Convergence of spectral measures and eigenvalue rigidity Elizabeth - PowerPoint PPT Presentation

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 n µ := 1 � δ λ k . n k = 1

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 n µ := 1 � δ λ k . n 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 E Z 2 E Y 2 1 < ∞ . E Y 1 = E Z 12 = 0 12 = 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 2 ✶ | x |≤ 2 dx . 2 π

♣ 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 ) , S ♣ ( 2 n ) 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 0 < | z | < √ α ; � 2 ( 1 − α ) α ( 1 −| z | 2 ) 2 , f α ( z ) = 0 , otherwise .

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 0 < | z | < √ α ; � 2 ( 1 − α ) α ( 1 −| z | 2 ) 2 , f α ( z ) = 0 , otherwise . α = 4 α = 2 5 5 Figures from “Truncations of random unitary matrices”, ˙ Zyczkowski–Sommers, J. Phys. A, 2000

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 t dt , 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 E µ n : � � fd ( E µ n ) := E fd 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 E µ n : � � fd ( E µ n ) := E fd µ n . One may prove that E µ n ⇒ ν , possibly via explicit bounds on d ( E µ n , ν ) in some metric d ( · , · ) .

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 , � � � � a . s . P fd µ n − → fd ν or fd µ n − − → fd ν.

Levels of randomness The quenched case: ◮ Convergence weakly in probability or weakly almost surely: for any bounded continuous test function f , � � � � a . s . P fd µ n − → fd ν or fd µ n − − → fd ν. ◮ The random variable d ( µ n , ν ) : Look for ǫ n such that with high probability (or even probability 1), d ( µ n , ν ) < ǫ 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 < 2 π be the eigenvalue angles of U p , where U is a Haar random matrix in U ( n ) . For each j and t > 0 ,     t 2 �� � � � θ j − 2 π j � > 4 π    . � � N t ≤ 4 exp  − min , t P � � � � N N + 1 p log   p

Concentration of empirical spectral measures 2-D Coulomb gases

Concentration of empirical spectral measures 2-D Coulomb gases Coulomb transport inequality (Chafa¨ ı–Hardy–Ma¨ ıda): Consider the 2-D Coulomb gas model with Hamiltonian n � � H n ( z 1 , . . . , z n ) = − log | z j − z k | + n V ( z j ); j � = k j = 1 let µ V denote the equilibrium measure. There is a constant C V such that 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 ) + Vd µ.

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 m 1 1 − m n | z j | 2 � n − m − 1 � � | z j − z k | 2 � d λ ( z 1 ) · · · d λ ( z n ) , ˜ c n , m 1 ≤ j < k ≤ m j = 1 which corresponds to a two-dimensional Coulomb gas with external potential  � − n − m − 1 1 − m n � n | z | 2 � log | z | < m ; .  m ˜ V n , m ( z ) = � 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 2 ( 1 − α ) � 0 < | z | < 1 ; ( 1 − α | z | 2 ) 2 , g α ( z ) = 0 , otherwise . then P [ d BL ( µ m , n , ν α ) > r ] ≤ e − C α m 2 r 2 + 2 m [ log ( m )+ C ′ α ] + e − cn , � � 1 where C α = min log ( α − 1 ) , 1 and � log ( 1 α ) , α → 0 ; C ′ α ∼ 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 �→ fd µ M 1 is a √ n -Lipschitz function of M .

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 �→ fd µ M 1 is a √ n -Lipschitz function of M . ⇒ For any reference measure ν , = M �→ W 1 ( µ M , ν ) 1 is √ n -Lipschitz

♣ 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 �� � ≤ Ce − cNt 2 . � > t � � F ( M ) − E F ( M ) P

♣ 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 �� � ≤ Ce − cNt 2 . � > t � � F ( M ) − E F ( M ) P Some Examples:

♣ 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 �� � ≤ Ce − cNt 2 . � > t � � F ( M ) − E F ( M ) P Some Examples: ◮ GUE; Wigner matrices in which the entries satisfy a c quadratic transportation cost inequality with constant N . √