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Decorrelation estimates for the eigenlevels of random operators in the localized regime F. Klopp Universit e Paris 13 and Institut Universitaire de France Conference on Spectral Theory Euler Institute, St Petersburg July 15th 2010 F.

  1. The Anderson model in the localized regime On ℓ 2 ( Z d ) , we consider the Anderson model  ω 1  1 0 ··· ··· 0 . H ω = − ∆ + V ω where V ω = ∑ γ ∈ Z d ω γ π γ and .   ω 2 1 1 .   − ∆ is the standard discrete Laplacian,  .  .   ω 3 0 1 1 .   π γ is the orthogonal projector on δ γ ,   . ... .   the random variables ( ω γ ) γ ∈ Z d are non . 0     ··· ω n − 1 0 0 1 1 trivial, i.i.d. bounded and admit a   ··· ··· ω n 0 0 1 bounded density. Well known : there exists a set, say I ⊂ R , such that, in I , the spectrum of H ω is localized. Pick E ∈ I and L ∈ N . Let Λ = Λ L = [ − L , L ] d ∩ Z d ⊂ Z d and H ω ( Λ ) = H ω | Λ (per. BC). Denote its eigenvalues by E 1 ( ω , Λ ) ≤ E 2 ( ω , Λ ) ≤ ··· ≤ E N ( ω , Λ ) . 1 Integrated density of states: N ( E ) = lim N max { j ; E j ( ω , Λ ) ≤ N } . N → ∞ Density of states ν ( E ) = N ′ ( E ) . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 3 / 16

  2. The Anderson model in the localized regime On ℓ 2 ( Z d ) , we consider the Anderson model  ω 1  1 0 ··· ··· 0 . H ω = − ∆ + V ω where V ω = ∑ γ ∈ Z d ω γ π γ and .   ω 2 1 1 .   − ∆ is the standard discrete Laplacian,  .  .   ω 3 0 1 1 .   π γ is the orthogonal projector on δ γ ,   . ... .   the random variables ( ω γ ) γ ∈ Z d are non . 0     ··· ω n − 1 0 0 1 1 trivial, i.i.d. bounded and admit a   ··· ··· ω n 0 0 1 bounded density. Well known : there exists a set, say I ⊂ R , such that, in I , the spectrum of H ω is localized. Pick E ∈ I and L ∈ N . Let Λ = Λ L = [ − L , L ] d ∩ Z d ⊂ Z d and H ω ( Λ ) = H ω | Λ (per. BC). Denote its eigenvalues by E 1 ( ω , Λ ) ≤ E 2 ( ω , Λ ) ≤ ··· ≤ E N ( ω , Λ ) . 1 Integrated density of states: N ( E ) = lim N max { j ; E j ( ω , Λ ) ≤ N } . N → ∞ Density of states ν ( E ) = N ′ ( E ) . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 3 / 16

  3. The Anderson model in the localized regime On ℓ 2 ( Z d ) , we consider the Anderson model  ω 1  1 0 ··· ··· 0 . H ω = − ∆ + V ω where V ω = ∑ γ ∈ Z d ω γ π γ and .   ω 2 1 1 .   − ∆ is the standard discrete Laplacian,  .  .   ω 3 0 1 1 .   π γ is the orthogonal projector on δ γ ,   . ... .   the random variables ( ω γ ) γ ∈ Z d are non . 0     ··· ω n − 1 0 0 1 1 trivial, i.i.d. bounded and admit a   ··· ··· ω n 0 0 1 bounded density. Well known : there exists a set, say I ⊂ R , such that, in I , the spectrum of H ω is localized. Pick E ∈ I and L ∈ N . Let Λ = Λ L = [ − L , L ] d ∩ Z d ⊂ Z d and H ω ( Λ ) = H ω | Λ (per. BC). Denote its eigenvalues by E 1 ( ω , Λ ) ≤ E 2 ( ω , Λ ) ≤ ··· ≤ E N ( ω , Λ ) . 1 Integrated density of states: N ( E ) = lim N max { j ; E j ( ω , Λ ) ≤ N } . N → ∞ Density of states ν ( E ) = N ′ ( E ) . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 3 / 16

  4. The Anderson model in the localized regime On ℓ 2 ( Z d ) , we consider the Anderson model  ω 1  1 0 ··· ··· 0 . H ω = − ∆ + V ω where V ω = ∑ γ ∈ Z d ω γ π γ and .   ω 2 1 1 .   − ∆ is the standard discrete Laplacian,  .  .   ω 3 0 1 1 .   π γ is the orthogonal projector on δ γ ,   . ... .   the random variables ( ω γ ) γ ∈ Z d are non . 0     ··· ω n − 1 0 0 1 1 trivial, i.i.d. bounded and admit a   ··· ··· ω n 0 0 1 bounded density. Well known : there exists a set, say I ⊂ R , such that, in I , the spectrum of H ω is localized. Pick E ∈ I and L ∈ N . Let Λ = Λ L = [ − L , L ] d ∩ Z d ⊂ Z d and H ω ( Λ ) = H ω | Λ (per. BC). Denote its eigenvalues by E 1 ( ω , Λ ) ≤ E 2 ( ω , Λ ) ≤ ··· ≤ E N ( ω , Λ ) . 1 Integrated density of states: N ( E ) = lim N max { j ; E j ( ω , Λ ) ≤ N } . N → ∞ Density of states ν ( E ) = N ′ ( E ) . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 3 / 16

  5. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  6. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  7. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  8. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  9. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  10. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  11. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  12. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  13. Local level statistics near E : N ∑ Ξ ( ξ , E , ω , Λ ) = δ ξ j ( E , ω , Λ ) ( ξ ) where ξ j ( E , ω , Λ ) = | Λ | ν ( E )( E j ( ω , Λ ) − E ) . j = 1 Theorem (Molchanov,Minami,Germinet-K.) Assume that ν ( E ) > 0 . When | Λ | → + ∞ , the point process Ξ ( , ω , Λ ) converges weakly to a Poisson process on R with intensity the Lebesgue measure. Question: pick E 0 ∈ I and E ′ 0 ∈ I such that E 0 � = E ′ 0 , ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0; Are the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) asymptotically independent? Not much known about this question for random Schr¨ odinger operators. Results for random matrices. The answer may be model dependent:   ω 1 0 0 ··· 0   ω 1 0 ··· 0 0 ω 1 + 1 0 ··· 0 .   .    . .  ω 2 0 0 . . .     ω 2 . 0 0 .  .    ... .    .  ... . 0 .     . 0   0 ··· 0 ω 2 n 0 0 ··· 0 ω n + 1 F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 4 / 16

  14. The independence Theorem (Ge-Kl,Kl) Assume that the dimension d = 1 .When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. That is, for U + ⊂ R and U − ⊂ R compact intervals and { k + , k − } ∈ N × N , one has �� � �� Λ → Z d e −| U + | | U + | k + · e −| U − | | U − | k − # { j ; ξ j ( E 0 , ω , Λ ) ∈ U + } = k + ω ; → . P # { j ; ξ j ( E ′ 0 , ω , Λ ) ∈ U − } = k − k + ! k − ! Theorem (Ge-Kl,Kl) Pick E 0 ∈ I and E ′ 0 ∈ I such that | E 0 − E ′ 0 | > 2 d, ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0 . When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 5 / 16

  15. The independence Theorem (Ge-Kl,Kl) Assume that the dimension d = 1 .When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. That is, for U + ⊂ R and U − ⊂ R compact intervals and { k + , k − } ∈ N × N , one has �� � �� Λ → Z d e −| U + | | U + | k + · e −| U − | | U − | k − # { j ; ξ j ( E 0 , ω , Λ ) ∈ U + } = k + ω ; → . P # { j ; ξ j ( E ′ 0 , ω , Λ ) ∈ U − } = k − k + ! k − ! Theorem (Ge-Kl,Kl) Pick E 0 ∈ I and E ′ 0 ∈ I such that | E 0 − E ′ 0 | > 2 d, ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0 . When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 5 / 16

  16. The independence Theorem (Ge-Kl,Kl) Assume that the dimension d = 1 .When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. That is, for U + ⊂ R and U − ⊂ R compact intervals and { k + , k − } ∈ N × N , one has �� � �� Λ → Z d e −| U + | | U + | k + · e −| U − | | U − | k − # { j ; ξ j ( E 0 , ω , Λ ) ∈ U + } = k + ω ; → . P # { j ; ξ j ( E ′ 0 , ω , Λ ) ∈ U − } = k − k + ! k − ! Theorem (Ge-Kl,Kl) Pick E 0 ∈ I and E ′ 0 ∈ I such that | E 0 − E ′ 0 | > 2 d, ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0 . When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 5 / 16

  17. The independence Theorem (Ge-Kl,Kl) Assume that the dimension d = 1 .When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. That is, for U + ⊂ R and U − ⊂ R compact intervals and { k + , k − } ∈ N × N , one has �� � �� Λ → Z d e −| U + | | U + | k + · e −| U − | | U − | k − # { j ; ξ j ( E 0 , ω , Λ ) ∈ U + } = k + ω ; → . P # { j ; ξ j ( E ′ 0 , ω , Λ ) ∈ U − } = k − k + ! k − ! Theorem (Ge-Kl,Kl) Pick E 0 ∈ I and E ′ 0 ∈ I such that | E 0 − E ′ 0 | > 2 d, ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0 . When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 5 / 16

  18. The independence Theorem (Ge-Kl,Kl) Assume that the dimension d = 1 .When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. That is, for U + ⊂ R and U − ⊂ R compact intervals and { k + , k − } ∈ N × N , one has �� � �� Λ → Z d e −| U + | | U + | k + · e −| U − | | U − | k − # { j ; ξ j ( E 0 , ω , Λ ) ∈ U + } = k + ω ; → . P # { j ; ξ j ( E ′ 0 , ω , Λ ) ∈ U − } = k − k + ! k − ! Theorem (Ge-Kl,Kl) Pick E 0 ∈ I and E ′ 0 ∈ I such that | E 0 − E ′ 0 | > 2 d, ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0 . When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 5 / 16

  19. The independence Theorem (Ge-Kl,Kl) Assume that the dimension d = 1 .When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. That is, for U + ⊂ R and U − ⊂ R compact intervals and { k + , k − } ∈ N × N , one has �� � �� Λ → Z d e −| U + | | U + | k + · e −| U − | | U − | k − # { j ; ξ j ( E 0 , ω , Λ ) ∈ U + } = k + ω ; → . P # { j ; ξ j ( E ′ 0 , ω , Λ ) ∈ U − } = k − k + ! k − ! Theorem (Ge-Kl,Kl) Pick E 0 ∈ I and E ′ 0 ∈ I such that | E 0 − E ′ 0 | > 2 d, ν ( E 0 ) > 0 and ν ( E ′ 0 ) > 0 . When | Λ | → + ∞ , the point processes Ξ ( E 0 , ω , Λ ) and Ξ ( E ′ 0 , ω , Λ ) converge weakly respectively to two independent Poisson processes on R with intensity the Lebesgue measure. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 5 / 16

  20. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  21. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  22. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  23. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  24. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  25. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  26. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  27. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  28. The decorrelation lemmas Lemma (Kl) For the discrete Anderson model , fix α ∈ ( 0 , 1 ) , β ∈ ( 1 / 2 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. | E 0 − E ′ 0 | > 2 d, for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, one has �� σ ( H ω ( Λ ℓ )) ∩ ( E 0 + L − d ( − 1 , 1 )) � = / �� 0 , ≤ C ( ℓ/ L ) 2 d e ( log L ) β . P σ ( H ω ( Λ ℓ )) ∩ ( E ′ 0 + L − d ( − 1 , 1 )) � = / 0 Lemma (Kl) Assume d = 1 . For the discrete Anderson model, for α ∈ ( 0 , 1 ) and { E 0 , E ′ 0 } ⊂ I s.t. 0 , for any c > 0 , there exists C > 0 such that, for L ≥ 3 and cL α ≤ ℓ ≤ L α / c, E 0 � = E ′ the result of the previous theorem holds. Another decorrelation estimate: the Minami estimate Theorem (Min, GV, BHS, CGK) For J ⊂ K, one has E [ tr [ 1 J ( H ω ( Λ ))] · ( tr [ 1 K ( H ω ( Λ ))] − 1 )] ≤ C | J || K || Λ | 2 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 6 / 16

  29. Basic idea of the proof of decorrelation lemmas Let J L = E 0 + L − d ( − 1 , 1 ) and J ′ L = E ′ 0 + L − d ( − 1 , 1 ) . By Minami’s estimate # [ σ ( H ω ( Λ ℓ )) ∩ J L ] ≥ 2 or # [ σ ( H ω ( Λ ℓ )) ∩ J ′ ≤ C ( ℓ/ L ) 2 d � � P L ] ≥ 2 # [ σ ( H ω ( Λ ℓ )) ∩ J L ] = 1 , # [ σ ( H ω ( Λ ℓ )) ∩ J ′ � � If P 0 = P L ] = 1 , suffices to show that P 0 ≤ C ( ℓ/ L ) 2 d e ( log L ) β . Let E j ( ω ) and E k ( ω ) be the eigenvalues resp. in J L and J ′ L . Need to show that they don’t vary “synchronously”. Basic idea: find random variables ( ω γ , ω γ ′ ) such that ψ : ( ω γ , ω γ ′ ) �→ ( E j ( ω ) , E k ( ω )) be a local diffeomorphism. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 7 / 16

  30. Basic idea of the proof of decorrelation lemmas Let J L = E 0 + L − d ( − 1 , 1 ) and J ′ L = E ′ 0 + L − d ( − 1 , 1 ) . By Minami’s estimate # [ σ ( H ω ( Λ ℓ )) ∩ J L ] ≥ 2 or # [ σ ( H ω ( Λ ℓ )) ∩ J ′ ≤ C ( ℓ/ L ) 2 d � � P L ] ≥ 2 # [ σ ( H ω ( Λ ℓ )) ∩ J L ] = 1 , # [ σ ( H ω ( Λ ℓ )) ∩ J ′ � � If P 0 = P L ] = 1 , suffices to show that P 0 ≤ C ( ℓ/ L ) 2 d e ( log L ) β . Let E j ( ω ) and E k ( ω ) be the eigenvalues resp. in J L and J ′ L . Need to show that they don’t vary “synchronously”. Basic idea: find random variables ( ω γ , ω γ ′ ) such that ψ : ( ω γ , ω γ ′ ) �→ ( E j ( ω ) , E k ( ω )) be a local diffeomorphism. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 7 / 16

  31. Basic idea of the proof of decorrelation lemmas Let J L = E 0 + L − d ( − 1 , 1 ) and J ′ L = E ′ 0 + L − d ( − 1 , 1 ) . By Minami’s estimate # [ σ ( H ω ( Λ ℓ )) ∩ J L ] ≥ 2 or # [ σ ( H ω ( Λ ℓ )) ∩ J ′ ≤ C ( ℓ/ L ) 2 d � � P L ] ≥ 2 # [ σ ( H ω ( Λ ℓ )) ∩ J L ] = 1 , # [ σ ( H ω ( Λ ℓ )) ∩ J ′ � � If P 0 = P L ] = 1 , suffices to show that P 0 ≤ C ( ℓ/ L ) 2 d e ( log L ) β . Let E j ( ω ) and E k ( ω ) be the eigenvalues resp. in J L and J ′ L . Need to show that they don’t vary “synchronously”. Basic idea: find random variables ( ω γ , ω γ ′ ) such that ψ : ( ω γ , ω γ ′ ) �→ ( E j ( ω ) , E k ( ω )) be a local diffeomorphism. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 7 / 16

  32. Basic idea of the proof of decorrelation lemmas Let J L = E 0 + L − d ( − 1 , 1 ) and J ′ L = E ′ 0 + L − d ( − 1 , 1 ) . By Minami’s estimate # [ σ ( H ω ( Λ ℓ )) ∩ J L ] ≥ 2 or # [ σ ( H ω ( Λ ℓ )) ∩ J ′ ≤ C ( ℓ/ L ) 2 d � � P L ] ≥ 2 # [ σ ( H ω ( Λ ℓ )) ∩ J L ] = 1 , # [ σ ( H ω ( Λ ℓ )) ∩ J ′ � � If P 0 = P L ] = 1 , suffices to show that P 0 ≤ C ( ℓ/ L ) 2 d e ( log L ) β . Let E j ( ω ) and E k ( ω ) be the eigenvalues resp. in J L and J ′ L . Need to show that they don’t vary “synchronously”. Basic idea: find random variables ( ω γ , ω γ ′ ) such that ψ : ( ω γ , ω γ ′ ) �→ ( E j ( ω ) , E k ( ω )) be a local diffeomorphism. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 7 / 16

  33. Basic idea of the proof of decorrelation lemmas Let J L = E 0 + L − d ( − 1 , 1 ) and J ′ L = E ′ 0 + L − d ( − 1 , 1 ) . By Minami’s estimate # [ σ ( H ω ( Λ ℓ )) ∩ J L ] ≥ 2 or # [ σ ( H ω ( Λ ℓ )) ∩ J ′ ≤ C ( ℓ/ L ) 2 d � � P L ] ≥ 2 # [ σ ( H ω ( Λ ℓ )) ∩ J L ] = 1 , # [ σ ( H ω ( Λ ℓ )) ∩ J ′ � � If P 0 = P L ] = 1 , suffices to show that P 0 ≤ C ( ℓ/ L ) 2 d e ( log L ) β . Let E j ( ω ) and E k ( ω ) be the eigenvalues resp. in J L and J ′ L . Need to show that they don’t vary “synchronously”. Basic idea: find random variables ( ω γ , ω γ ′ ) such that ψ : ( ω γ , ω γ ′ ) �→ ( E j ( ω ) , E k ( ω )) be a local diffeomorphism. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 7 / 16

  34. Basic idea of the proof of decorrelation lemmas Let J L = E 0 + L − d ( − 1 , 1 ) and J ′ L = E ′ 0 + L − d ( − 1 , 1 ) . By Minami’s estimate # [ σ ( H ω ( Λ ℓ )) ∩ J L ] ≥ 2 or # [ σ ( H ω ( Λ ℓ )) ∩ J ′ ≤ C ( ℓ/ L ) 2 d � � P L ] ≥ 2 # [ σ ( H ω ( Λ ℓ )) ∩ J L ] = 1 , # [ σ ( H ω ( Λ ℓ )) ∩ J ′ � � If P 0 = P L ] = 1 , suffices to show that P 0 ≤ C ( ℓ/ L ) 2 d e ( log L ) β . Let E j ( ω ) and E k ( ω ) be the eigenvalues resp. in J L and J ′ L . Need to show that they don’t vary “synchronously”. Basic idea: find random variables ( ω γ , ω γ ′ ) such that ψ : ( ω γ , ω γ ′ ) �→ ( E j ( ω ) , E k ( ω )) be a local diffeomorphism. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 7 / 16

  35. Basic idea of the proof of decorrelation lemmas Let J L = E 0 + L − d ( − 1 , 1 ) and J ′ L = E ′ 0 + L − d ( − 1 , 1 ) . By Minami’s estimate # [ σ ( H ω ( Λ ℓ )) ∩ J L ] ≥ 2 or # [ σ ( H ω ( Λ ℓ )) ∩ J ′ ≤ C ( ℓ/ L ) 2 d � � P L ] ≥ 2 # [ σ ( H ω ( Λ ℓ )) ∩ J L ] = 1 , # [ σ ( H ω ( Λ ℓ )) ∩ J ′ � � If P 0 = P L ] = 1 , suffices to show that P 0 ≤ C ( ℓ/ L ) 2 d e ( log L ) β . Let E j ( ω ) and E k ( ω ) be the eigenvalues resp. in J L and J ′ L . Need to show that they don’t vary “synchronously”. Basic idea: find random variables ( ω γ , ω γ ′ ) such that ψ : ( ω γ , ω γ ′ ) �→ ( E j ( ω ) , E k ( ω )) be a local diffeomorphism. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 7 / 16

  36. Basic idea of the proof of decorrelation lemmas Let J L = E 0 + L − d ( − 1 , 1 ) and J ′ L = E ′ 0 + L − d ( − 1 , 1 ) . By Minami’s estimate # [ σ ( H ω ( Λ ℓ )) ∩ J L ] ≥ 2 or # [ σ ( H ω ( Λ ℓ )) ∩ J ′ ≤ C ( ℓ/ L ) 2 d � � P L ] ≥ 2 # [ σ ( H ω ( Λ ℓ )) ∩ J L ] = 1 , # [ σ ( H ω ( Λ ℓ )) ∩ J ′ � � If P 0 = P L ] = 1 , suffices to show that P 0 ≤ C ( ℓ/ L ) 2 d e ( log L ) β . Let E j ( ω ) and E k ( ω ) be the eigenvalues resp. in J L and J ′ L . Need to show that they don’t vary “synchronously”. Basic idea: find random variables ( ω γ , ω γ ′ ) such that ψ : ( ω γ , ω γ ′ ) �→ ( E j ( ω ) , E k ( ω )) be a local diffeomorphism. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 7 / 16

  37. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  38. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  39. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  40. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  41. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  42. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  43. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  44. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  45. Problem: even if | Jac ψ | ≍ 1, one has L − 2 d ≍ ℓ 4 d / L 2 d . Proba ≤ ∑ j , k ∑ γ , γ ′ We need to reduce the volume of the cube Λ ℓ . Reduction to localization boxes: This can be done using localization. Lemma There exists C > 0 such that for L sufficiently large P 0 ≤ C ( ℓ/ L ) 2 d + C ( ℓ/ ˜ ℓ ) d P 1 where J ′ ℓ )) ∩ ˜ ℓ )) ∩ ˜ P 1 : = P ( # [ σ ( H ω ( Λ ˜ J L ] = # [ σ ( H ω ( Λ ˜ L ] = 1 ) ˜ ˜ J L = J L +[ − L − d , L − d ] J ′ ˜ L = J ′ L +[ − L − d , L − d ] ℓ ≍ log L, and Idea of proof: if e.v. distinct loc. centers, use Wegner and spacial independence. As localization boxes of size ˜ ℓ , remains to estimate P 1 . F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 8 / 16

  46. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  47. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  48. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  49. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  50. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  51. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  52. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  53. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  54. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  55. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  56. Analysis on a localization box Let ω �→ E ( ω ) be the e.v of H ω ( Λ ˜ ℓ ) in J L . E ( ω ) being simple, ω �→ E ( ω ) and the ass. eigenvect. ω �→ ϕ ( ω ) analytic; 1 ∂ ω γ E ( ω ) = � π γ ϕ ( ω ) , ϕ ( ω ) � ≥ 0 ; hence � ∇ ω E ( ω ) � ℓ 1 = 1; 2 ℓ ) − E ( ω )) − 1 ψ γ ( ω ) , ψ β ( ω ) � Hess ω E ( ω ) = (( h γβ )) γ , β , h γ , β = − 2Re � ( H ω ( Λ ˜ 3 where ◮ ψ γ = Π ( ω ) π γ ϕ ( ω ) , ◮ Π ( ω ) is the orthogonal projector on the orthogonal to ϕ ( ω ) . Lemma C � Hess ω ( E ( ω )) � ℓ ∞ → ℓ 1 ≤ ℓ )) \{ E ( ω ) } ) . dist ( E ( ω ) , σ ( H ω ( Λ ˜ Hence, by Minami’s estimate Lemma ℓ 2 d L − d + P ε where P ε = P ( Ω 0 ( ε )) and For ε ∈ ( 4 L − d , 1 ) , one has P 1 ≤ C ε ˜ � � ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ J L = { E ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ( E − C ε , E + C ε ) , Ω 0 ( ε ) = ω ; ℓ )) ∩ ( E ′ − C ε , E ′ + C ε ) J ′ L = { E ′ ( ω ) } = σ ( H ω ( Λ ˜ ℓ )) ∩ ˜ σ ( H ω ( Λ ˜ F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 9 / 16

  57. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  58. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  59. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  60. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  61. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  62. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  63. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  64. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  65. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  66. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  67. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  68. To estimate the Jac ( ψ ) , need to show that ∇ ω E ( ω ) and ∇ ω E ′ ( ω ) not colinear as Lemma 2 � � 1 u j u k Pick ( u , v ) ∈ ( R + ) 2 n such that � u � 1 = � v � 1 = 1 . Then max 2 n 3 � u − v � 2 � � ≥ 1 . � � v j v k j � = k � � Difficulty : gradient may be colinear e.g. for ω = 0. The fundamental estimate: Lemma In any dimension d: for ∆ E > 2 d, if the random variables ( ω γ ) γ ∈ Λ are bounded 1 by K, for E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E, one has � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 ; K in dimension 1: fix E < E ′ and β > 1 / 2 ; let P denote the probability that there 2 exists E j ( ω ) and E k ( ω ) , simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ e − L β and such that � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≤ e − L β ; P ≤ e − cL 2 β . then, there exists c > 0 such that F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 10 / 16

  69. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  70. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  71. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  72. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  73. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  74. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  75. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  76. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  77. Completing the proof of the decorrelation lemma One now has P ε ≤ ∑ γ � = γ ′ P ( Ω γ , γ ′ 0 , ν ( ε ))+ P r where Ω γ , γ ′ � ω ; | J γ , γ ′ ( E ( ω ) , E ′ ( ω )) | ≥ e − ˜ ℓ β � 0 , ν ( ε ) = Ω 0 ( ε ) ∩ ; � � ∂ ω γ E ( ω ) ∂ ω γ ′ E ( ω ) � � J γ , γ ′ ( E ( ω ) , E ′ ( ω )) = � ; � � ∂ ω γ E ′ ( ω ) ∂ ω γ ′ E ′ ( ω ) � � � ℓ 2 β , thus, P r ≤ L − 2 d ; in dimension 1, we have P r ≤ Ce − c ˜ in dimension d , as by assumption ∆ E > 2 d , one has P r = 0. The estimate of Jacobian and picking ε ≍ L − d ˜ ℓ ν + 1 yields ℓ β . P ( Ω γ , γ ′ 0 , ν ( ε )) ≤ CL − 2 d e 2 ˜ Summing over ( γ , γ ′ ) ∈ Λ 2 ℓ , we obtain ˜ P ε ≤ CL − 2 d e 4 ˜ ℓ β Proof is complete. F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 11 / 16

  78. The proof of the fundamental estimate: case 1 E j ( ω ) and E k ( ω ) simple evs of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E > 2 d . Then, ω �→ E j ( ω ) and ω �→ E k ( ω ) are real analytic functions. Let ω �→ ϕ j ( ω ) and ω �→ ϕ k ( ω ) be normalized eigenvec. ass. resp. to E j ( ω ) and E k ( ω ) . Differentiating the eigenvalue equation in ω , one computes ω · ∇ ω ( E j ( ω ) − E k ( ω )) = � V ω ϕ j ( ω ) , ϕ j ( ω ) �−� V ω ϕ k ( ω ) , ϕ k ( ω ) � = E j ( ω ) − E k ( ω )+ �− ∆ ϕ k ( ω ) , ϕ k ( ω ) �−�− ∆ ϕ j ( ω ) , ϕ j ( ω ) � . So ∆ E − 2 d ≤ | E j ( ω ) − E k ( ω ) |− 2 d ≤ | ω · ∇ ω ( E j ( ω ) − E k ( ω )) | . Hence, � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 . K F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 12 / 16

  79. The proof of the fundamental estimate: case 1 E j ( ω ) and E k ( ω ) simple evs of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E > 2 d . Then, ω �→ E j ( ω ) and ω �→ E k ( ω ) are real analytic functions. Let ω �→ ϕ j ( ω ) and ω �→ ϕ k ( ω ) be normalized eigenvec. ass. resp. to E j ( ω ) and E k ( ω ) . Differentiating the eigenvalue equation in ω , one computes ω · ∇ ω ( E j ( ω ) − E k ( ω )) = � V ω ϕ j ( ω ) , ϕ j ( ω ) �−� V ω ϕ k ( ω ) , ϕ k ( ω ) � = E j ( ω ) − E k ( ω )+ �− ∆ ϕ k ( ω ) , ϕ k ( ω ) �−�− ∆ ϕ j ( ω ) , ϕ j ( ω ) � . So ∆ E − 2 d ≤ | E j ( ω ) − E k ( ω ) |− 2 d ≤ | ω · ∇ ω ( E j ( ω ) − E k ( ω )) | . Hence, � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 . K F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 12 / 16

  80. The proof of the fundamental estimate: case 1 E j ( ω ) and E k ( ω ) simple evs of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E > 2 d . Then, ω �→ E j ( ω ) and ω �→ E k ( ω ) are real analytic functions. Let ω �→ ϕ j ( ω ) and ω �→ ϕ k ( ω ) be normalized eigenvec. ass. resp. to E j ( ω ) and E k ( ω ) . Differentiating the eigenvalue equation in ω , one computes ω · ∇ ω ( E j ( ω ) − E k ( ω )) = � V ω ϕ j ( ω ) , ϕ j ( ω ) �−� V ω ϕ k ( ω ) , ϕ k ( ω ) � = E j ( ω ) − E k ( ω )+ �− ∆ ϕ k ( ω ) , ϕ k ( ω ) �−�− ∆ ϕ j ( ω ) , ϕ j ( ω ) � . So ∆ E − 2 d ≤ | E j ( ω ) − E k ( ω ) |− 2 d ≤ | ω · ∇ ω ( E j ( ω ) − E k ( ω )) | . Hence, � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 . K F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 12 / 16

  81. The proof of the fundamental estimate: case 1 E j ( ω ) and E k ( ω ) simple evs of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E > 2 d . Then, ω �→ E j ( ω ) and ω �→ E k ( ω ) are real analytic functions. Let ω �→ ϕ j ( ω ) and ω �→ ϕ k ( ω ) be normalized eigenvec. ass. resp. to E j ( ω ) and E k ( ω ) . Differentiating the eigenvalue equation in ω , one computes ω · ∇ ω ( E j ( ω ) − E k ( ω )) = � V ω ϕ j ( ω ) , ϕ j ( ω ) �−� V ω ϕ k ( ω ) , ϕ k ( ω ) � = E j ( ω ) − E k ( ω )+ �− ∆ ϕ k ( ω ) , ϕ k ( ω ) �−�− ∆ ϕ j ( ω ) , ϕ j ( ω ) � . So ∆ E − 2 d ≤ | E j ( ω ) − E k ( ω ) |− 2 d ≤ | ω · ∇ ω ( E j ( ω ) − E k ( ω )) | . Hence, � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 . K F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 12 / 16

  82. The proof of the fundamental estimate: case 1 E j ( ω ) and E k ( ω ) simple evs of H ω ( Λ L ) such that | E k ( ω ) − E j ( ω ) | ≥ ∆ E > 2 d . Then, ω �→ E j ( ω ) and ω �→ E k ( ω ) are real analytic functions. Let ω �→ ϕ j ( ω ) and ω �→ ϕ k ( ω ) be normalized eigenvec. ass. resp. to E j ( ω ) and E k ( ω ) . Differentiating the eigenvalue equation in ω , one computes ω · ∇ ω ( E j ( ω ) − E k ( ω )) = � V ω ϕ j ( ω ) , ϕ j ( ω ) �−� V ω ϕ k ( ω ) , ϕ k ( ω ) � = E j ( ω ) − E k ( ω )+ �− ∆ ϕ k ( ω ) , ϕ k ( ω ) �−�− ∆ ϕ j ( ω ) , ϕ j ( ω ) � . So ∆ E − 2 d ≤ | E j ( ω ) − E k ( ω ) |− 2 d ≤ | ω · ∇ ω ( E j ( ω ) − E k ( ω )) | . Hence, � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 ≥ ∆ E − 2 d L − d / 2 . K F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 12 / 16

  83. The proof of the fundamental estimate: case 2 Let us now assume d = 1. We prove a weaker result. Theorem Fix ν > 8 . For the discrete Anderson model in dimension 1, there exists ∆ E of total measure such that, for E − E ′ ∈ ∆ E , for L sufficiently large, if E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − ν then � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≥ L − ν ; Fix E < E ′ . Pick E j ( ω ) and E k ( ω ) , simple evs s.t. | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − α . Then, 4 L − 2 ν ≥ � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 γ ( ω ) | 2 ·| ϕ j 2 = ∑ | ϕ j γ ( ω ) − ϕ k γ ( ω )+ ϕ k γ ( ω ) | 2 γ ∈ Λ L there exists a partition of Λ L , say P ⊂ Λ L and Q ⊂ Λ L s.t. for γ ∈ P , | ϕ j γ ( ω ) − ϕ k γ ( ω ) | ≤ L − ν ; for γ ∈ Q , | ϕ j γ ( ω ) | ≤ L − ν . γ ( ω )+ ϕ k Introduce the orthogonal projectors P and Q defined by P = ∑ Q = ∑ | γ �� γ | | γ �� γ | . and γ ∈ P γ ∈ Q F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 13 / 16

  84. The proof of the fundamental estimate: case 2 Let us now assume d = 1. We prove a weaker result. Theorem Fix ν > 8 . For the discrete Anderson model in dimension 1, there exists ∆ E of total measure such that, for E − E ′ ∈ ∆ E , for L sufficiently large, if E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − ν then � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≥ L − ν ; Fix E < E ′ . Pick E j ( ω ) and E k ( ω ) , simple evs s.t. | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − α . Then, 4 L − 2 ν ≥ � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 γ ( ω ) | 2 ·| ϕ j 2 = ∑ | ϕ j γ ( ω ) − ϕ k γ ( ω )+ ϕ k γ ( ω ) | 2 γ ∈ Λ L there exists a partition of Λ L , say P ⊂ Λ L and Q ⊂ Λ L s.t. for γ ∈ P , | ϕ j γ ( ω ) − ϕ k γ ( ω ) | ≤ L − ν ; for γ ∈ Q , | ϕ j γ ( ω ) | ≤ L − ν . γ ( ω )+ ϕ k Introduce the orthogonal projectors P and Q defined by P = ∑ Q = ∑ | γ �� γ | | γ �� γ | . and γ ∈ P γ ∈ Q F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 13 / 16

  85. The proof of the fundamental estimate: case 2 Let us now assume d = 1. We prove a weaker result. Theorem Fix ν > 8 . For the discrete Anderson model in dimension 1, there exists ∆ E of total measure such that, for E − E ′ ∈ ∆ E , for L sufficiently large, if E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − ν then � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≥ L − ν ; Fix E < E ′ . Pick E j ( ω ) and E k ( ω ) , simple evs s.t. | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − α . Then, 4 L − 2 ν ≥ � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 γ ( ω ) | 2 ·| ϕ j 2 = ∑ | ϕ j γ ( ω ) − ϕ k γ ( ω )+ ϕ k γ ( ω ) | 2 γ ∈ Λ L there exists a partition of Λ L , say P ⊂ Λ L and Q ⊂ Λ L s.t. for γ ∈ P , | ϕ j γ ( ω ) − ϕ k γ ( ω ) | ≤ L − ν ; for γ ∈ Q , | ϕ j γ ( ω ) | ≤ L − ν . γ ( ω )+ ϕ k Introduce the orthogonal projectors P and Q defined by P = ∑ Q = ∑ | γ �� γ | | γ �� γ | . and γ ∈ P γ ∈ Q F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 13 / 16

  86. The proof of the fundamental estimate: case 2 Let us now assume d = 1. We prove a weaker result. Theorem Fix ν > 8 . For the discrete Anderson model in dimension 1, there exists ∆ E of total measure such that, for E − E ′ ∈ ∆ E , for L sufficiently large, if E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − ν then � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≥ L − ν ; Fix E < E ′ . Pick E j ( ω ) and E k ( ω ) , simple evs s.t. | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − α . Then, 4 L − 2 ν ≥ � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 γ ( ω ) | 2 ·| ϕ j 2 = ∑ | ϕ j γ ( ω ) − ϕ k γ ( ω )+ ϕ k γ ( ω ) | 2 γ ∈ Λ L there exists a partition of Λ L , say P ⊂ Λ L and Q ⊂ Λ L s.t. for γ ∈ P , | ϕ j γ ( ω ) − ϕ k γ ( ω ) | ≤ L − ν ; for γ ∈ Q , | ϕ j γ ( ω ) | ≤ L − ν . γ ( ω )+ ϕ k Introduce the orthogonal projectors P and Q defined by P = ∑ Q = ∑ | γ �� γ | | γ �� γ | . and γ ∈ P γ ∈ Q F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 13 / 16

  87. The proof of the fundamental estimate: case 2 Let us now assume d = 1. We prove a weaker result. Theorem Fix ν > 8 . For the discrete Anderson model in dimension 1, there exists ∆ E of total measure such that, for E − E ′ ∈ ∆ E , for L sufficiently large, if E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − ν then � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≥ L − ν ; Fix E < E ′ . Pick E j ( ω ) and E k ( ω ) , simple evs s.t. | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − α . Then, 4 L − 2 ν ≥ � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 γ ( ω ) | 2 ·| ϕ j 2 = ∑ | ϕ j γ ( ω ) − ϕ k γ ( ω )+ ϕ k γ ( ω ) | 2 γ ∈ Λ L there exists a partition of Λ L , say P ⊂ Λ L and Q ⊂ Λ L s.t. for γ ∈ P , | ϕ j γ ( ω ) − ϕ k γ ( ω ) | ≤ L − ν ; for γ ∈ Q , | ϕ j γ ( ω ) | ≤ L − ν . γ ( ω )+ ϕ k Introduce the orthogonal projectors P and Q defined by P = ∑ Q = ∑ | γ �� γ | | γ �� γ | . and γ ∈ P γ ∈ Q F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 13 / 16

  88. The proof of the fundamental estimate: case 2 Let us now assume d = 1. We prove a weaker result. Theorem Fix ν > 8 . For the discrete Anderson model in dimension 1, there exists ∆ E of total measure such that, for E − E ′ ∈ ∆ E , for L sufficiently large, if E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − ν then � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≥ L − ν ; Fix E < E ′ . Pick E j ( ω ) and E k ( ω ) , simple evs s.t. | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − α . Then, 4 L − 2 ν ≥ � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 γ ( ω ) | 2 ·| ϕ j 2 = ∑ | ϕ j γ ( ω ) − ϕ k γ ( ω )+ ϕ k γ ( ω ) | 2 γ ∈ Λ L there exists a partition of Λ L , say P ⊂ Λ L and Q ⊂ Λ L s.t. for γ ∈ P , | ϕ j γ ( ω ) − ϕ k γ ( ω ) | ≤ L − ν ; for γ ∈ Q , | ϕ j γ ( ω ) | ≤ L − ν . γ ( ω )+ ϕ k Introduce the orthogonal projectors P and Q defined by P = ∑ Q = ∑ | γ �� γ | | γ �� γ | . and γ ∈ P γ ∈ Q F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 13 / 16

  89. The proof of the fundamental estimate: case 2 Let us now assume d = 1. We prove a weaker result. Theorem Fix ν > 8 . For the discrete Anderson model in dimension 1, there exists ∆ E of total measure such that, for E − E ′ ∈ ∆ E , for L sufficiently large, if E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − ν then � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≥ L − ν ; Fix E < E ′ . Pick E j ( ω ) and E k ( ω ) , simple evs s.t. | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − α . Then, 4 L − 2 ν ≥ � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 γ ( ω ) | 2 ·| ϕ j 2 = ∑ | ϕ j γ ( ω ) − ϕ k γ ( ω )+ ϕ k γ ( ω ) | 2 γ ∈ Λ L there exists a partition of Λ L , say P ⊂ Λ L and Q ⊂ Λ L s.t. for γ ∈ P , | ϕ j γ ( ω ) − ϕ k γ ( ω ) | ≤ L − ν ; for γ ∈ Q , | ϕ j γ ( ω ) | ≤ L − ν . γ ( ω )+ ϕ k Introduce the orthogonal projectors P and Q defined by P = ∑ Q = ∑ | γ �� γ | | γ �� γ | . and γ ∈ P γ ∈ Q F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 13 / 16

  90. The proof of the fundamental estimate: case 2 Let us now assume d = 1. We prove a weaker result. Theorem Fix ν > 8 . For the discrete Anderson model in dimension 1, there exists ∆ E of total measure such that, for E − E ′ ∈ ∆ E , for L sufficiently large, if E j ( ω ) and E k ( ω ) are simple eigenvalues of H ω ( Λ L ) such that | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − ν then � ∇ ω ( E j ( ω ) − E k ( ω )) � 1 ≥ L − ν ; Fix E < E ′ . Pick E j ( ω ) and E k ( ω ) , simple evs s.t. | E k ( ω ) − E | + | E j ( ω ) − E ′ | ≤ L − α . Then, 4 L − 2 ν ≥ � ∇ ω ( E j ( ω ) − E k ( ω )) � 2 γ ( ω ) | 2 ·| ϕ j 2 = ∑ | ϕ j γ ( ω ) − ϕ k γ ( ω )+ ϕ k γ ( ω ) | 2 γ ∈ Λ L there exists a partition of Λ L , say P ⊂ Λ L and Q ⊂ Λ L s.t. for γ ∈ P , | ϕ j γ ( ω ) − ϕ k γ ( ω ) | ≤ L − ν ; for γ ∈ Q , | ϕ j γ ( ω ) | ≤ L − ν . γ ( ω )+ ϕ k Introduce the orthogonal projectors P and Q defined by P = ∑ Q = ∑ | γ �� γ | | γ �� γ | . and γ ∈ P γ ∈ Q F. Klopp (Universit´ e Paris 13) Decorrelation estimates Euler Institute, St Petersburg 13 / 16

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