Convergence of spectral measures and eigenvalue rigidity

Convergence of spectral measures and eigenvalue rigidity Elizabeth Meckes Case Western Reserve University ICERM, March 1, 2018

Macroscopic scale: the empirical spectral measure

Macroscopic scale: the empirical spectral measure Suppose that M is an n n random matrix with eigenvalues λ 1,..., λ n. The empirical spectral measure µ of M is the (random) measure µ := 1 n n δ λk. k=1

Wigner s Theorem

Wigner s Theorem For each n N, let {Y i } 1 i, {Z ij } 1 i<j be independent collections of i.i.d. random variables, with EY 1 = EZ 12 = 0 EZ 2 12 = 1 EY 2 1 <. Let M n be the symmetric random matrix with diagonal entries Y i and off-diagonal entries Z ij or Z ji. The empirical spectral measure µ n of 1 n M n is close, for large n, to the semi-circular law: 1 4 x 2π 2 1 x 2 dx.

Other examples

Other examples The circular law (Ginibre): The empirical spectral measure of a large random matrix with i.i.d. Gaussian entries is approximately uniform on a disc.

Other examples The circular law (Ginibre): The empirical spectral measure of a large random matrix with i.i.d. Gaussian entries is approximately uniform on a disc. The classical compact groups (Diaconis Shahshahani): The empirical spectral measure of a uniform random matrix in O (n), U (n), Sp (2n) is approximately uniform on the unit circle when n is large.

Other examples Truncations of random unitary matrices (Petz Reffy): Let U m be the upper-left m m block of a uniform random matrix in U (n), and let α = m n.

Other examples Truncations of random unitary matrices (Petz Reffy): Let U m be the upper-left m m block of a uniform random matrix in U (n), and let α = m n. For large n, the empirical spectral measure of U m is close to the measure with density f α (z) = { 2(1 α), α(1 z 2 ) 2 0 < z < α; 0, otherwise.

Other examples Brownian motion on U (n) (Biane): Let {U t } t 0 be a Brownian motion on U (n); i.e., a solution to du t = U t dw t 1 2 U tdt, with U 0 = I and W t a standard B.M. on u(n). There is a deterministic family of measures {ν t } t 0 on the unit circle such that the spectral measure of U t converges weakly almost surely to ν t.

Other examples Brownian motion on U (n):

Levels of randomness

Levels of randomness Let µ n be the (random) spectral measure of an n n random matrix, and let ν be some deterministic measure which supposedly approximates µ n.

Levels of randomness Let µ n be the (random) spectral measure of an n n random matrix, and let ν be some deterministic measure which supposedly approximates µ n. The annealed case: The ensemble-averaged spectral measure is Eµ n : fd(eµ n ) := E fdµ n.

Levels of randomness The quenched case:

Levels of randomness The quenched case: Convergence weakly in probability or weakly almost surely: for any bounded continuous test function f, P a.s. f dµ n f dν or f dµ n f dν.

Levels of randomness The quenched case: Convergence weakly in probability or weakly almost surely: for any bounded continuous test function f, P a.s. f dµ n f dν or f dµ n f dν. The random variable d(µ n, ν): Look for ɛ n such that with high probability (or even probability 1), d(µ n, ν) < ɛ n.

Microscopic scale: eigenvalue rigidity

Microscopic scale: eigenvalue rigidity In many settings, eigenvalues concentrate strongly about predicted locations.

Microscopic scale: eigenvalue rigidity In many settings, eigenvalues concentrate strongly about predicted locations. 1.0 0.5-1.0-0.5 0.5 1.0-0.5-1.0

The eigenvalues of U m for m = 1, 5, 20, 45, 80, for U a realization of a random 80 80 unitary matrix.

Theorem (E. M. M. Meckes) Let 0 θ 1 < θ 2 < < θ n < 2π be the eigenvalue angles of U p, where U is a Haar random matrix in U (n). For each j and t > 0, P [ θ j 2πj N > 4π ] N t t 4 exp min ( 2 p log N p ) + 1, t.

Concentration of empirical spectral measures 2-D Coulomb gases

Concentration of empirical spectral measures 2-D Coulomb gases Coulomb transport inequality (Chafaï Hardy Maïda): Consider the 2-D Coulomb gas model with Hamiltonian H n (z 1,..., z n ) = j k log z j z k + n n V (z j ); let µ V denote the equilibrium measure. There is a constant C V such that j=1 d BL (µ, µ V ) 2 W 1 (µ, µ V ) 2 C V [E V (µ) E V (µ V )], where E V is the modified energy functional E V (µ) = E(µ) + Vdµ.

Truncations of random unitary matrices Let U be distributed according to Haar measure in U (n) and let n 1 m n. Let U m denote the top-left m m block of m U. The eigenvalue density of U m is given by 1 c n,m 1 j<k m z j z k 2 m j=1 ( 1 m n z j 2) n m 1 dλ(z1 ) dλ(z n ), which corresponds to a two-dimensional Coulomb gas with external potential n m 1 m log ( 1 m n Ṽ n,m (z) = z 2) n. z < m ; n, z m.

Truncations of random unitary matrices Theorem (M. Lockwood) Let µ m,n be the spectral measure of the top-left m m block of n m U, where U is a random n n unitary matrix and 1 m n 2 log(n). Let α = m n, and let ν α have density then g α (z) = { 2(1 α), (1 α z 2 ) 2 0 < z < 1; 0, otherwise. P[d BL (µ m,n, ν α ) > r] e Cαm2 r 2 +2m[log(m)+C α] + e cn, { } 1 where C α = min log(α 1 ), 1 and C α { log( 1 α ), α 0; log(1 α), α 1.

Concentration of empirical spectral measures Ensembles with concentration properties

Concentration of empirical spectral measures Ensembles with concentration properties If M is an n n normal matrix with spectral measure µ M and f : C R is 1-Lipschitz, it follows from the Hoffman-Wielandt inequality that M fdµ M is a 1 n -Lipschitz function of M.

Concentration of empirical spectral measures Many random matrix ensembles satisfy the following concentration property: Let F : S M N R be 1-Lipschitz with respect to H.S.. Then [ F(M) ] P EF(M) > t Ce cnt2. Some Examples: GUE; Wigner matrices in which the entries satisfy a quadratic transportation cost inequality with constant Wishart (sort of) c N. Haar measure and heat kernel measure on the compact classical groups: SO (N), U (N), SU (N), Sp (2N)

Typical vs. average

Typical vs. average In ensembles with the concentration property, W 1 (µ n, ν), this means P[W 1 (µ n, ν) > EW 1 (µ n, ν) + t] Ce cn2 t 2.

Typical vs. average In ensembles with the concentration property, W 1 (µ n, ν), this means P[W 1 (µ n, ν) > EW 1 (µ n, ν) + t] Ce cn2 t 2. To show W 1 (µ n, ν) is typically small, it s enough to show that EW 1 (µ n, ν) is small.

Average distance to average One approach: consider the stochastic process X f := fdµ n E fdµ n.

Average distance to average One approach: consider the stochastic process X f := fdµ n E fdµ n. Under the concentration hypothesis, {X f } f satisfies a sub-gaussian increment condition: P [ X f X g > t ] 2e cn 2 t 2 f g 2 L. Dudley s entropy bound together with approximation theory, truncation arguments, etc., can lead to a bound on ( ) EW 1 (µ n, Eµ n ) = E sup X f. f L 1

Theorem (M. Melcher) Let µ N t be the spectral measure of U t, where {U t } t 0 is a Brownian motion on U (n) with U 0 = I.

Theorem (M. Melcher) Let µ N t be the spectral measure of U t, where {U t } t 0 is a Brownian motion on U (n) with U 0 = I. 1. For any t, x > 0, ( P W 1 (µ N t, µ N t ) > c ( ) t 1/3 N 2 + x) 2e N2 x 2 t.

Theorem (M. Melcher) Let µ N t be the spectral measure of U t, where {U t } t 0 is a Brownian motion on U (n) with U 0 = I. 1. For any t, x > 0, ( P W 1 (µ N t, µ N t ) > c ( ) t 1/3 N 2 + x) 2e N2 x 2 t. 2. There are constants c, C such that for T 0 and x c T 2/5 log(n) P ( N 2/5, sup W 1 (µ N t, ν t ) > x 0 t T ) ( ) T C x 2 + 1 e N2 x 2 T. In particular, with probability one for N sufficiently large sup W 1 (µ N t, ν t ) c T 2/5 log(n) 0 t T N 2/5.

Second approach: Eigenvalue rigidity!

Second approach: Eigenvalue rigidity! The set of eigenvalues of many types of random matrices are determinantal point processes with symmetric kernels:

Second approach: Eigenvalue rigidity! The set of eigenvalues of many types of random matrices are determinantal point processes with symmetric kernels: GUE U (N) Complex Ginibre j=0 K N (x, y) n 1 h j (x)h j (y)e (x2 +y 2 ) 2 R 1 π N 1 j=0 N 1 j=0 (zw) j j! Λ e ij(x y) [0, 2π) e ( z 2 + w 2 ) 2 { z = 1}

The gift of determinantal point processes

The gift of determinantal point processes Theorem (Hough/Krishnapur/Peres/Virág) Let K : Λ Λ C be the kernel of a determinantal point process, and suppose the corresponding integral operator is self-adjoint, nonnegative, and locally trace-class. For D Λ, let N D denote the number of particles of the point process in D. Then d N D = ξ k, where {ξ k } is a collection of independent Bernoulli random variables. k

Concentration of the counting function

Concentration of the counting function Since N D is a sum of i.i.d. Bernoullis, Bernstein s inequality applies: ( { }) P [ N D EN D > t] 2 exp min where σd 2 = Var N D. t 2 4σ 2 D, t, 2

Rigidity of individual eigenvalues

Rigidity of individual eigenvalues If U is a random unitary matrix, then U has eigenvalues {e iθ k } N k=1, for 0 θ 1 < θ 2 < < θ N < 2π.

Rigidity of individual eigenvalues If U is a random unitary matrix, then U has eigenvalues {e iθ k } N k=1, for 0 θ 1 < θ 2 < < θ N < 2π. We define the predicted locations to be {e 2πik N } N k=1. concentration of N [0,θ] concentration of e iθ k about e ik N

Co-authors Kathryn Lockwood (Ph.D. student, CWRU): truncations of random unitary matrices Tai Melcher (UVA): Brownian motion on U (n) Mark Meckes (CWRU): most of the rest