0 < α < 1. Then we have the convergence in distribution (b N (ω 1 ),...,b N (ω m )) {d B(ω 1 ),...,d B(ω m )}, N Continuum (mesoscopic) analogue of Diaconis-Shahshahani.

81 Universality of mesoscopic fluctuations - Wigner matrices Consider a general Wigner matrix W consisting of i.i.d. random variables with EW ij = 0 and E W ij 2 = 1. Assume: distribution of W ij has sub-gaussian decay..

82 Universality of mesoscopic fluctuations - Wigner matrices Consider a general Wigner matrix W consisting of i.i.d. random variables with EW ij = 0 and E W ij 2 = 1. Assume: distribution of W ij has sub-gaussian decay. Theorem (Lodhia and Simm 15) Fix E ( 1+δ,1 δ) and let z t := E + t+iη d N. Assume that 1 d N N 1/3. Then the sequence of random functions W (η) N (t) = log det(z t W) +log det(z 0 W) converges as N in the sense of finite-dimensional distributions to B (η) 0 (t): ( W N (t 1 ),..., W N (t k )) (B (η) 0 (t 1),...,B (η) 0 (t k)), k = 1,2,3,....

83 Universality of mesoscopic fluctuations - Wigner matrices Consider a general Wigner matrix W consisting of i.i.d. random variables with EW ij = 0 and E W ij 2 = 1. Assume: distribution of W ij has sub-gaussian decay. Theorem (Lodhia and Simm 15) Fix E ( 1+δ,1 δ) and let z t := E + t+iη d N. Assume that 1 d N N 1/3. Then the sequence of random functions W (η) N (t) = log det(z t W) +log det(z 0 W) converges as N in the sense of finite-dimensional distributions to B (η) 0 (t): ( W N (t 1 ),..., W N (t k )) (B (η) 0 (t 1),...,B (η) 0 (t k)), k = 1,2,3,... Boutet de Monvel, Khorunzhy: 1 d N N 1/8..

84 Universality of mesoscopic fluctuations - Wigner matrices Consider a general Wigner matrix W consisting of i.i.d. random variables with EW ij = 0 and E W ij 2 = 1. Assume: distribution of W ij has sub-gaussian decay. Theorem (Lodhia and Simm 15) Fix E ( 1+δ,1 δ) and let z t := E + t+iη d N. Assume that 1 d N N 1/3. Then the sequence of random functions W (η) N (t) = log det(z t W) +log det(z 0 W) converges as N in the sense of finite-dimensional distributions to B (η) 0 (t): ( W N (t 1 ),..., W N (t k )) (B (η) 0 (t 1),...,B (η) 0 (t k)), k = 1,2,3,... Boutet de Monvel, Khorunzhy: 1 d N N 1/8. Four moments: If W ij s match moments with the GUE to order 4, one gets all of 1 d N N (using arguments of Tao and Vu 12).

85 Contents Introduction Regularization 1 Regularization 2 Distribution of the maximum value

86 Statistics of the maximum Now we ask a different question. What about the statistics of the highest peak of the characteristic polynomial? Figure : A single realization of p N (x) e Elog p N(x). What are the statistics of the maximum value? (see the red dot) x

87 Analytical approach Step 1. Random partition function: Z N (β) = 1 1 e βv N(x) ρ sc (x)dx with potential V N (x) = 2(log p N (x) Elog p N (x) ).

88 Analytical approach Step 1. Random partition function: Z N (β) = 1 1 e βv N(x) ρ sc (x)dx with potential V N (x) = 2(log p N (x) Elog p N (x) ). Free energy F(β) = β 1 logz N (β) satisfies lim F(β) = min V N(x) β x [ 1,1]

89 Analytical approach Step 1. Random partition function: Z N (β) = 1 1 e βv N(x) ρ sc (x)dx with potential V N (x) = 2(log p N (x) Elog p N (x) ). Free energy F(β) = β 1 logz N (β) satisfies Step 2. Moments of Z N (β) EZ N (β) K = lim F(β) = min V N(x) β x [ 1,1] [ 1,1] K E (e β K k=1 V N(x k ) ) K k=1 Integrand: Use Krasovsky 07 asymptotics (but...) ρ sc (x k )dx k

90 Large N limit of the moments Inserting Krasovsky 07 asymptotics: K E(Z N (β) K ) cβ K NKβ2 dx k (1 xk 2 )a/2 Selberg s integral! [ 1,1] K k=1 x i x j 2β2 1 i<j K

91 Large N limit of the moments Inserting Krasovsky 07 asymptotics: E(Z N (β) K ) c K β NKβ2 [ 1,1] K k=1 K dx k (1 xk 2 )a/2 Selberg s integral! For β 2 < 1 the latter equals K 1 c K β N Kβ2 j=0 1 i<j K Γ(a+1 jβ 2 ) 2 Γ(1 (j +1)β 2 ) Γ(2a+2 (k +j 2)β 2 ) x i x j 2β2

92 Large N limit of the moments Inserting Krasovsky 07 asymptotics: E(Z N (β) K ) c K β NKβ2 [ 1,1] K k=1 K dx k (1 xk 2 )a/2 Selberg s integral! For β 2 < 1 the latter equals K 1 c K β N Kβ2 j=0 1 i<j K Γ(a+1 jβ 2 ) 2 Γ(1 (j +1)β 2 ) Γ(2a+2 (k +j 2)β 2 ) x i x j 2β2 Problem: Asymptotics are not uniform in x 1,...,x k. Claeys and Krasovsky 14, Claeys and Fahs 15: rigorous transition asymptotics for the determinants with K = 2.

93 Large N limit of the moments Inserting Krasovsky 07 asymptotics: E(Z N (β) K ) c K β NKβ2 [ 1,1] K k=1 K dx k (1 xk 2 )a/2 Selberg s integral! For β 2 < 1 the latter equals K 1 c K β N Kβ2 j=0 1 i<j K Γ(a+1 jβ 2 ) 2 Γ(1 (j +1)β 2 ) Γ(2a+2 (k +j 2)β 2 ) x i x j 2β2 Problem: Asymptotics are not uniform in x 1,...,x k. Claeys and Krasovsky 14, Claeys and Fahs 15: rigorous transition asymptotics for the determinants with K = 2. Main task: analytic continuation of Selberg s integral to complex K.

94 Freezing and temperature duality: β 1/β The obtained analytic continuation of the moments turns out to be invariant under β 1/β.

95 Freezing and temperature duality: β 1/β The obtained analytic continuation of the moments turns out to be invariant under β 1/β. Freezing hypothesis: Thermodynamic quantities which for β < 1 are invariant under β 1/β retain for all β > 1 the value at the critical point β = 1.

96 Freezing and temperature duality: β 1/β The obtained analytic continuation of the moments turns out to be invariant under β 1/β. Freezing hypothesis: Thermodynamic quantities which for β < 1 are invariant under β 1/β retain for all β > 1 the value at the critical point β = 1. Hypothesis supported by heuristic reasoning, numerics and more recently by rigorous calculation. e.g. Subag and Zeitouni 2014, Arguin, Belius and Bourgade 2015.

97 Freezing and temperature duality: β 1/β The obtained analytic continuation of the moments turns out to be invariant under β 1/β. Freezing hypothesis: Thermodynamic quantities which for β < 1 are invariant under β 1/β retain for all β > 1 the value at the critical point β = 1. Hypothesis supported by heuristic reasoning, numerics and more recently by rigorous calculation. e.g. Subag and Zeitouni 2014, Arguin, Belius and Bourgade This philosophy allows us to obtain the β limit by setting β = 1.

98 Freezing and temperature duality: β 1/β The obtained analytic continuation of the moments turns out to be invariant under β 1/β. Freezing hypothesis: Thermodynamic quantities which for β < 1 are invariant under β 1/β retain for all β > 1 the value at the critical point β = 1. Hypothesis supported by heuristic reasoning, numerics and more recently by rigorous calculation. e.g. Subag and Zeitouni 2014, Arguin, Belius and Bourgade This philosophy allows us to obtain the β limit by setting β = 1. Rigorously proving these conjectures represents an ongoing and significant mathematical challenge.

99 Conjecture

100 Conjecture Consider the random variable V N := max x [ 1,1] (2log p N(x) E2log p N (x) ).

101 Conjecture Consider the random variable V N := max x [ 1,1] (2log p N(x) E2log p N (x) ). Conjecture (Fyodorov and Simm 15) We have the convergence in distribution V N 2log(N)+ 3 2 log(log(n)) d = u, N, where u is a continuous random variable characterized by E(e us ) = 1 C 2 4s Γ(s +1)Γ(s +3)G(s +7/2) 2 π s G(s +1)G(s +6) and G(s) is the Barnes-G function satisfying G(s +1) = Γ(s)G(s), G(1) = 1

102 Recent developments: Unitary matrices Theorem (Arguin, Belius and Bourgade 15. Freezing.) Let U U(N) be chosen from the unitary group with Haar measure and P N (θ) = det(e iθ U). Then the following limits hold in probability lim N ( 1 N 2π ) βlogn log P N (θ) 2β dθ = 2π 0 { β + 1 β, β < 1 2, β 1

103 Recent developments: Unitary matrices Theorem (Arguin, Belius and Bourgade 15. Freezing.) Let U U(N) be chosen from the unitary group with Haar measure and P N (θ) = det(e iθ U). Then the following limits hold in probability lim N ( 1 N 2π ) βlogn log P N (θ) 2β dθ = 2π 0 { β + 1 β, β < 1 2, β 1 and max θ [0,2π] 2log P N (θ) lim = 2 N log(n)

104 Recent developments: Unitary matrices Theorem (Arguin, Belius and Bourgade 15. Freezing.) Let U U(N) be chosen from the unitary group with Haar measure and P N (θ) = det(e iθ U). Then the following limits hold in probability lim N ( 1 N 2π ) βlogn log P N (θ) 2β dθ = 2π 0 { β + 1 β, β < 1 2, β 1 and max θ [0,2π] 2log P N (θ) lim = 2 N log(n) Open questions: Correction term (3/2) log(log(n))? The fluctuating O(1) term?

105 Y.V. Fyodorov, B.A. Khoruzhenko and N.J. Simm. Fractional Brownian motion with Hurst index H = 0 and the Gaussian Unitary Ensemble. arxiv: A. Lodhia and N.J. Simm. Mesoscopic linear statistics of Wigner matrices. arxiv: Y.V. Fyodorov and N.J. Simm. On the distribution of the maximum of GUE characteristic polynomials. arxiv: Thank you!

106 Proof: Mesoscopic Riemann-Hilbert problem

107 Proof: Mesoscopic Riemann-Hilbert problem Proof is based on the RHP for orthogonal polynomials (extending Krasovsky s approach to mesoscopic regime).

108 Proof: Mesoscopic Riemann-Hilbert problem Proof is based on the RHP for orthogonal polynomials (extending Krasovsky s approach to mesoscopic regime). Problem: control error term R when singularities come very close to the real axis. Figure : N-dependent contour for the R Riemann-Hilbert problem

109 Proof: Mesoscopic Riemann-Hilbert problem Proof is based on the RHP for orthogonal polynomials (extending Krasovsky s approach to mesoscopic regime). Problem: control error term R when singularities come very close to the real axis. Figure : N-dependent contour for the R Riemann-Hilbert problem RHP very useful for resolving this construction down to almost optimal scales d N = o(n/log(n)).