The largest eigenvalue of finite rank deformation of large Wigner - PowerPoint PPT Presentation

The largest eigenvalue of finite rank deformation of large Wigner matrices: convergence and non-universality of the fluctuations M. Capitaine, C. Donati-Martin, D. F´ eral I M T Univ Toulouse 3 and CNRS, Equipe de Statistique et Probabilit´ es UPMC Univ Paris 06 and CNRS, Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires

√ Step 1: Inclusion of the spectrum of M N = W N / N + A N P [ Spect ( M N ) ⊂ K σ ( θ 1 , · · · , θ J ) + ( − ε ; ε ) for large N ] = 1 . � � � � 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 . ρ θ i = θ i + σ 2 θ i if | θ i | > σ .

√ Step 1: Inclusion of the spectrum of M N = W N / N + A N P [ Spect ( M N ) ⊂ K σ ( θ 1 , · · · , θ J ) + ( − ε ; ε ) for large N ] = 1 . � � � � 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 . � ρ θ i = θ i + σ 2 1 1 θ i if | θ i | > σ . θ i = g σ ( ρ θ i ) , g σ ( z ) = z − t d µ sc ( t ).

√ Step 1: Inclusion of the spectrum of M N = W N / N + A N P [ Spect ( M N ) ⊂ K σ ( θ 1 , · · · , θ J ) + ( − ε ; ε ) for large N ] = 1 . � � � � 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 . � ρ θ i = θ i + σ 2 1 1 θ i if | θ i | > σ . θ i = g σ ( ρ θ i ) , g σ ( z ) = z − t d µ sc ( t ). −∞ − 2 σ + ∞ x ρ θ J 2 σ ρ θ 2 ρ θ 1 − σ + ∞ ր ր θ 1 ր 1 θ J θ 2 g σ ( x ) ր ր −∞ σ

Step 2: Exact separation phenomenon 1 1 [ a , b ] gap in Spect( M N ) ← → [ g σ ( a ) , g σ ( b ) ] gap in Spect( A N ) ✲ 1 1 σ θ 3 g σ ( b ) θ 2 θ 1 g σ ( a ) � �� � � �� � N-l eigenvalues of A N l eigenvalues of A N ✲ 2 σ ρ θ 3 a b ρ θ 2 ρ θ 1 � �� � � �� � N-l eigenvalues of M N l eigenvalues of M N

Theorem Exact separation phenomenon 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 . c K σ ( θ 1 , . . . , θ J ) , i N ∈ { 0 , . . . , N } s.t [ a , b ] ⊂ 1 1 λ i N +1 ( A N ) < and λ i N ( A N ) > g σ ( a ) g σ ( b ) ( λ 0 := + ∞ and λ N +1 := −∞ ). Then P [ λ i N +1 ( M N ) < a and λ i N ( M N ) > b , for large N ] = 1 .

Assume that θ 1 > σ . λ 1 ( M N ) → ρ θ 1 a.s.?

Assume that θ 1 > σ . λ 1 ( M N ) → ρ θ 1 a.s.? P [ Spect ( M N ) ⊂ K σ ( θ 1 , · · · , θ J ) + ( − ε ; ε ) for large N ] = 1 . � � � � 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 . ⇒ P [ λ 1 ( M N ) < ρ θ 1 + ǫ for large N ] = 1 . =

Assume that θ 1 > σ . λ 1 ( M N ) → ρ θ 1 a.s.? P [ Spect ( M N ) ⊂ K σ ( θ 1 , · · · , θ J ) + ( − ε ; ε ) for large N ] = 1 . � � � � 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 . ⇒ P [ λ 1 ( M N ) < ρ θ 1 + ǫ for large N ] = 1 . = By the exact separation phenomenon with [ a ; b ] = [ θ 2 + η ; θ 1 − η ], 1 ( θ 1 − η = g σ ( ρ θ 1 − ǫ ) ) P [ λ 1 ( M N ) > ρ θ 1 − ǫ, for large N ] = 1 .

Assume that θ 1 > σ . λ 1 ( M N ) → ρ θ 1 a.s.? P [ Spect ( M N ) ⊂ K σ ( θ 1 , · · · , θ J ) + ( − ε ; ε ) for large N ] = 1 . � � � � 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 . ⇒ P [ λ 1 ( M N ) < ρ θ 1 + ǫ for large N ] = 1 . = By the exact separation phenomenon with [ a ; b ] = [ θ 2 + η ; θ 1 − η ], 1 ( θ 1 − η = g σ ( ρ θ 1 − ǫ ) ) P [ λ 1 ( M N ) > ρ θ 1 − ǫ, for large N ] = 1 . = ⇒ P [ ρ θ 1 − ǫ < λ 1 ( M N ) < ρ θ 1 + ǫ for large N ] = 1 .

Theorem A N = diag ( θ, 0 , · · · , 0) with θ > σ . Then � � � � √ → (1 − σ 2 D N λ 1 ( M N ) − ρ θ − θ 2 ) µ ∗ N (0 , v θ ) . � m 4 − 3 σ 4 � σ 4 v θ = t + t θ 2 − σ 2 θ 2 4 2 with t = 4 (resp. t = 2 ) if W N is real (resp. complex) and � x 4 d µ ( x ) . m 4 := ⇒ NON-UNIVERSALITY OF THE FLUCTUATIONS OF λ 1 ( M N ) since they do depend on µ

Theorem A N = diag ( θ, 0 , · · · , 0) with θ > σ . Then � � � � √ → (1 − σ 2 D N λ 1 ( M N ) − ρ θ − θ 2 ) µ ∗ N (0 , v θ ) . � m 4 − 3 σ 4 � σ 4 v θ = t + t θ 2 − σ 2 θ 2 4 2 with t = 4 (resp. t = 2 ) if W N is real (resp. complex) and � x 4 d µ ( x ) . m 4 := ⇒ NON-UNIVERSALITY OF THE FLUCTUATIONS OF λ 1 ( M N ) since they do depend on µ In the other particular case ( A N ) ij = θ N ∀ 1 ≤ i , j ≤ N , µ symmetric with sub-gaussian moments, D. F´ eral and S. P´ ech´ e: if θ > σ, then � � � √ D 1 − σ 2 → N (0 , σ 2 N λ 1 ( M N ) − ρ θ − θ ), σ θ = σ θ 2 .

� M N − 1 : the N − 1 × N − 1 matrix obtained from M N removing the √ N − 1 � N first row and the first column. ⇒ M N − 1 is a non-Deformed √ Wigner matrix associated with the measure µ . t (( M N ) 21 , . . . , ( M N ) N 1 ) . ˇ M · 1 =   θ + ( W N ) 11 ˇ M ∗ √ · 1 N   M N =   ˇ � M · 1 M N − 1 M N − 1 , ˇ � M · 1 , ( W N ) 11 are independent.

� M N − 1 : the N − 1 × N − 1 matrix obtained from M N removing the √ N − 1 � N first row and the first column. ⇒ M N − 1 is a non-Deformed √ Wigner matrix associated with the measure µ . t (( M N ) 21 , . . . , ( M N ) N 1 ) . ˇ M · 1 =   θ + ( W N ) 11 ˇ M ∗ √ · 1 N   M N =   ˇ � M · 1 M N − 1 M N − 1 , ˇ � M · 1 , ( W N ) 11 are independent. V = t ( v 1 , . . . , v N ) eigenvector relative to λ 1 := λ 1 ( M N ). t ( v 2 , . . . , v N ) � V =

� M N − 1 : the N − 1 × N − 1 matrix obtained from M N removing the √ N − 1 � N first row and the first column. ⇒ M N − 1 is a non-Deformed √ Wigner matrix associated with the measure µ . t (( M N ) 21 , . . . , ( M N ) N 1 ) . ˇ M · 1 =   θ + ( W N ) 11 ˇ M ∗ √ · 1 N   M N =   ˇ � M · 1 M N − 1 M N − 1 , ˇ � M · 1 , ( W N ) 11 are independent. V = t ( v 1 , . . . , v N ) eigenvector relative to λ 1 := λ 1 ( M N ). t ( v 2 , . . . , v N ) � V = � λ 1 v 1 = ( θ + ( W N ) 11 N ) v 1 + ˇ · 1 � M ∗ V √ M N V = λ 1 V ⇐ ⇒ V = v 1 ˇ λ 1 � M · 1 + � M N − 1 � V

0 < δ < ρ θ − 2 σ . ( ρ θ > 2 σ ) � 4 � λ 1 ( � M N − 1 ) ≤ 2 σ + δ ; λ N − 1 ( � Ω N = M N − 1 ) ≥ − 2 σ − δ ; λ 1 ( M N ) ≥ ρ θ − δ lim N → + ∞ P (Ω N ) = 1 .

0 < δ < ρ θ − 2 σ . ( ρ θ > 2 σ ) � 4 � λ 1 ( � M N − 1 ) ≤ 2 σ + δ ; λ N − 1 ( � Ω N = M N − 1 ) ≥ − 2 σ − δ ; λ 1 ( M N ) ≥ ρ θ − δ lim N → + ∞ P (Ω N ) = 1 . On Ω N , � G ( λ 1 ) := ( λ 1 I N − 1 − � M N − 1 ) − 1

0 < δ < ρ θ − 2 σ . ( ρ θ > 2 σ ) � 4 � λ 1 ( � M N − 1 ) ≤ 2 σ + δ ; λ N − 1 ( � Ω N = M N − 1 ) ≥ − 2 σ − δ ; λ 1 ( M N ) ≥ ρ θ − δ lim N → + ∞ P (Ω N ) = 1 . On Ω N , � G ( λ 1 ) := ( λ 1 I N − 1 − � M N − 1 ) − 1 � λ 1 v 1 = ( θ + ( W N ) 11 N ) v 1 + ˇ M ∗ · 1 � V √ V = v 1 ˇ λ 1 � M · 1 + � M N − 1 � V � θ v 1 + ( W N ) 11 N v 1 + v 1 ˇ · 1 � G ( λ 1 ) ˇ M ∗ λ 1 v 1 = M · 1 . √ ⇔ � v 1 � G ( λ 1 ) ˇ V = M · 1 .

0 < δ < ρ θ − 2 σ . ( ρ θ > 2 σ ) � 4 � λ 1 ( � M N − 1 ) ≤ 2 σ + δ ; λ N − 1 ( � Ω N = M N − 1 ) ≥ − 2 σ − δ ; λ 1 ( M N ) ≥ ρ θ − δ lim N → + ∞ P (Ω N ) = 1 . On Ω N , � G ( λ 1 ) := ( λ 1 I N − 1 − � M N − 1 ) − 1 � λ 1 v 1 = ( θ + ( W N ) 11 N ) v 1 + ˇ M ∗ · 1 � V √ V = v 1 ˇ λ 1 � M · 1 + � M N − 1 � V � θ v 1 + ( W N ) 11 N v 1 + v 1 ˇ · 1 � G ( λ 1 ) ˇ M ∗ λ 1 v 1 = M · 1 . √ ⇔ � v 1 � G ( λ 1 ) ˇ V = M · 1 . ⇒ λ 1 = θ + ( W N ) 11 + ˇ G ( λ 1 ) ˇ M ∗ · 1 � √ M · 1 N

√ √ M · 1 − σ 2 N ( ˇ G ( λ 1 ) ˇ M ∗ · 1 � N ( λ 1 − ρ θ ) = ( W N ) 11 + θ )

√ √ M · 1 − σ 2 N ( ˇ G ( λ 1 ) ˇ M ∗ · 1 � N ( λ 1 − ρ θ ) = ( W N ) 11 + θ ) √ M · 1 − σ 2 N ( ˇ G ( ρ θ ) ˇ M ∗ · 1 � = ( W N ) 11 + θ ) � � √ N ˇ ˇ M ∗ G ( λ 1 ) − � � + G ( ρ θ ) M · 1 · 1

√ √ M · 1 − σ 2 N ( ˇ G ( λ 1 ) ˇ M ∗ · 1 � N ( λ 1 − ρ θ ) = ( W N ) 11 + θ ) √ M · 1 − σ 2 N ( ˇ G ( ρ θ ) ˇ M ∗ · 1 � = ( W N ) 11 + θ ) � √ � σ 2 + σ 2 − θ 2 + o (1) N ( λ 1 − ρ θ )

√ √ M · 1 − σ 2 N ( ˇ G ( λ 1 ) ˇ M ∗ · 1 � N ( λ 1 − ρ θ ) = ( W N ) 11 + θ ) √ N ( ˇ · 1 � G ( ρ θ ) ˇ M · 1 − σ 2 tr N − 1 � M ∗ = ( W N ) 11 + G ( ρ θ )) � √ � σ 2 N ( λ 1 − ρ θ )+ o (1) + σ 2 − θ 2 + o (1) (using g σ ( ρ θ ) = 1 θ )