Cusp Local Law and Complex Hermitian Universality joint work with Johannes Alt, László Erdős and Torben Krüger Feburary 19, 2019 Workshop on Statistical Mechanics, Les Diablerets Dominik Schröder † † Partially supported by ERC Advanced Grant No. 338804
aa (matrix of all zeros, except for a 1 in the a a -th entry), Correlated Wigner Random Matrices . 1 aa Var h ab 2 E w ab E w ba w ab bb aa For R A random matrix h NN h N 1 N . . ... . h 11 h 1 N . . . . . H = H ∗ = ∈ C N × N . . . is called a correlated Wigner random matrix if it has the following properties: • Bounded expectation A . . = E H : � A � ≤ C , • Decaying correlations Cov ( h ij , h kl ) = E w ij w kl , where W . . = H − A , • Flat covariance operator S [ R ] . . = E WRW , i.e. c � R � ≤ S [ R ] ≤ C � R � , � R � . . = 1 N Tr R for R = R ∗ ≥ 0.
Correlated Wigner Random Matrices . A random matrix h NN h N 1 . h 11 . h 1 N . . . ... . . . H = H ∗ = ∈ C N × N . . . is called a correlated Wigner random matrix if it has the following properties: • Bounded expectation A . . = E H : � A � ≤ C , • Decaying correlations Cov ( h ij , h kl ) = E w ij w kl , where W . . = H − A , • Flat covariance operator S [ R ] . . = E WRW , i.e. c � R � ≤ S [ R ] ≤ C � R � , � R � . . = 1 N Tr R for R = R ∗ ≥ 0. For R = ∆ aa (matrix of all zeros, except for a 1 in the ( a , a ) -th entry), ( S [∆ aa ]) bb = E w ba w ab = E | w ab | 2 = Var h ab , � ∆ aa � = 1 N .
Optimal Local Law Theorem ((Erdős, Krüger, S. 2017), (Alt, Erdős, Krüger, S. 2018), (Erdős, Krüger, S. 2018)) Previous results for less general ensembles by (Adhikari, Ajanki, Che, He, Knowles, Lee, Wigner-type matrix, then (1) holds true also in the cusp regime. (1) O’Rourke, Schlein, Schnelli, Stetler, Rosenthal, Tau, Vu, Yau, Yin, …) matrix Dyson equation (MDE) . = ( H − z ) − 1 is well approximated by the solution M = M ( z ) to the The resolvent G = G ( z ) . ℑ M = M − M ∗ 1 = ( A − S [ M ] − z ) M , ℑ z > 0 . > 0 , 2 i Let H be a real symmetric or complex Hermitian correlated Wigner matrix such that Cov ( h ab , h cd ) � ( 1 + | ( a , b ) − ( c , d ) | ) − 12 − ǫ and M is bounded. Then the resolvent G ( z ) at z = E + i η satisfies �� � ρ ( E ) � u , ( G ( z ) − M ( z )) v � � N ǫ � u � � v � N η + 1 , N η � X [ G ( z ) − M ( z )] � � N ǫ � X � 1 N η for all η ≫ η f ( E ) above the fluctuation scale η f and E in the edge and bulk regime. If H is a
Derivation of the MDE
A . W . 1 WG G G W G 0 H W z 1 G 0 H W z Derivation of the MDE A E G GWG Assuming G E G , thus 1 A G z G This motivates studying the solution M M z to the MDE 1 A M E A E G E E WGWG . Gaussian integration by parts E xf x E x E f x Var x E f x E Hf H A E f H E E W W f W E H . H A Application to HG with G G H H z 1 : E HG A E G E E W W G z M 1 = HG − zG
1 WG Derivation of the MDE GWG H W z 1 G 0 H W z Assuming G W G E G , thus 1 A G z G This motivates studying the solution M M z to the MDE 1 A M 0 G G z M 1 : Gaussian integration by parts Application to HG with G E H z G H E HG A E G E E W W G A E G E E WGWG A E G 1 = HG − zG E xf ( x ) = ( E x )( E f ( x )) + ( Var x )( E f ′ ( x )) E Hf ( H ) = A E f ( H ) + E � E � W f ( W ) , . = E H , . = H − A . W ∂ � A . W .
Derivation of the MDE WG M A 1 M z to the MDE This motivates studying the solution M z G G A 1 E G , thus Assuming G WG z M Gaussian integration by parts 1 = HG − zG E xf ( x ) = ( E x )( E f ( x )) + ( Var x )( E f ′ ( x )) E Hf ( H ) = A E f ( H ) + E � E � W f ( W ) , . = E H , . = H − A . W ∂ � A . W . Application to HG with G = G ( H ) = ( H − z ) − 1 : E HG = A E G + E � E � W G = A E G − E � E � WG � W ∂ � WG = A E G − E S [ G ] G W − z ) − 1 − G ( H + ǫ � ( H + ǫ � W − z ) − 1 ǫ � = − G � ∂ � W G = lim = − lim ǫ ǫ ǫ → 0 ǫ → 0
Derivation of the MDE WG M A 1 M z to the MDE This motivates studying the solution M WG z M Gaussian integration by parts 1 = HG − zG E xf ( x ) = ( E x )( E f ( x )) + ( Var x )( E f ′ ( x )) E Hf ( H ) = A E f ( H ) + E � E � W f ( W ) , . = E H , . = H − A . W ∂ � A . W . Application to HG with G = G ( H ) = ( H − z ) − 1 : E HG = A E G + E � E � W G = A E G − E � E � WG � W ∂ � WG = A E G − E S [ G ] G W − z ) − 1 − G ( H + ǫ � ( H + ǫ � W − z ) − 1 ǫ � = − G � ∂ � W G = lim = − lim ǫ ǫ ǫ → 0 ǫ → 0 Assuming G ≈ E G , thus 1 ≈ ( A − S [ G ] − z ) G .
Derivation of the MDE WG WG Gaussian integration by parts 1 = HG − zG E xf ( x ) = ( E x )( E f ( x )) + ( Var x )( E f ′ ( x )) E Hf ( H ) = A E f ( H ) + E � E � W f ( W ) , . = E H , . = H − A . W ∂ � A . W . Application to HG with G = G ( H ) = ( H − z ) − 1 : E HG = A E G + E � E � W G = A E G − E � E � WG � W ∂ � WG = A E G − E S [ G ] G W − z ) − 1 − G ( H + ǫ � ( H + ǫ � W − z ) − 1 ǫ � = − G � ∂ � W G = lim = − lim ǫ ǫ ǫ → 0 ǫ → 0 Assuming G ≈ E G , thus 1 ≈ ( A − S [ G ] − z ) G . This motivates studying the solution M = M ( z ) to the MDE 1 = ( A − S [ M ] − z ) M .
Density of States = Empirical Distribution of Eigenvalues 0 2 4 0 0 . 2 0 . 1 − 4 − 2
Density of States = Empirical Distribution of Eigenvalues 1 0 2 4 0 i 0 . 2 0 . 1 − 4 − 2 Density of states ρ can be found by solving the matrix Dyson equation ℑ M = M − M ∗ − M ( z ) − 1 = z − A + S [ M ( z )] , > 0 , 2 i for ℑ z > 0, to obtain the Stieltjes transform N − 1 Tr M ( z ) and by Stieltjes inversion � ℑ M ii ( E + i η ) . ρ ( E ) . . = lim π N η ց 0
Density of States = Empirical Distribution of Eigenvalues 0 2 4 0 merge 0 . 2 0 . 1 − 4 − 2 Complete classification of singularities of DOS ρ achieved in (Alt, Erdős, Krüger 2018) • Only square-root edges and cubic root cusps • Cusps are not as ubiquitous as edges but arise naturally when two support intervals
f for 1 for z 1 Local law (on mesoscopic scales ): G M 1 z G f z Optimal local law: G M N N for M Global and Mesoscopic Scales Global law: bulk Typical scaling cusp edge Define the fluctuation scale η f � η f ( E ) ρ ( E + y ) d y = 1 N . − η f ( E ) N − 1 N − 3 / 4 η f = N − 2 / 3
Global and Mesoscopic Scales Global law: for cusp bulk Local law (on mesoscopic scales ): for Optimal local law: edge for Typical scaling Define the fluctuation scale η f � η f ( E ) ρ ( E + y ) d y = 1 N . − η f ( E ) N − 1 N − 3 / 4 η f = N − 2 / 3 |� G − M �| ≪ 1 η = ℑ z = O ( 1 ) |� G − M �| ≪ 1 η = ℑ z ≫ η f ( ℜ z ) |� G − M �| � N ǫ η ≫ η f N η
Proof of the Optimal Local Law
On the complement Q . Optimal Local Law Krüger, S. 2017) bulk, edge, 2 cusp. High probability error bound In all spectral regimes, and also for correlated matrices for general X (Erdős, XD Scaling N X N For indep. matrices improved cusp bound (Erdős, Krüger, S. 2018) VMD N 2 1 1 C , V , Stability analysis (Alt, Erdős, Krüger 2018) The operator B has a smallest eigenvalue V with spectral projection P V and eigenmatrix B V . 1 P , we have the bound BQ N Central idea: The resolvent G ( z ) = ( H − z ) − 1 almost fulfils the MDE, i.e. − 1 = ( z − A + S [ M ]) M , − 1 = ( z − A + S [ G ]) G + D , . = WG + S [ G ] G D . G − M = ( 1 − M S [ · ] M ) − 1 [ MD ] + . . . , . = 1 − M S [ · ] M stability operator B .
Optimal Local Law XD edge, cusp. High probability error bound In all spectral regimes, and also for correlated matrices for general X (Erdős, Krüger, S. 2017) N 1 X N For indep. matrices improved cusp bound (Erdős, Krüger, S. 2018) VMD N 2 bulk, N Stability analysis (Alt, Erdős, Krüger 2018) the bound Central idea: The resolvent G ( z ) = ( H − z ) − 1 almost fulfils the MDE, i.e. − 1 = ( z − A + S [ M ]) M , − 1 = ( z − A + S [ G ]) G + D , . = WG + S [ G ] G D . G − M = ( 1 − M S [ · ] M ) − 1 [ MD ] + . . . , . = 1 − M S [ · ] M stability operator B . • The operator B has a smallest eigenvalue β and eigenmatrix B [ V ] = β V with spectral projection P [ V ] = V , • On the complement Q . . = 1 − P , we have � ( BQ ) − 1 � � � ≤ C , • Scaling | β | ∼ ρ ρ 2
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