From Wigner-Dyson to Pearcey: Universality of the Local Eigenvalue - PowerPoint PPT Presentation

From Wigner-Dyson to Pearcey: Universality of the Local Eigenvalue Statistics of Random Matrices at the Cusp László Erdős IST Austria (supported by ERC Advanced Grant RANMAT) Workshop on Statistical Mechanics Les Diablerets, Switzerland February 18-19, 2019 László Erdős From Wigner-Dyson to Pearcey 1

Based upon several joint works with Oskari Ajanki Torben Krüger (Bonn) Dominik Schröder Johannes Alt (Geneva) Giorgio Cipolloni László Erdős From Wigner-Dyson to Pearcey 2

Introduction . From Wigner-Dyson to Pearcey László Erdős Analogy: Central limit theorem: eigenvalues? . . . What can be said about the . . . . . 3 statistical properties of the eigenvalues of a large random matrix? Eugene Wigner (1954) Do some universal patterns emerge?   h 11 h 12 . . . h 1 N   h 21 h 22 . . . h 2 N   H =   = ⇒ ( λ 1 , λ 2 , . . . , λ N )   h N 1 h N 2 . . . h NN N = size of the matrix, will go to infjnity. 1 N ( X 1 + X 2 + . . . + X N ) ∼ N (0 , σ 2 ) √

Wigner matrix ensemble possible. No exact formula beyond Gaussian. From Wigner-Dyson to Pearcey László Erdős Goal: Statistics of eigenvalues 4 H = ( h jk ) 1 ≤ j,k ≤ N complex hermitian or real symmetric N × N matrix h jk = ¯ h kj (for j < k ) are indep. random variables with normalization E | h jk | 2 = 1 E h jk = 0 , N . The eigenvalues λ 1 ≤ λ 2 ≤ . . . ≤ λ N are of order one: (on average) � � E 1 i = E 1 N Tr H 2 = 1 E | h ij | 2 = 1 λ 2 N N i ij If h ij is Gaussian, then GUE, GOE – (hard) explicit calculations are

Observation scales: Macro/Meso/Micro eigenvalues behave on the scale From Wigner-Dyson to Pearcey László Erdős that eigenvalues form a strongly correlated point process. Very difgerent behavior than for independent (Poisson) points. Indicates (Wigner-Dyson universality) of spacing? 5 local semicircle law ) Size of the spectrum O(1). Typical ev. spacing (gap) ≈ 1 N (bulk) • Macro scale: Semicircle Law • Meso scale: LLN holds in the entire mesoscopic regime ( = ⇒ • Micro scale: How do individual

Wigner semicircle law László Erdős From Wigner-Dyson to Pearcey 6 Density of eigenvalues on the global (macroscopic) scale: Let H be a Wigner matrix with eigenvalues λ 1 , λ 2 , . . . , λ N . � b � 1 ̺ ( x ) = 1 (4 − x 2 ) + N # { λ i ∈ [ a, b ] } → ̺ ( x )d x, 2 π a holds for any fjxed [ a, b ] interval. 1 N ≪ | b − a | ≪ 1 ? Local Law ! What about shrinking intervals as

Typical meso observable: Stieltjes transform and Resolvent Inversion formula From Wigner-Dyson to Pearcey László Erdős (weakly) 7 Def: Let µ be a probability measure on R . Its Stieltjes transform at spectral parameter z ∈ C + is given by � d µ ( x ) m µ ( z ) := x − z R Meaning: m µ ( E + iη ) resolves the measure µ around E on scale η � 1 δ η ( x − E )d µ ( x ) π Im m µ ( z ) := ( δ η ⋆ µ )( E ) = R where δ η is an approximate delta fn. on scale η � δ η ( x ) := 1 η x 2 + η 2 , δ η ( x )d x = 1 π 1 lim π Im m µ ( E + iη ) = µ ( E )d E η → 0+

Stieltjes transform & Resolvent (cont’d) Fact: From Wigner-Dyson to Pearcey László Erdős [edge] Local law tells us that the eigenvalues are uniformly distributed down to This limit is the Stieltjes transform of the density of states (DOS) 8 Stieltjes transform of its empirical spectral density: Obvious: The trace of the resolvent G of a hermitian matrix H is the � N � ̺ N ( E ) := 1 N Tr G ( z ) = 1 1 1 δ ( λ α − E ) , λ α − z = m ̺ N ( z ) N N α α =1 Recall: η = Im z is the resolution scale. 1 N Tr G ( z ) becomes deterministic as long as η ≫ (ev spacing). In fact, it holds for G ( z ) itself [Mesoscopic LLN or local law] the smallest possible scales η above the local ev spacing: η ≫ N − 1 η ≫ N − 2 / 3 [bulk] ,

Typical micro observable: Wigner surmise and level repulsion László Erdős From Wigner-Dyson to Pearcey 9 � � ≃ π 2 s e − πs 2 / 4 d s P GOE N̺ ( λ i )( λ i +1 − λ i ) = s + d s � � ≃ 32 π 2 s 2 e − 4 s 2 /π d s , P GUE N̺ ( λ i )( λ i +1 − λ i ) = s + d s for λ i in the bulk (repulsive correlation!) 1.0 0.8 0.6 0.4 0.2 0.0 0.5 1.0 1.5 2.0 2.5 Histogram of the rescaled gaps for GUE matrix with N = 3000

Wigner-Dyson universality The universality of microscopic ev. gap fmuctuation is expected to hold for From Wigner-Dyson to Pearcey László Erdős Wigner’s revolutionary observation: The global density may be model 10 Physically the Wigner matrix is a special toy model, but it is the basic In particular, it can be determined from the Gaussian case (GUE/GOE). symmetric) and not on other details of the RM ensemble. robust, it depends only on the symmetry class (complex hermitian or real dependent, but the gap statistics (i.e. micro-scale fmuctuation) is very prototype of a disordered quantum system in the mean fjeld regime models very far from mean fjeld, most prominent example is the Anderson model in the conducting regime: � on ℓ 2 (Λ) , with Λ = [ − L, L ] d ∩ Z d H = ∆ x + v i δ ( x − i ) i ∈ Λ where { v i : i ∈ Λ } is i.i.d., L → ∞ . Big open question!

Models of increasing complexity [E-Schlein-Yau-Yin, 2009–2011], [Tao-Vu, 2009] [Ajanki-E-Krüger, 2015] [Ajanki-E-Krüger ’15-’16] [Che ’16], [E-Krüger-Schröder ’17], [Alt-E-Krüger-Schröder ’18] László Erdős From Wigner-Dyson to Pearcey 11 • Wigner matrix: i.i.d. entries, s ij := E | h ij | 2 are constant (= 1 N ) (Density = semicircle; G ≈ diagonal, G xx ≈ G yy ) (Same if � j s ij = 1 , for all i – [E-Yau-Yin, ’11] [E-Knowles-Yau-Yin, ’12] ) • Wigner type matrix: indep. entries, s ij arbitrary (Density � = semicircle; G ≈ diagonal, G xx �≈ G yy ) • Correlated Wigner matrix: correlated entries, s ij arbitrary (Density � = semicircle; G �≈ diagonal)

Typical correlation structure: like 2d random fjeld Cov From Wigner-Dyson to Pearcey László Erdős + Matching bounds for higher cumulants (smallest spanning tree) 12 A ) � � � C K �∇ φ � ∞ �∇ ψ � ∞ φ ( W A ) , ψ ( W B ) [1 + dist ( A, B )] 12 for any A, B ⊂ S × S , assuming the usual metric on the set S = { 1 , 2 , . . . , N } of indices. Here W A = { W ij : ( i, j ) ∈ A } . ( d ( A, B ) A B B

Mean fjeld quantum Hamiltonian with correlation Non-trivial spatial structure changes the density of states but not the micro statistics! László Erdős From Wigner-Dyson to Pearcey 13 H is viewed as a Σ × Σ matrix (operator) acting on ℓ 2 (Σ) . Equip the confjguration space Σ with a metric to have ”nearby” states. It is reasonable to assume that h xy and h xy ′ are correlated if y and y ′ are close with a decaying correlation as dist ( y, y ′ ) increases.

Other extensions of the original Wigner model Girko, Bai, Tao-Vu-Krishnapur, Bordenave-Chafai, Fyodorov, Bourgade-Yau-Yin, Alt-E-Kruger, From Wigner-Dyson to Pearcey László Erdős Many other directions and references are left out, apologies... Bourgade-E-Yau-Yin, E-Bao, Bourgade-Yau-Yin etc. Fyodorov-Mirlin, Disertori-Pinson-Spencer, Schenker, Sodin, E-Knowles-Yau, T. Shcherbina, E-Kruger-Renfrew, Bourgade-Dubach E-Knowles-Yau-Yin, Huang-Landon-Yau, Bauerschmidt-Huang-Knowles-Yau, etc. O’Rourke-Vu, Lee-Schnelli-Stetler-Yau, He-Knowles-Rosenthal Johansson, Guionnet-Bordenave, Götze-Naumov-Tikhomirov, Benaych-Peche, Aggarwal Deift et. al., Valko-Virag, Pastur-Shcherbina, Bourgade-E-Yau, Bekerman-Guionnet-Figalli 14 � � • Invariant ensembles: P ( H ) ∼ exp − βN Tr V ( H ) • Low moment assumptions, heavy tails • Deformed models, general expectation • Sparse matrices, Erdős-Rényi and d-regular graphs • Nonhermitian matrices • Band matrices

Matrix Dyson Equation to compute density of states Self-consistent DOS From Wigner-Dyson to Pearcey László Erdős 15 G ( z ) = ( H − z ) − 1 , where H = H ∗ has a correlation structure given S [ R ] := E ( H − A ) R ( H − A ) , A := E H Theorem [AEK, EKS] In the bulk spectrum, ̺ ( ℜ z ) ≥ c , we have � � 1 1 � � G ( z ) − M ( z ) � � � √ | G xy ( z ) − M xy ( z ) | � , N Im z N Im z with very high probability, where � A � := 1 N Tr A . M is given by the solution of the Matrix Dyson Equation (MDE) Im M := M − M ∗ − 1 M = z − A + S [ M ] , ≥ 0 , Im z > 0 2 i ρ ( E ) := π − 1 � Im M ( E + i 0) � Depends only on the fjrst two moments of H .

Rigidity Local law on optimal scale implies rigidity of the eigenvalues on the From Wigner-Dyson to Pearcey László Erdős with very high probability. Lemma: Optimal local law (in averaged form) implies 16 of the density. optimal scale, i.e. that eigenvalues are close to the corresponding quantiles Given a density ρ and x ∈ R , defjne the local spacing η f ( x ) as � x + η f ρ ( y )d y = 2 N x − η f Let γ i = γ ( N ) be the i -th N -quantile of ρ : i � γ i ρ ( y )d y = i N −∞ | λ i − γ i | ≤ N ε η f ( γ i )