Extreme eigenvalue fluctutations for GUE

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1 Extreme eigenvalue fluctutations for GUE C. Donati-Martin 204 Program Women and Mathematics, IAS Introduction andom matrices were introduced in multivariate statistics, in the thirties by Wishart [Wis] and in theoretical physics by Wigner [Wig] in the fifties. Since then, the theory developed in a wide range of mathematics fields and physical mathematics. These lectures give a brief introduction to a well studied model : the Gaussian Unitary ensemble (GUE). The GUE is both a Wigner matrix (independent entries) and a model invariant by unitary conjugation. The Gaussian structure enable to compute explicitely some quantities leading to a complete description of the global and local behavior of the spectrum. In particular, in the asymptotics N where N is the size of the matrix, we shall study the fluctuation of the largest eigenvalue around its deterministic limit and prove a central limit theorem towards the so called Tracy-Widom distribution. 2 The Gaussian unitary ensemble - Definition Let H N be the space of Hermitian matrices of size N i.e. matrices M such that M = M. H N is a real vector space of dimension N 2. Definition 2. X N is a gaussian unitary matrix of size N, variance σ 2 (denoted by GUE(N; σ 2 )) if : - X N H N, - The entries of X N satisfy :

2 X N (i, i), i N, 2X N (j, k); j < k, 2IX N (j, k); j < k are independent and distributed as N(0, σ 2 ). One can give an equivalent definition (exercice): Definition 2.2 The distribution P N,σ 2 of GUE(N; σ 2 ) is given by dp N,σ 2(M) = exp( Z N,σ 2 2σ Tr(M 2 ))dm (2.) 2 where dm denotes the Lebesgue measure on H N given by and Z N,σ 2 dm = N i= dm ii is a normalizing constant. i<j N dm ij dim ij We are interesting in the behavior of GUE(N; σ 2 ) in the asymptotics N. The good normalization to see something at the limit is to take σ 2 =. N In the following, we take σ 2 = and denote P N N, by P N. N emark 2. ) GUE(N; ) is a Wigner matrix (independent entries above N the diagonal). Proposition 2. The distribution P N is invariant under unitary conjugation : if X N is P N distributed, then UX N U is also P N distributed, for all unitary matrix U. Proof: We make the change of variable in (2.) : T U : M U MU, for U unitary. T U is an isometry on H N since : T U (M) 2 = Tr((T U (M)) 2 ) = Tr(U MUU MU) = Tr(M 2 ) = M 2. Thus det(t U ) =. On the other hand, exp( 2σ 2 Tr((U MU) 2 ) = exp( 2σ 2 Tr(M 2 )). We conclude, by the change of variable formula, that : E(f(UX N U )) = E(f(X N )). 2

3 2. Distribution of the eigenvalues of GUE(N, N ) Let X N a random matrix distributed as GUE(N, )) and we denote by N λ (X N ) λ N (X N ) the ranked eigenvalues of X N. Proposition 2.2 The joint distribution of the eigenvalues λ (X N ) λ N (X N ) has a density with respect to Lebesgue measure equal to p N (x) = ZN i<j N (x j x i ) 2 exp( N 2 N x 2 i ) x x N (2.2) i= (x) = i<j N (x j x i ) is called the Vandermonde determinant and equals det(x j i ) i,j N. We refer to Mehta [Me, Chap. 3], Deift [D, Chap. 5], Anderson-Guionnet- Zeitouni [AGZ, Chap. 2] for the proof of this proposition. It relies on the expression of the N 2 components of M in (2.) in terms of the N eigenvalues (x i ) and N(N ) independent parameters (p i ) which parametrize the unitary matrix U in the decomposition M = Udiag(x)U. Heuristically, the term exp( N N 2 i= x2 i ) comes from the exp( N Tr(M 2 )) in P 2 N and the square of the Vandermonde determinant comes from the Jacobian of the map M ((x i ), U) after integration on U on the unitary group. Corollary 2. If f is a bounded function of H N, invariant by the unitary transformations, that is f(m) = f(umu ) for all unitary matrix U then f(m) = f(λ (M),, λ N (M)) is a symmetric function of the eigenvalues and E[f(X N )] = Z N H N f(m) exp( 2 N Tr(M 2 ))dm = ZN x x N f(x,, x N ) = N! Z N N f(x,, x N ) i<j N i<j N (x i x j ) 2 exp( N 2 (x i x j ) 2 exp( N k-point correlation functions of the GUE N x 2 i )d N x i= N x 2 i )d N x Let ρ N a symmetric density distribution on N, considered as the distribution of N particles X i. 3 i=

4 Definition 2.3 Let k N. The k- point correlation functions of (X i ) are defined by N! N,k (x, x k ) = ρ N (x, x N )dx k+ dx N. (2.3) (N k)! N k The correlation functions are, up to a constant, the marginal distributions of ρ N. Heuristically, N,k is the probability of finding a particle at x,... N! a particle at x k. The factor comes from the choice of the k particles (N k)! and the symmetry of ρ N (see the computation below). We have, using the symmetry of ρ N, N N E[ ( + f(x i )] = E[ f(x i ) f(x ik )] i= k=0 i < <i k N = E[ f(x i ) f(x ik )] i < <i k and thus, [ N ] E ( + f(x i )) = i= N k=0 = = k=0 N k=0 N k=0 ( N k k! ) E[f(X ) f(x k )] N! (N k)! E[f(X ) f(x k )] f(x ) f(x k ) N,k (x, x N )dx dx k. k! k (2.4) The correlation functions enables to express probabilistic quantities as: ) The hole probability: Take f(x) = \I where I is a Borel set of. Then, the left-hand side of (2.4) is the probability of having no particles in I. Therefore, P( i, X i I) = N ( ) k k=0 k! I k N,k (x, x k )dx dx k. In particular, for I =]a, + [, N ( ) k P(max X i a) = N,k (x, x k )dx dx k. (2.5) k! [a, [ k k=0 4

5 2) the density of state: E[ N f(x i )] = N N i= f(x) N, (x)dx that is N N,(x)dx represents the expectation of the empirical distribution E[ N N i= δ X i ]. We now compute the correlation functions associated to the symmetric density of the (unordered) eigenvalues of the GUE ρ N (x) = N! Z N i<j N (x i x j ) 2 exp( N 2 N x 2 i ). i= Proposition 2.3 The correlation functions of the eigenvalues of GUE(N, N ) are given by N,k(x, xk) = det(kn(xi, xj)) i,j k (2.6) where the kernel K N is given by where K N (x, y) = exp( N N 4 (x2 + y 2 )) q l (x)q l (y) (2.7) q l (x) = where h l are the Hermite polynomials. l=0 ( ) /4 N 2π 2l l! h l( N/2 x) (2.8) The process of the eigenvalues of GUE is said to be a determinantal process. Proof: Since the value of a determinant does not change if we replace a column by the column + a linear combination of the others, we have that the Vandermonde determinant (x) = det(p j (x i )) if P j denotes a polynomial of degree j with higher coefficient equal to. Let w(x) = exp( N 2 x2 ) and define the orthonormal polynomials q l with respect to w such that: - q l is of degree l, q l (x) = a l x l +... with a l > 0. 5

6 - q l(x)q p (x)w(x)dx = δ pl. (q l ) l also depends on N and is up to a scaling factor the family of Hermite polynomials (to be discussed later). Thus, (x) = C N det(q j (x i )) and, using (det(a)) 2 = det(a) det(a T ), we have: ρ N (x) = ZN w(x i ) (det(q j (x i ))) 2 where i N = ZN i N ( N ) w(x i ) det q l (x i )q l (x j ) l= = ZN det (K N (x i, x j )) i,j N K N (x, y) = w(x) w(y) N q l (x)q l (y) = l= N l=0 i,j N φ l (x)φ l (y) where φ l (x) = w(x)q l (x). The sequence (φ l ) l is orthornormal for the Lebesgue measure dx. From the orthonormality of (φ l ), it is easy to show that the kernel K N satisfies the properties: K N (x, x)dx = N K N (x, y)k N (y, z)dy = K N (x, z). This proves (2.6) for k = N (up to a constant). The general case follows from the Lemma: Lemma 2. Let J N = (J ij ) a matrix of size N of the form J ij = f(x i, x j ) with f satisfying:. f(x, x)dx = C 2. f(x, y)f(y, z)dy = f(x, z) Then, det(j N )dx N = (C N + ) det(j N ) where J N is a matrix of size N obtained from J N by removing the last row and column containing x N. 6

7 Proof of Lemma 2.: Exercice Hint: use the formula giving the determinant : det(j N ) = σ Σ N ɛ(σ) N f(x i, x σ(i) ) where Σ N is the set of permutations on {,..., N} and ɛ stands for the signature of a permutation. Next, consider two cases for σ : σ(n) = N and σ(n) N. In the case of GUE, J = (K N (x i, x j )) satisfies the hypothesis of the lemma with C = N. det(k N (x i, x j )) i,j N dx N = (N N ) det(k N (x i, x j )) i,j N det(k N (x i, x j )) i,j N dx N = (N N 2) det(k N (x i, x j )) i,j N 2. Integrating over all the variables gives: det(k N (x i, x j )) i,j N dx... dx N = N! and therefore, Z N = N!. Integrating over the N k variables dx k+,... dx N gives: i= and det(k N (x i, x j )) i,j N dx... dx N = (N k)! det(k N (x i, x j )) i,j k N! k,n (x,... x k ) = ρ N (x,... x N )dx k+... dx N (N k)! = det(k N (x i, x j )) i,j N dx k+... dx N (N k)! = det(k N (x i, x j )) i,j k This proves (2.6) and (2.7). It remains to determine the polynomials q l. Let h l the Hermite polynomial of degree l defined by: ( ) l d h l (x) = ( ) l e x2 (e x2 ). dx 7

8 These polynomials (see [Sz]) are orthogonal with respect to e x2 dx, h2 l (x)e x2 dx = 2 l l! π and the coefficient of x l in h l is 2 l. Then, it is easy to see that q l given by (2.8) are orthonormal with respect to exp( N 2 x2 )dx. Corollary 2.2 Let µ HN (dx) = E[µ HN (dx)] where µ HN is the spectral distribution of GUE(N, ), then µ N H N (dx) is absolutely continuous with respect to Lebesgue measure with density f N given by: f N (x) = N N,(x, x) = N K N(x, x), x. f N is called the density of state. 2.3 The local regime Let us denote, for I a Borel set of, ν N (I) = #{i N; λ i I} = Nµ HN (I) where λ i are the eigenvalues of GUE(N, ). From Wigner s theorem, as N N, ν N (I) N f I sc(x)dx) a.s. where f sc is the density of the semicircular distribution µ sc. The spacing between eigenvalues is of order /N. In the local regime, we consider an interval I N whose size tends to 0 as N. Two cases have to be considered. a) Inside the bulk: Take I N = [u ε N, u + ε N ] with u such that f sc (u) > 0 that is u ] 2, 2[. Then, ν N (I N ) has the order of a constant for ε N. This suggest to introduce new random variables (renormalisation) N l i by l i λ i = u +, i =,... N. Nf sc (u) The mean spacing between the rescaled eigenvalues l i is. Straightforward computations give: Lemma 2.2 The correlation functions bulk of the distribution of (l,..., l N ) are given in terms of the correlation functions of the (λ i ) by bulk N,k (y,..., y k ) = (Nf sc (u)) k N,k(u + y Nf sc (u),..., u + y k ). (2.9) Nf sc (u) We shall see in the next subsection the asymptotic of the correlation functions bulk (or the kernel K N ). 8

9 b) At the edge of the spectrum: u = 2 (or -2). f sc (u) = 0. ν N ([2 ε N, 2]) = N 2π 2 2 ε 4 x2 dx = N 2π So the normalisation at the edge is ε = correlation functions by: edge N,k (y,..., y k ) = N 2/3 ε (N 2/3 ) k N,k(2 + y N 2/3,..., y y2 dy CNε 3/2. and we define the rescaled y k ). (2.0) N 2/3 From (2.5) and (2.0), P[N 2/3 (λ max 2) a] = N ( ) k k=0 k! [a, [ k edge N,k (x, x k )dx dx k. (2.) where λ max is the maximal eigenvalue of the GUE. The asymptotic of edge will be given in the next section. 2.4 Limit kernel The asymptotic of the correlation functions relies on asymptotic formulas for the orthonormal polynomials q l for l N. We have the following: Proposition 2.4 (Plancherel - otach formulas, [Sz]) Let (h n ) n denote the Hermite polynomials. ) If x = 2n + cos(φ) with ε Φ π ε, exp( x 2 /2)h n (x) = b n (sin(φ)) /2 {sin[( n )(sin(2φ) 2Φ)+3π/4]+O( n )} where b n = 2 n/2+/4 (n!) /2 (πn) /4. 2) If x = 2n /2 n /6 t, t bounded in C, exp( x 2 /2)h n (x) = π /4 2 n/2+/4 (n!) /2 (n) /2 {Ai(t) + O( n )} (2.2) where Ai is Airy s function, that is the solution of the differential equation y = xy with y(x) x + 2 π x /4 exp( 2 3 x3/2 ). 9

10 From these formulas, one can show: Theorem 2. lim N bulk n,k (y,..., y k ) = det(k bulk (y i, y j )) i,j k (2.3) where where K bulk (x, y) = sin(π(x y)) π(x y) (2.4) lim N edge n,k (y,..., y k ) = det(k edge (y i, y j )) i,j k (2.5) K edge (x, y) = Ai(x)Ai (y) Ai (x)ai(y) (x y) (2.6) Sketch of Proof of (2.3): From (2.9), (2.6), we may find the limit of Nf sc (u) K N(u + s Nf sc (u), u + t Nf sc (u) ). We express the kernel K N given by (2.7) thanks to Cristoffel-Darboux formula (see Appendix) K N (X, Y ) = k N k N K N (X, Y ) = q N (X)q N (Y ) q N (Y )q N (X) X Y exp( N 4 (X2 + Y 2 )) 2 N (N )! h N ( N/2 X)h N ( N/2 Y ) h N ( N/2 Y )h N ( N/2 X) π X Y exp( N 4 (X2 + Y 2 )) with k N the highest coefficient in q N. Then, set X = u + s, Y = Nf sc(u) u + t, u = 2 cos(φ). Then, f Nf sc(u) sc(u) = sin(φ) and π x = N/2X = 2N(cos(Φ) + πs 2N sin(φ) ). In order to use Plancherel-otach formulas, we express x as x = 2N + cos(φ N ). 0

11 A development gives Φ N = Φ + with a = 2 tan(φ) πs sin 2 (Φ). Then, and a 2N + O( N ) 2 sin(2φ N ) 2Φ N = (sin(2φ) 2Φ) + a (cos(2φ) ) + O( N N ) 2 Formula (2.2) gives: (sin(φ N )) /2 = (sin(φ)) /2 ( + O(/N)). exp( x 2 /2)h N (x) = b N (sin(φ)) /2 {sin[( N )(sin(2φ) 2Φ)+a 2 (cos(2φ) )+3π 4 ]+O(/N)} We make the same transformations for e x2 /2 h N (x), e y2 /2 h N (y), e y2 /2 h N (y) giving φ N, Ψ N and Ψ N associated respectively to: a = 2 tan(φ) πs sin 2 (Φ), b = 2 tan(φ) πt sin 2 (Φ), b = 2 tan(φ) πt sin 2 (Φ). Then, we replace in the product h N (x)h N (y) the product of two sinus by a trigonometric formula and then in the difference, we obtain a linear combination of cosinus, The difference of two of them cancels using that a + b = a + b. Then, we use again a trigonometric formula. After some computations, the kernel K bulk appears. The Airy kernel appears, using the second formula of Plancherel-otach. Corollary 2.3 (Fluctuations of λ max ) The fluctuations of the largest eigenvalue of the GUE around 2 are given by: P(N 2/3 (λ max 2) x) = F 2 (x) where F 2 is called the Tracy-Widom distribution and is given by F 2 (x) = ( ) k k=0 k! ]x, [ k det(k edge (y i, y j )) i,j k d k y. F 2 can be written F 2 (x) = det(i K) L 2 (x, ) where K is the integral operator on L 2 with kernel K edge (x, y) and the det is the Fredholm determinant.

12 2.5 Comments. The computation of the correlation functions which have a determinantal form is specific to the unitary case and do not hold for the GOE case. 2. We refer to [D], [Me] for others computations involving correlation functions such as the spacing distribution. 3. The Tracy-Widom distribution can also be expressed as ( ) F 2 (x) = exp (y x)q 2 (y)dy) x where q (x) = xq(x) + q 3 (x) with q(x) = Ai(x)( + o()) as x. The function q is called the solution of Painlevé II equation (see [TW]). 4. One of the important ideas of the theory is that of universality. This idea is that the asymptotic distribution of some statistic of the eigenvalues in the local scale does not depend very much on the ensemble (like in the TCL), that is the sine kernel (2.4) or the Airy kernel is universal and appears in other models of Hermitian random matrices. This has been shown for - Hermitian Wigner matrices: Soshnikov [So] (for the edge), Johansson [J] for a particular class of matrices. The universality in the bulk was proved in great generality independently by two teams : Erdos, Schlein, Yau [ESY] and their collaborators on one hand and Tao, Vu [TV] on the other hand. - unitary invariant ensemble of the form P N (dm) = C N exp( N Tr(V (M)))dM for a weight V satisfying some assumptions. See [DKMVZ], [PS]. Note that the GUE corresponds to the quadratic weight V (x) = 2 x2. For example, for the Wishart ensemble (associated to the Laguerre polynomials), we have the same asymptotic kernel as in the GUE, while the density of state is not universal (semicircular for GUE and Marchenko- Pastur distribution for Wishart). The main difficulty for general V is to derive the asymptotics of orthogonal polynomials. This can be done using iemann-hilbert techniques (see [D]). 2

13 3 Appendix 3. Change of variable formula Let U and V two open sets in d, g a C diffeomorphism from U to V. If φ is a measurable function on V, positive or Lebesgue integrable, then φ(g(x)) Dg Dx (x) dx = f(y)dy U where Dg (x) is the Jacobian of g, that is the determinant of the Jacobian Dx matrix ( ) gi x j. i,j d 3.2 Van der Monde determinant ecall the van der Monde determinant: det(x j i ) i,j n = i<j(x j x i ). V 3.3 Orthogonal polynomials (see [D], [Sz]) Let w(x) a positive function on such that x m w(x)dx < for all m 0. On the space of real polynomials P [X], we consider the scalar product (P Q) = P (x)q(x)w(x)dx. Then the orthogonalisation procedure of Schmidt enables to construct of sequence of orthogonal polynomials (p l ): p l is of degree l and p m (x)p n (x)w(x)dx = 0 if m n. We denote by a l the coefficient of x l in p l (x) and d l = p l(x) 2 w(x)dx. Example: If w(x) = exp( x 2 ), the Hermite polynomials h l are orthogonal with a l = 2 l and d l = 2 l l! π. 3

14 Christoffel-Darboux formula: We consider a family of orthonormal polynomials (p l ) (d l = ) for the weight w. We denote by K n the kernel defined by: n K n (x, y) = p l (x)p l (y). l=0 K n is the kernel associated to the orthogonal projection in the space of polynomials of degree less than n. This kernel has a simple expression based upon a three terms recurrence relation between the (p l ): xp n (x) = α n p n+ (x) + β n p n (x) + α n p n (x) for some coefficients α n = an a n+ and β n ( depending on a n and the coefficient b n of x n in p n ). From this relation, one obtains: K n (x, y) = α n p n (x)p n (y) p n (y)p n (x) x y (3.) For the orthonormal polynomials q l defined in (2.8), and α N =. a l (= a N,l ) = 3.4 Fredholm determinant ( ) /4 ( ) l N N 2π l! Let K(x, y) a bounded measurable kernel on a space (X, µ) where µ is a finite measure on X. The Fredholm determinant of K is defined by ( λ) k D(λ) = det(i λk) := + det(k(x i, x j )) i,j k µ(dx )... µ(dx k ). k! X k k= The serie converges for all λ. 4

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16 [TW] Tracy, C.A and Widom, H. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 59 (994), no., [Wig] E. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) [Wis] J. Wishart The generalized product moment distribution in samples from a normal multivariate population. Biometrika 20 (928),