The largest eigenvalue of finite rank deformation of Wigner matrices - PDF document

The largest eigenvalue of finite rank deformation of Wigner matrices M. Capitaine, C. Donati-Martin and D. F´ eral 1 The model 1 √ M N = X N + A N := W N + A N N where W N is a N × N Hermitian matrix such that ( W N ) ii , √ √ 2 Re (( W N ) ij ) i<j , 2 Im (( W N ) ij ) i<j are iid with com- mon distribution µ . µ is assumed to be symmetric with variance σ 2 and it satisfies a Poincar´ e inequality. A N is a deterministic, Hermitian matrix. Example : µ = N (0; σ 2 ), X N ∼ GUE ( N, σ 2 N ). 2 Some known result in the non deformed case ( A N = 0 ) • Convergence of the spectral measure µ X n := 1 � i δ λ i ( X N ) √ N 4 σ 2 − x 2 1 [ − 2 σ, 2 σ ] . 1 to the Wigner distribution µ sc = 2 πσ 2 1

• Convergence a.s. of λ max ( X N ) to 2 σ (the right end- point of the support of the limiting distribution) • Fluctuations (Tracy-Widom, Soshnikov) σ − 1 N 2 / 3 ( λ max ( X N ) − 2 σ ) L − → T-W distribution F 2 where the distribution F 2 can be expressed with the Fredholm determinant of an operator associated to the Airy kernel. 3 The deformation A N Hermitian of finite rank r (independent of N) with eigenvalues θ i of multiplicity k i ; θ 1 > θ 2 > . . . > θ J . Convergence of the spectral measure to the semicircular distribution µ sc . What about the extremal eigenvalues? 2

1) The Gaussian case (P´ ech´ e) Ex: θ 1 with multiplicity 1. σ − 1 N 2 / 3 ( λ max ( M N ) − 2 σ ) L 1) si 0 ≤ θ 1 < σ , − → F 2 σ − 1 N 2 / 3 ( λ max ( M N ) − 2 σ ) L 2) si θ 1 = σ , − → F 3 N 1 / 2 ( λ max ( M N ) − ρ θ 1 ) L → N (0 , σ 2 − 3) si θ 1 > σ , θ 1 ) with ρ θ 1 = θ 1 + σ 2 θ 1 > 2 σ . 2) The non Gaussian case for a particular A N (F´ eral-P´ ech´ e) θ A N is the deformation defined by ( A N ) ij = N , so that r = 1 and θ 1 = θ . Same TCL as in the Gaussian case, universality of the fluctuations (independent of µ , the distribution of the entries). 3

3) The non Gaussian case, A N general Theorem 1 a.s. behaviour of the spectrum of M N . Let J + σ (resp. J − σ ) be the number of j’s such that θ j > σ (resp. θ j < − σ ). (a) ∀ 1 ≤ j ≤ J + σ , ∀ 1 ≤ i ≤ k j , λ k 1 + ··· + k j − 1 + i ( M N ) − → ρ θ j a.s., λ k 1 + ··· + k J + σ +1 ( M N ) − → 2 σ (b) a.s., λ k 1 + ··· + k J − J − σ ( M N ) − → − 2 σ (c) a.s., (d) ∀ j ≥ J − J − σ + 1 , ∀ 1 ≤ i ≤ k j , λ k 1 + ··· + k j − 1 + i ( M N ) − → ρ θ j a.s. Remark: Same result as in the sample covariance ma- trices (Bai-Silverstein, Baik-Silverstein) 4

4 Elements of Proof of Theorem 1 Step 1 Prove that a.s. Spect ( M N ) ⊂ K σ ( θ 1 , . . . θ J ) + [ − ǫ, + ǫ ] (1) for N large, where K σ ( θ 1 , · · · , θ J ) := � � � � ρ θ J ; · · · ; ρ θ J − J − σ +1 ∪ [ − 2 σ ; 2 σ ] ∪ ρ θ J + σ ; · · · ; ρ θ 1 . Tool: The Stieltjes transform: for z ∈ C \ R , define g N ( z ) = tr N ( G N ( z )) where G N ( z ) = ( zI N − M N ) − 1 is the resolvent of M N . We set h N ( z ) = E [ g N ( z )]. � 1 � 1 g N ( z ) = z − xdµ M N ( x ); h σ ( z ) = z − xdµ sc ( x ) . Aim : Obtain a precise estimate h σ ( z ) − h N ( z ) + 1 N L σ ( z ) = O ( 1 N 2 ) (2) where L σ is the Stieltjes transform of a distribution η with compact support in K σ . With the help of the inverse Stieltjes transform, � ϕ ( x ) dµ sc ( x )+ 1 � ϕ ( x ) dη ( x )+ O ( 1 E [tr N ( ϕ ( M N ))] = N 2 ) , N for ϕ smooth with compact support; and some variance estimates, we deduce from (2) 4 tr N 1 c K ε σ ( θ 1 , ··· ,θ J ) ( M N ) = O (1 /N 3 ) a.s. 5

and therefore the inclusion of the spectrum (1). Proof of (2): 1) The Gaussian Case: • The Gaussian integration by parts formula: φ : R → C , ξ standard Gaussian ′ ( ξ )) . E ( ξφ ( ξ )) = E ( φ Φ : H n ( C ) → C , H ∈ H n ( C ) , N ′ ( X N ) · H ] σ 2 E [Tr( X N H )Φ( X N )] = E [Φ Apply it for Φ( X N ) = [( zI N − X N − A N ) − 1 ] kl = G N ( z ) kl and H = E kl ; then sum over k and l . N ( z )] − z E [ g N ( z )]+1+ 1 → σ 2 E [ g 2 N E [Tr( G N ( z ) A N )] = 0 N ( z ) − zh N ( z )+1+ 1 N E [Tr( G N ( z ) A N )] = O ( 1 → σ 2 h 2 N 2 ) Recall that σ 2 h 2 σ ( z ) − zh σ ( z ) + 1 = 0. 6

Estimate for E [Tr( G N ( z ) A N )] : A N = U ∗ Λ U where Λ is a diagonal matrix with entries λ i � = 0 for i ≤ r , λ i = 0, i > r . We can show using • The Gaussian integration by parts formula • Some variance estimates • h N ( z ) = h σ ( z ) + O ( 1 N ) the estimate r z − λ i − σ 2 h σ ( z ) + O ( 1 λ i � E [Tr( G N ( z ) A N )] = N ) i =1 Set r λ i θ i R A N � � G ( z ) = z − λ i − σ 2 h σ ( z ) = k i z − θ i − σ 2 h σ ( z ) . i =1 θ i � =0 Then, N ( z ) − zh N ( z ) + 1 + 1 G ( z ) = O ( 1 N R A N σ 2 h 2 N 2 ) leading to h N ( z ) − h σ ( z ) + 1 N L ( z ) = O ( 1 N 2 ) σ ( z ) E [( z − sc ) − 2 ] R A N where L ( z ) = h − 1 G ( z ). 7

Question: - L Stieltjes transform of a distribution ? - Support of this distribution? ← → Analyticity of L (+ conditions); set of singular points. If | θ i | > σ , x ∈ R \ [ − 2 σ, 2 σ ], ⇒ x = θ i + σ 2 x − θ i − σ 2 h σ ( x ) = 0 ⇐ := ρ θ i . θ i 2) The non Gaussian case GIP replaced by: (Khorunzhy, Khoruzhenko, Pastur) Lemma 1 Let ξ be a real-valued rv such that E ( | ξ | p +2 ) < ∞ . Let φ : R → C such that the first p +1 derivatives are continuous and bounded. Then, p κ a +1 � a ! E ( φ ( a ) ( ξ )) + ǫ E ( ξφ ( ξ )) = a =0 where κ a are the cumulants of ξ , | ǫ | ≤ C sup t | φ ( p +1) ( t ) | E ( | ξ | p +2 ) . Apply to ξ = Re (( X N ) ij ) , Im (( X N ) ij ) , ( X N ) ii , the odd cumulants vanish ( µ symmetric). One must consider the third derivative of Φ = ( G N ( z )) kl . 8

One obtains: N ( z ) − zh N ( z ) + 1 + 1 N R ( z ) = O ( 1 σ 2 h 2 N 2 ) where R ( z ) = R A N G ( z ) + κ 4 R 0 Φ ′′′ ( z ). R 0 Φ ′′′ ( z ) is analytic on C \ [ − 2 σ, 2 σ ]. 9