Local law of addition of random matrices Kevin Schnelli 1 IST - PowerPoint PPT Presentation

Local law of addition of random matrices Kevin Schnelli 1 IST Austria Joint work with Zhigang Bao and L´ aszl´ o Erd˝ os 1 Supported by ERC Advanced Grant RANMAT No. 338804

Spectrum of sum of random matrices Question : Given A = diag ( a 1 , . . . , a N ) and B = diag ( b 1 , . . . , b N ) , what is the eigenvalue density of the random matrix H = A + UBU ∗ if U is a Haar unitary and N is large? Answer : [Voiculescu ‘91] N N � � µ A := 1 µ B := 1 Let δ a i , δ b i . N N i =1 i =1 Then for large N the empirical spectral distribution of A + UBU ∗ , � N µ H := 1 δ λ i , λ i : eigenvalues of H , N i =1 is close to µ A ⊞ µ B , the free additive convolution of µ A and µ B . Of course, we choose neither A nor B to be multiples of the identity matrix. Wlog: Tr A = Tr B = 0 .

Stieltjes transform Definition: For any probability measure ν , its Stieltjes transform m ν ( z ) is defined by � 1 z ∈ C + . m ν ( z ) := x − z d ν ( x ) , R Observe: m ν : C + → C + , analytic and lim η ր∞ i η m ν (i η ) = − 1 . Define (negative) reciprocal Stieltjes transform: 1 z ∈ C + . F ν ( z ) := − m ν ( z ) , F ν (i η ) Observe: F ν : C + → C + , analytic and lim = 1 . i η η ր∞

Free additive convolution Analytic definition via subordination functions: Symmetric binary operation on the set of probability measures uniquely characterized by the following result: Theorem (Belinschi-Bercovici ‘07, Chistyakov-G¨ otze ‘11). Given µ A and µ B (thus also F µ A and F µ B ), there exist unique analytic ω A , ω B : C + → C + , such that ω A (i η ) ω B (i η ) (1) Im ω A ( z ) , Im ω B ( z ) ≥ Im z and lim = lim = 1 ; i η i η η ր∞ η ր∞ (2) � F µ A ( ω B ( z )) = ω A ( z ) + ω B ( z ) − z self-consistent equation (SCE) for ω A , ω B . F µ B ( ω A ( z )) = ω A ( z ) + ω B ( z ) − z By (2) : F µ A ( ω B ( z )) = F µ B ( ω A ( z ))=: F ( z ) . By (1) : F ( z ) is the reciprocal Stieltjes transform of a probability measure: µ A ⊞ µ B . Algebraic definition: Addition of free random variables [Voiculescu ‘86]. Subordination phenomenon: [Voiculescu ‘93], [Biane ‘98].

Examples I semicircle ⊞ semicircle 0.2 0.2 0.2 0.1 0.1 0.1 = ⊞ - 2 - 1 0 1 2 - 2 - 1 0 1 2 - 2 - 1 0 1 2 semicircle ⊞ Bernoulli 1 / 2 1 / 2 0.2 0.2 0.1 0.1 ⊞ = - 2 - 1 0 1 2 - 1 1 - 3 - 2 - 1 0 1 2 3

Examples II 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 / 4 1 / 2 1 / 4 1 / 4 1 / 2 1 / 4 2 1 = ⊞ - 2 - 1 0 1 2 - 1 1 - 1 1 Definition: Regular bulk: Free additive convolution admits a finite and strictly positive density. Lemma: Inside the regular bulk, η ց 0 Im ω A ( E + i η ) > 0 , lim η ց 0 Im ω B ( E + i η ) > 0 . lim

Theorem (Voiculescu ‘91). � N Let H = A + UBU ∗ and µ H := 1 δ λ i , with ( λ i ) the eigenvalues of H . N i =1 For any fixed interval I ⊂ R , | µ H ( I ) − µ A ⊞ µ B ( I ) | a.s. − − → 0 , N → ∞ . |I| Alternative proofs: [Speicher ‘93], [Biane ‘98], [Pastur-Vasilchuk ‘00], [Collins ‘03],... Question 1 (local law): Does the convergence still hold if |I| = o (1) , and how small can |I| be? Question 2 (convergence rate): What is the convergence rate, as N ր ∞ , of � � µ H ( I ) − µ A ⊞ B ( I ) � � . sup I⊂ R Questions 1 and 2 are related.

Main result: Theorem (Bao-Erd˝ os-S. ‘15b). N � Let H = A + UBU ∗ and µ H := 1 δ λ i , with ( λ i ) the eigenvalues of H . N i =1 Fix any γ > 0 . For any compact interval I in the regular bulk with |I| ≥ N − 1+ γ , | µ H ( I ) − µ A ⊞ µ B ( I ) | 1 � ≺ , |I| N |I| for N sufficiently large.

Main result: Theorem (Bao-Erd˝ os-S. ‘15b). Fix any γ > 0 . For any compact interval I in the regular bulk with |I| ≥ N − 1+ γ , we have | µ H ( I ) − µ A ⊞ µ B ( I ) | 1 � ≺ , |I| N |I| for N sufficiently large. Remarks: ◦ Technical assumption: � A � , � B � ≤ C . ◦ Typical eigenvalue spacing in the regular bulk is order 1 /N . ◦ Special case: Entries of A and B are supported at two points (Bernoulli). ◦ Previous results: | µ H ( I ) − µ A ⊞ µ B ( I ) | 1 |I| ≥ N − 1 / 7+ γ ≺ N |I| 7 , [Kargin ‘12-‘15] |I| | µ H ( I ) − µ A ⊞ µ B ( I ) | 1 |I| ≥ N − 2 / 3+ γ ≺ N |I| 3 / 2 , [Bao-Erd˝ os-S. ‘15a] |I|

Main technical result: Local law Local law is mostly stated in terms of the Green function G ( z ) := ( H − z ) − 1 . Link with � N Stieltjes transform m H ≡ m µ H : tr G ( z ) = 1 1 tr := 1 λ i − z = m H ( z ) , N Tr. N i =1 Theorem (Bao-Erd˝ os-S. ‘15b). Choose any compact interval I in the regular bulk of µ A ⊞ µ B , and set S I ( γ ) := { z = E + i η : E ∈ I , N − 1+ γ ≤ η < ∞} . For any (small) γ > 0 , we have � � � � 1 � m H ( z ) − m µ A ⊞ µ B ( z ) � ≺ √ Nη , � � � δ ij � 1 � G ij ( z ) − � ≺ √ Nη , uniformly on S I ( γ ) . a i − ω B ( z ) � N Recall: m µ A ⊞ µ B ( z ) = m µ A ( ω B ( z )) = 1 1 a i − ω B ( z ) . N i =1

About local laws in RMT Local laws for the spectrum of random matrices have been widely studied since the works by Erd˝ os-Schlein-Yau-Yin etc.. It serves as an input for proving the universality of local statistics. Some reference: (on optimal scale) ◦ (Wigner type matrices) [Erd˝ os-Schlein-Yau ‘07-‘09], [Tao-Vu ‘09-‘12], [Erd˝ os-Yau-Yin ‘10-‘12], [Erd˝ os-Knowles-Yau-Yin ‘13], [Ajanki-Erd˝ os-Kr¨ uger ‘15], [G˝ otze-Naumov-Tikhomirov ‘15 ], .... Remarks: ◦ Schur complement is used, which expresses G ii in terms of a ∗ i G ( i ) a i , where a i is a column of the matrix and G ( i ) (a submatrix of G ) is independent of a i .

Local stability of SCE � F µ A ( ω 2 ) − ω 1 − ω 2 + z � Let Φ µ A ,µ B ( ω 1 , ω 2 , z ) := . F µ B ( ω 1 ) − ω 1 − ω 2 + z SCE for ω A , ω B : Φ µ A ,µ B ( ω A ( z ) , ω B ( z ) , z ) = 0 . Local Stability: [Bao-Erd˝ os-S. ‘15a] Fix z ∈ S I ( γ ) . Assume ω c A , ω c B , r satisfy Im ω c A ( z ) , Im ω c B ( z ) > 0 and Φ µ A ,µ B ( ω c A ( z ) , ω c B ( z ) , z ) = r ( z ) , and that there is a small δ > 0 such that | ω c | ω c A ( z ) − ω A ( z ) | ≤ δ , B ( z ) − ω B ( z ) | ≤ δ . Then we have, in the regular bulk, uniformly in Im z ≥ 0 , | ω c | ω c A ( z ) − ω A ( z ) | ≤ C � r ( z ) � , B ( z ) − ω B ( z ) | ≤ C � r ( z ) � . Previous results: Local stability with an additional condition [Kargin ‘13].

Perturbed SCE for random matrix 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 ) Since ( A + UBU ∗ − z ) G ( z ) = I , we have 1 m H ( z ) = ω c A ( z ) + ω c − B ( z ) − z . Our aim: Show that 1 � Φ µ A ,µ B ( ω c A ( z ) , ω c B ( z ) , z ) � ≺ √ Nη , z = E + i η , which is equivalent to � � 1 m H ( z ) = m µ A ( ω c B ( z )) + O ≺ , √ Nη � � 1 m H ( z ) = m µ B ( ω c A ( z )) + O ≺ √ Nη .

Main task: Prove � � 1 1 G ii ( z ) = B ( z ) + O ≺ √ Nη . a i − ω c Non-optimal way: Using the full randomness of U at once Full expectation E [ G ii ] + Gromov-Milman concentration for G ii − E [ G ii ] . Optimal way: Separating some partial randomness v i from U Partial expectation E v i [ G ii ] + Concentration for G ii − E v i [ G ii ] . Remark: Shorthand E i := E v i . In general, identifying E [ · ] is easier than identifying E i [ · ] , while estimating (Id − E )[ · ] is harder than estimating (Id − E i )[ · ] .

Householder reflection as partial randomness Proposition (Diaconis-Shahshahani ‘87). U Haar distributed on U ( N ) , � 1 � := − e i θ 1 R 1 U � 1 � , U = − e i θ 1 ( I − 2 r 1 r ∗ 1 ) U 1 e 1 + e − i θ 1 v 1 r 1 := . � e 1 + e − i θ 1 v 1 � 2 v 1 denotes the first column of U , v 1 is uniformly distributed on S N − 1 , C U 1 is Haar on U ( N − 1) , v 1 and U 1 are independent. Remark 1: − e i θ 1 R 1 is the Householder reflection sending e 1 to v 1 . Remark 2: Analogously, we have an independent pair v i and U i for all i . Remark 3: Independence between v i and U i enables us to work with the partial expectation E v i [ G ii ] .

Concentration of Green function elements Lemma. For all z ∈ S I ( γ ) , � � G ii ( z ) − E i [ G ii ( z )] � 1 � ≺ √ Nη , z = E + i η . Proof: Use resolvent expansions to write ii + Ψ i G ii = G [ i ] , Ξ i G [ i ] : a matrix independent of v i ; Ψ i , Ξ i : polynomials of quadratic forms x ∗ i G [ i ] y i , with x i , y i = e i , v i . Then concentration of quadratic forms, e.g. � � � ≺ � G [ i ] � 2 � � i G [ i ] v i ] = tr G [ i ] , � v ∗ i G [ i ] v i − E i [ v ∗ i G [ i ] v i ] E i [ v ∗ , N implies concentration of G ii .

Green function entries Aim: B ( z ) = z − tr � 1 BG ( z ) ω c B := UBU ∗ � G ii ≈ B ( z ) , tr G ( z ) , a i − ω c From ( H − z ) G ( z ) = 1 , we have ( a i − z ) G ii = − ( � BG ) ii + 1 , so that 1 G ii = . a i − z + ( � BG ) ii G ii We shall show: Proposition. For all i = 1 , 2 , . . . , N , BG ) ii ≈ tr � BG ( � tr G G ii .