Spectral rigidity for addition of random matrices at the regular edge Zhigang Bao HKUST Random Matrices and Related Topics KIAS, Seoul, Korea May 6 -10, 2019 Joint with L´ aszl´ o Erd˝ os and Kevin Schnelli 1
Addition of random matrices Given real A = diag( a 1 , . . . , a N ) and B = diag( b 1 , . . . , b N ), Matrix model: consider the model H = A + UBU ∗ where U is a Haar unitary matrix. Global spectral distribution [Voiculescu ’91]: � � µ A = 1 µ B = 1 Let i δ a i i δ b i N N When N is large, The empirical spectral distribution of H � µ H = 1 δ λ i , λ 1 � . . . � λ N : eigenvalues of H N i is close to the free additive convolution µ A ⊞ µ B . We choose neither A nor B to be multiples of identity. 2
Our questions Theorem [Voiculescu ’91] For any fixed interval I ⊂ R , | µ H ( I ) − µ A ⊞ µ B ( I ) | µ H ( I ) = |{ i : λ i ∈ I}| a.s. − − → 0 , N → ∞ . |I| N Alternative proofs [Speicher’93, Biane’98, Collins’03, Pastur-Vasilchuk’00] Question 1 (local law) Does the convergence still hold if |I| = o (1), and how 1 small can |I| be? ( Answer : N ) Question 2 (convergence rate) What is the convergence rate of � � 1 � µ H ( I ) − µ A ⊞ µ B ( I ) � ( Answer : N ) sup I⊂ R Question 3 (Spectral rigidity) What is the size of | λ i − γ i | where γ i is the N − i + 1-th N -quantile of µ A ⊞ µ B . 3
Stieltjes transform Definition: For any probability measure µ , its Stieltjes transform m µ ( z ) is � 1 z ∈ C + . m µ ( z ) = λ − z d µ ( λ ) , Inverse formula: one to one correspondence between measure and its Stielt- jes transform: density of µ given by ρ ( E ) = 1 π lim η ↓ 0 Im m µ ( E + i η ) . Notation: For α = A, B, and A ⊞ B , we will use m α ( z ) to denote the Stieltjes � δ a i and transfrom of µ A , µ B and µ A ⊞ µ B , respectively. Note that for µ A = 1 N � δ b i , we have µ B = 1 N � � m A ( z ) = 1 1 m B ( z ) = 1 1 a i − z, b i − z. N N 4
Analytic definition of free additive convolution Theoerm [Belinschi-Bercovici ‘06, Chistyakov-G¨ otze ‘05] There exist unique analytic ω A , ω B : C + → C + , s.t. ℑ ω k ( z ) � ℑ z and lim η ↑∞ ω k (i η ) = 1 i η for k = A, B , such that − [ m A ( ω B ( z ))] − 1 = ω A ( z ) + ω B ( z ) − z. m A ( ω B ( z )) = m B ( ω A ( z )) , • ω A ( z ) , ω B ( z ): subordination functions Let m ( z ) := m A ( ω B ( z )) = m B ( ω A ( z )). Claim : m ( z ) is a Stieltjes transform of a probability measure: µ A ⊞ µ B . • Algebraic definition: Addition of freely independent random variables [Voiculescu ‘86]. • Subordination phenomenon: [Voiculescu ‘93], [Biane ‘98]. 5
examples semicircle ⊞ semicircle 0.2 0.2 = ⊞ 0.1 0.1 - 2 - 1 - 2 - 1 0 1 2 0 1 2 semicircle ⊞ Bernoulli 1 / 2 1 / 2 0.2 0.2 ⊞ = 0.1 0.1 - 2 - 1 0 1 2 - 3 - 2 - 1 0 1 2 3 - 1 1 6
Bernoulli ⊞ Bernoulli 2 1 / 2 1 / 2 1 / 2 1 / 2 ⊞ = 1 - 2 - 1 0 1 2 - 1 1 - 1 1 three point masses ⊞ three point masses 3 1 / 2 1 / 4 1 / 2 1 / 4 1 / 4 1 / 4 2 ⊞ = 1 - 2 - 1 0 1 2 - 1 1 - 1 1 regular bulk: where the density is positive and finite regular edge: where the density vanishes as a square root 7
Optimal local law for the regular bulk Assumption: � A � , � B � � C ; µ A ⇒ µ α , µ B ⇒ µ β ; µ α , µ β not one point mass Theorem [B-Erd˝ os-Schnelli ’15b] local law for Stieltjes transform � � 1 � � N − 1+ γ � η � 1 , � m H ( E + i η ) − m A ⊞ B ( E + i η ) � ≺ Nη, E ∈ bulk , where m H is the Stieltjes transform of µ H . ⇓ Theorem [B-Erd˝ os-Schnelli ’15b] local law for spectral distribution | µ H ( I ) − µ A ⊞ µ B ( I ) | 1 N − 1+ γ � |I| � 1 , ≺ N |I| , I ⊂ bulk |I| [Kargin’12] ( η � (log N ) − 1 / 2 ), [Kargin’15] ( η � N − 1 / 7 ), Previous works : os-Schnelli’15a] ( η � N − 2 / 3 ). [B.-Erd˝ Notation A ≺ B : | A | � N ε | B | with high probability for any given ε > 0. 8
Extension to the edge: Assumption Assumption: � A � , � B � � C ; µ A ⇒ µ α , µ B ⇒ µ β (sufficiently fast), with µ α , µ β Jacobi type, i.e., µ α and µ β are a.c. with densities ρ α , ρ β supported on [ E α − , E α + ] and [ E β − , E β + ], respectively, and such that for some C � 1, ρ α ( x ) C − 1 � x ∈ [ E α − , E α + − x ) α + � C, a.e. + ] ( x − E α − ) α − ( E α ρ β ( x ) C − 1 � x ∈ [ E β − , E β + − x ) β + � C, a.e. + ] ( x − E β − ) β − ( E β with exponents − 1 < α ± , β ± < 1 . Theorem [B.-Erd˝ os-Schnelli ’18] Let µ α and µ β be of Jacobi type. Then supp µ α ⊞ µ β = [ E − , E + ] for some E − < E + ∈ R , and the density ρ α ⊞ β of µ α ⊞ µ β satisfies ρ α ⊞ β ( x ) C − 1 � � E + − x � C, a.e. x ∈ [ E − , E + ] . √ x − E − Similar problem was considered in [Olver-Nadakuditi ’12]. 9
Extension to the edge: Results Assumption: � A � , � B � � C ; µ A ⇒ µ α , µ B ⇒ µ β (sufficiently fast), with µ α , µ β Jacobi type Theorem [B.-Erd˝ os-Schnelli ’16-’18] Under the above assumption (i)(local law) For any fixed γ > 0, and any compact interval I ⊂ R with |I| � N − 1+ γ , | µ H ( I ) − µ A ⊞ µ B ( I ) | 1 ≺ |I| N |I| (ii) (convergence rate) � � � ≺ 1 � µ H ( I ) − µ A ⊞ µ B ( I ) sup N I⊂ R (iii) (rigidity) For any i = 1 , . . . , N , | λ i − γ i | ≺ max { i − 1 3 , ( N − i + 1) − 1 3 } N − 2 3 10
Local law of Green function Green function : G ( z ) := ( H − z ) − 1 , note � � m H ( z ) = 1 λ i − z = tr G ( z ) = 1 1 tr = 1 G ii ( z ) , N Tr . N N Theorem [B.-Erd˝ os-Schnelli ’16-’18] Let z = E + i η . Under the previous assumption, for any N − 1+ γ � η � 1 with any small γ > 0 (i) (Green function subordination) � � δ ij 1 � � max � G ij ( z ) − � ≺ √ Nη a i − ω B ( z ) i,j (ii) (Local law for Stieltjes transform) � � 1 � � � m H ( z ) − m A ⊞ B ( z ) � ≺ Nη (iii) (Improvement of (ii) outside the support) � � 1 � � � m H ( z ) − m A ⊞ B ( z ) κ := dist( E, ∂ supp( µ A ⊞ µ B )) � ≺ N ( κ + η ) , when E ∈ R \ supp( µ A ⊞ µ B ) and κ � N − 2 3 + ε . 11
Local laws in RMT Local laws for Wigner type matrices were widely studied in the last ten years. A key difference for the additive model is the complicated dependence struc- ture of the entries of the Haar unitary. For the model discussed: Universality of local bulk eigenvalue statistics was proved in [Che-Landon ’17] Some reference (on optimal scale) • (Wigner type) [Erd¨ os-Schlein-Yau ’07-’09], [Tao-Vu ’09-’12], [Erd¨ os-Yau- Yin ’10-’12], [Erd¨ os-Knowles-Yau-Yin ’13], [G¨ otze-Naumov-Tikhomirov-Timushev ’16], [G¨ otze-Naumov-Tikhomirov ’15-’19],... • (Addition of Wigner type) [Lee-Schnelli ’13], [Knowles-Yin ’14], [He-Knowles- Rosenthal ’16], [Ajanki-Erd¨ os-Kr¨ uger ’16], [Erd¨ os, Kr¨ uger, Schr¨ oder, ’18]... • (Random d -regular graph) [Bauerschmidt-Knowles-Yau ’15]... 12
Perturbed subordination equation for random matrix Subordination equation: Φ µ A ,µ B ( ω A ( z ) , ω B ( z ) , z ) = 0, where � � − ( m A ( ω 2 )) − 1 − ω 1 − ω 2 + z Φ µ A ,µ B ( ω 1 , ω 2 , z ) := − ( m B ( ω 1 )) − 1 − ω 1 − ω 2 + z Approximate subordination functions B ( z ) := z − tr UBU ∗ G ( z ) A ( z ) := z − tr AG ( z ) ω c ω c m H ( z ) , . m H ( z ) By ( A + UBU ∗ − z ) G = I , we have ( m H ( z )) − 1 = − ω c A ( z ) − ω c B ( z ) + z. Observe that � � − 1 − � � − 1 m A ( ω c m H ( z ) B ( z )) = Φ µ A ,µ B ( ω c A , ω c B , z ) � � − 1 − � � − 1 m B ( ω c m H ( z ) A ( z )) 13
Denote by Λ i ( z ) := ω c i ( z ) − ω i ( z ) , i = A, B, In order to estimate Λ i ( z ), we need two ingredients: (i): A stability analysis of the equation Φ µ A ,µ B ( ω A ( z ) , ω B ( z ) , z ) = 0. (ii): An estimate of Φ µ A ,µ B ( ω c A , ω c B , z ) = (Φ c 1 , Φ c 2 ) T , where 1 = ( m H ) − 1 − ( m A ( ω c 2 = ( m H ) − 1 − ( m B ( ω c Φ c B )) − 1 , Φ c A )) − 1 . 14
Local stability for subordination equation Expansion of the perturbed subordination eq. around ( ω A ( z ) , ω B ( z ) , z ) gives S Λ A + T A Λ 2 A + · · · = Φ c 1 + ( F ′ A ( ω B ) − 1)Φ c 2 S Λ B + T B Λ 2 B + · · · = Φ c 2 + ( F ′ B ( ω A ) − 1)Φ c 1 where F i ( · ) = − 1 /m i ( · ) , i = A, B are the negative reciprocal Stieltjes trans- forms, and � �� � F ′ F ′ S = A ( ω B ( z )) − 1 B ( ω A ( z )) − 1 − 1 � �� � � 2 + F ′′ � T A = 1 F ′′ F ′ F ′ A ( ω B ( z )) B ( ω A ( z )) − 1 B ( ω A ( z )) A ( ω B ( z )) − 1 2 and T B is defined analogously. Basic facts: � S ( z ) ∼ κ + η, T A ( z ) ∼ 1 , T B ( z ) ∼ 1 15
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