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Local Universality of Repulsive Particle Systems and Random Matrices Friedrich Gtze joint with M.Venker, A.Naumov and A.Tikhomirov Bielefeld University www.math.uni-bielefeld.de/ goetze Workshop Random Matrices and their

  1. Global Marginal Distributions � ρ h , k R N − k P h N , Q ( x 1 , . . . , x k ) := N , Q ( x ) dx k + 1 . . . dx N : the k -th correlation function of P h N , Q . Thm (G.-Venker ’12) For all h above exist α h > 0 s.th. for all Q above with min t ∈ R Q ′′ ( t ) > α h , there exists µ h Q , p-measure with compact support s.th. � � ⊗ k as N → ∞ , ( k -th correlation measure of P h µ h N , Q ) ⇒ Q F. Götze (Bielefeld) Local/Global Universality October 9, 2012 6 / 25

  2. Global Marginal Distributions � ρ h , k R N − k P h N , Q ( x 1 , . . . , x k ) := N , Q ( x ) dx k + 1 . . . dx N : the k -th correlation function of P h N , Q . Thm (G.-Venker ’12) For all h above exist α h > 0 s.th. for all Q above with min t ∈ R Q ′′ ( t ) > α h , there exists µ h Q , p-measure with compact support s.th. � � ⊗ k as N → ∞ , ( k -th correlation measure of P h µ h N , Q ) ⇒ Q i.e. for g ∈ C b ( R k ) F. Götze (Bielefeld) Local/Global Universality October 9, 2012 6 / 25

  3. Global Marginal Distributions � ρ h , k R N − k P h N , Q ( x 1 , . . . , x k ) := N , Q ( x ) dx k + 1 . . . dx N : the k -th correlation function of P h N , Q . Thm (G.-Venker ’12) For all h above exist α h > 0 s.th. for all Q above with min t ∈ R Q ′′ ( t ) > α h , there exists µ h Q , p-measure with compact support s.th. � � ⊗ k as N → ∞ , ( k -th correlation measure of P h µ h N , Q ) ⇒ Q i.e. for g ∈ C b ( R k ) � � R k g ( t ) ρ h , k N , Q ( t ) d k t = R k g ( t )( µ h Q ) ⊗ k ( dt ) . lim N →∞

  4. Global Marginal Distributions � ρ h , k R N − k P h N , Q ( x 1 , . . . , x k ) := N , Q ( x ) dx k + 1 . . . dx N : the k -th correlation function of P h N , Q . Thm (G.-Venker ’12) For all h above exist α h > 0 s.th. for all Q above with min t ∈ R Q ′′ ( t ) > α h , there exists µ h Q , p-measure with compact support s.th. � � ⊗ k as N → ∞ , ( k -th correlation measure of P h µ h N , Q ) ⇒ Q i.e. for g ∈ C b ( R k ) � � R k g ( t ) ρ h , k N , Q ( t ) d k t = R k g ( t )( µ h Q ) ⊗ k ( dt ) . lim N →∞ F. Götze (Bielefeld) Local/Global Universality October 9, 2012 6 / 25

  5. Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q , h and α h > 0: F. Götze (Bielefeld) Local/Global Universality October 9, 2012 7 / 25

  6. Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q , h and α h > 0: For k ≥ 1 and a ∈ supp ( µ h Q ) ◦ F. Götze (Bielefeld) Local/Global Universality October 9, 2012 7 / 25

  7. Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q , h and α h > 0: For k ≥ 1 and a ∈ supp ( µ h Q ) ◦ density µ h Q ( a ) > 0, uniformly on compacts in t 1 , . . . , t k F. Götze (Bielefeld) Local/Global Universality October 9, 2012 7 / 25

  8. Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q , h and α h > 0: For k ≥ 1 and a ∈ supp ( µ h Q ) ◦ density µ h Q ( a ) > 0, uniformly on compacts in t 1 , . . . , t k � � 1 t 1 t k Q ( a ) k ρ h , k lim a + Q ( a ) , . . . , a + N , Q µ h N µ h N µ h Q ( a ) N →∞ F. Götze (Bielefeld) Local/Global Universality October 9, 2012 7 / 25

  9. Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q , h and α h > 0: For k ≥ 1 and a ∈ supp ( µ h Q ) ◦ density µ h Q ( a ) > 0, uniformly on compacts in t 1 , . . . , t k � � 1 t 1 t k Q ( a ) k ρ h , k lim a + Q ( a ) , . . . , a + N , Q µ h N µ h N µ h Q ( a ) N →∞ � sin ( π ( t i − t j )) � = det . π ( t i − t j ) 1 ≤ i , j ≤ k F. Götze (Bielefeld) Local/Global Universality October 9, 2012 7 / 25

  10. Extensions to β -Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ ( 0 ) = 0 and ϕ ( t ) > 0 for t � = 0. F. Götze (Bielefeld) Local/Global Universality October 9, 2012 8 / 25

  11. Extensions to β -Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ ( 0 ) = 0 and ϕ ( t ) > 0 for t � = 0. Assume that for some β > 0 and c > 0 F. Götze (Bielefeld) Local/Global Universality October 9, 2012 8 / 25

  12. Extensions to β -Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ ( 0 ) = 0 and ϕ ( t ) > 0 for t � = 0. Assume that for some β > 0 and c > 0 ϕ ( ε ) | ε | β = c . lim ε → 0 F. Götze (Bielefeld) Local/Global Universality October 9, 2012 8 / 25

  13. Extensions to β -Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ ( 0 ) = 0 and ϕ ( t ) > 0 for t � = 0. Assume that for some β > 0 and c > 0 ϕ ( ε ) | ε | β = c . lim ε → 0 Let Q be a strictly convex function of sufficient growth at infinity and define N , Q as the probability measure on R N with density P ϕ,β F. Götze (Bielefeld) Local/Global Universality October 9, 2012 8 / 25

  14. Extensions to β -Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ ( 0 ) = 0 and ϕ ( t ) > 0 for t � = 0. Assume that for some β > 0 and c > 0 ϕ ( ε ) | ε | β = c . lim ε → 0 Let Q be a strictly convex function of sufficient growth at infinity and define N , Q as the probability measure on R N with density P ϕ,β � 1 ϕ ( x i − x j ) e − N � N P ϕ,β j = 1 Q ( x j ) dx . N , Q ( x ) := Z ϕ,β N , Q i < j F. Götze (Bielefeld) Local/Global Universality October 9, 2012 8 / 25

  15. Extensions to β -Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ ( 0 ) = 0 and ϕ ( t ) > 0 for t � = 0. Assume that for some β > 0 and c > 0 ϕ ( ε ) | ε | β = c . lim ε → 0 Let Q be a strictly convex function of sufficient growth at infinity and define N , Q as the probability measure on R N with density P ϕ,β � 1 ϕ ( x i − x j ) e − N � N P ϕ,β j = 1 Q ( x j ) dx . N , Q ( x ) := Z ϕ,β N , Q i < j We conjecture that P ϕ,β N , Q has the same bulk local k -correlation, say ρ k β , as the Gaussian- β ensemble � N ( x ) := 1 | x k − x j | β e − N � N j = 1 x 2 P β j . Z β N j < k F. Götze (Bielefeld) Local/Global Universality October 9, 2012 8 / 25

  16. β -Ensembles Theorem ( Venker 2012) Write ϕ ( x ) := | x | β exp { h } , h real analytic and even Schwartz function, α h ≥ 0 s. th. for all real analytic, strongly convex and even Q with α Q > α h : F. Götze (Bielefeld) Local/Global Universality October 9, 2012 9 / 25

  17. β -Ensembles Theorem ( Venker 2012) Write ϕ ( x ) := | x | β exp { h } , h real analytic and even Schwartz function, α h ≥ 0 s. th. for all real analytic, strongly convex and even Q with α Q > α h :The correlation measure ρ h , 1 N , Q ,β converges weakly to a compactly supported p.m. µ h Q ,β . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 9 / 25

  18. β -Ensembles Theorem ( Venker 2012) Write ϕ ( x ) := | x | β exp { h } , h real analytic and even Schwartz function, α h ≥ 0 s. th. for all real analytic, strongly convex and even Q with α Q > α h :The correlation measure ρ h , 1 N , Q ,β converges weakly to a compactly supported p.m. µ h Q ,β . Conditions on Q as above, there is µ Q ,β of compact support, semicircular for Q ( x ) = x 2 F. Götze (Bielefeld) Local/Global Universality October 9, 2012 9 / 25

  19. β -Ensembles Theorem ( Venker 2012) Write ϕ ( x ) := | x | β exp { h } , h real analytic and even Schwartz function, α h ≥ 0 s. th. for all real analytic, strongly convex and even Q with α Q > α h :The correlation measure ρ h , 1 N , Q ,β converges weakly to a compactly supported p.m. µ h Q ,β . Conditions on Q as above, there is µ Q ,β of compact support, semicircular for Q ( x ) = x 2 and a scaled deformed correlation function F. Götze (Bielefeld) Local/Global Universality October 9, 2012 9 / 25

  20. β -Ensembles Theorem ( Venker 2012) Write ϕ ( x ) := | x | β exp { h } , h real analytic and even Schwartz function, α h ≥ 0 s. th. for all real analytic, strongly convex and even Q with α Q > α h :The correlation measure ρ h , 1 N , Q ,β converges weakly to a compactly supported p.m. µ h Q ,β . Conditions on Q as above, there is µ Q ,β of compact support, semicircular for Q ( x ) = x 2 and a scaled deformed correlation function � � 1 t 1 t k Q ,β ( a ) k ρ h , k a + Q ,β ( a ) , . . . , a + , (1) N , Q ,β µ h N µ h N µ h Q ,β ( a ) where a ∈ supp ( µ h Q ,β ( a )) ◦ . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 9 / 25

  21. β -Ensembles Theorem ( Venker 2012) Write ϕ ( x ) := | x | β exp { h } , h real analytic and even Schwartz function, α h ≥ 0 s. th. for all real analytic, strongly convex and even Q with α Q > α h :The correlation measure ρ h , 1 N , Q ,β converges weakly to a compactly supported p.m. µ h Q ,β . Conditions on Q as above, there is µ Q ,β of compact support, semicircular for Q ( x ) = x 2 and a scaled deformed correlation function � � 1 t 1 t k Q ,β ( a ) k ρ h , k a + Q ,β ( a ) , . . . , a + , (1) N , Q ,β µ h N µ h N µ h Q ,β ( a ) where a ∈ supp ( µ h Q ,β ( a )) ◦ . For h = 0 and Q = x 2 , F. Götze (Bielefeld) Local/Global Universality October 9, 2012 9 / 25

  22. β -Ensembles Theorem ( Venker 2012) Write ϕ ( x ) := | x | β exp { h } , h real analytic and even Schwartz function, α h ≥ 0 s. th. for all real analytic, strongly convex and even Q with α Q > α h :The correlation measure ρ h , 1 N , Q ,β converges weakly to a compactly supported p.m. µ h Q ,β . Conditions on Q as above, there is µ Q ,β of compact support, semicircular for Q ( x ) = x 2 and a scaled deformed correlation function � � 1 t 1 t k Q ,β ( a ) k ρ h , k a + Q ,β ( a ) , . . . , a + , (1) N , Q ,β µ h N µ h N µ h Q ,β ( a ) where a ∈ supp ( µ h Q ,β ( a )) ◦ . For h = 0 and Q = x 2 , the limit N → ∞ exists for Q ( x ) = G ( x ) := x 2 and h = 0 by Valko-Virag (09). F. Götze (Bielefeld) Local/Global Universality October 9, 2012 9 / 25

  23. Universality of Deformed (Averaged) Local Correlations of β -Ensembles Compare local correlations of P N , h M η with those of the Gaussian β -Ensemble P N , G ,β : F. Götze (Bielefeld) Local/Global Universality October 9, 2012 10 / 25

  24. Universality of Deformed (Averaged) Local Correlations of β -Ensembles Compare local correlations of P N , h M η with those of the Gaussian β -Ensemble P N , G ,β : Theorem ( Venker 2012, arxiv 1209.317) h and Q as above. Let 0 < ξ ≤ 1 / 2 and s N := N − 1 + ξ . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 10 / 25

  25. Universality of Deformed (Averaged) Local Correlations of β -Ensembles Compare local correlations of P N , h M η with those of the Gaussian β -Ensemble P N , G ,β : Theorem ( Venker 2012, arxiv 1209.317) h and Q as above. Let 0 < ξ ≤ 1 / 2 and s N := N − 1 + ξ . Q ,β ) ◦ , any a ′ supp ( µ G ,β ) ◦ , any smooth For k = 1 , 2 , . . . , any a ∈ supp ( µ h function f : R k − → R with compact support F. Götze (Bielefeld) Local/Global Universality October 9, 2012 10 / 25

  26. Universality of Deformed (Averaged) Local Correlations of β -Ensembles Compare local correlations of P N , h M η with those of the Gaussian β -Ensemble P N , G ,β : Theorem ( Venker 2012, arxiv 1209.317) h and Q as above. Let 0 < ξ ≤ 1 / 2 and s N := N − 1 + ξ . Q ,β ) ◦ , any a ′ supp ( µ G ,β ) ◦ , any smooth For k = 1 , 2 , . . . , any a ∈ supp ( µ h function f : R k − → R with compact support � dt k f ( t ) lim N →∞ � � a + s N � � 1 t 1 t k Q ,β ( a ) k ρ h , k u + Q ,β ( a ) , . . . , u + µ h N , Q ,β N µ h N µ h Q ,β ( a ) a − s N F. Götze (Bielefeld) Local/Global Universality October 9, 2012 10 / 25

  27. Universality of Deformed (Averaged) Local Correlations of β -Ensembles Compare local correlations of P N , h M η with those of the Gaussian β -Ensemble P N , G ,β : Theorem ( Venker 2012, arxiv 1209.317) h and Q as above. Let 0 < ξ ≤ 1 / 2 and s N := N − 1 + ξ . Q ,β ) ◦ , any a ′ supp ( µ G ,β ) ◦ , any smooth For k = 1 , 2 , . . . , any a ∈ supp ( µ h function f : R k − → R with compact support � dt k f ( t ) lim N →∞ � � a + s N � � 1 t 1 t k Q ,β ( a ) k ρ h , k u + Q ,β ( a ) , . . . , u + µ h N , Q ,β N µ h N µ h Q ,β ( a ) a − s N � � � a ′ + s N � 1 t 1 t k du µ G ,β ( a ′ ) k ρ k − u + N µ G ,β ( a ′ ) , . . . , u + N , G ,β N µ G ,β ( a ′ ) 2 s N a ′ − s N = 0 . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 10 / 25

  28. Universality of Deformed (Averaged) Local Correlations of β -Ensembles Compare local correlations of P N , h M η with those of the Gaussian β -Ensemble P N , G ,β : Theorem ( Venker 2012, arxiv 1209.317) h and Q as above. Let 0 < ξ ≤ 1 / 2 and s N := N − 1 + ξ . Q ,β ) ◦ , any a ′ supp ( µ G ,β ) ◦ , any smooth For k = 1 , 2 , . . . , any a ∈ supp ( µ h function f : R k − → R with compact support � dt k f ( t ) lim N →∞ � � a + s N � � 1 t 1 t k Q ,β ( a ) k ρ h , k u + Q ,β ( a ) , . . . , u + µ h N , Q ,β N µ h N µ h Q ,β ( a ) a − s N � � � a ′ + s N � 1 t 1 t k du µ G ,β ( a ′ ) k ρ k − u + N µ G ,β ( a ′ ) , . . . , u + N , G ,β N µ G ,β ( a ′ ) 2 s N a ′ − s N = 0 . Local correlation limits using relaxation flow methods of Bourgade, Erd˝ os, B. Schlein, and H.-T. Yau (2011,2012) instead of potential theory. F. Götze (Bielefeld) Local/Global Universality October 9, 2012 10 / 25

  29. Proof: Simple Example h ( x ) := − x 2 and γ > 0 � Z − 1 N ,α,γ P GUE ( x i − x j ) 2 } , P γ N ,α ( x ) := N ,α ( x ) exp { γ i < j F. Götze (Bielefeld) Local/Global Universality October 9, 2012 11 / 25

  30. Proof: Simple Example h ( x ) := − x 2 and γ > 0 � Z − 1 N ,α,γ P GUE ( x i − x j ) 2 } , P γ N ,α ( x ) := N ,α ( x ) exp { γ i < j � � � � 1 � 2 exp {− α N � x i − x j P GUE x 2 N ,α ( x ) := j } , Z N ,α 1 ≤ i < j ≤ N j F. Götze (Bielefeld) Local/Global Universality October 9, 2012 11 / 25

  31. Proof: Simple Example h ( x ) := − x 2 and γ > 0 � Z − 1 N ,α,γ P GUE ( x i − x j ) 2 } , P γ N ,α ( x ) := N ,α ( x ) exp { γ i < j � � � � 1 � 2 exp {− α N � x i − x j P GUE x 2 N ,α ( x ) := j } , Z N ,α 1 ≤ i < j ≤ N j � ( x i − x j ) 2 } exp {− γ N M 2 ( x ) } exp { γ M 1 ( x ) 2 } , exp { γ = i < j F. Götze (Bielefeld) Local/Global Universality October 9, 2012 11 / 25

  32. Proof: Simple Example h ( x ) := − x 2 and γ > 0 � Z − 1 N ,α,γ P GUE ( x i − x j ) 2 } , P γ N ,α ( x ) := N ,α ( x ) exp { γ i < j � � � � 1 � 2 exp {− α N � x i − x j P GUE x 2 N ,α ( x ) := j } , Z N ,α 1 ≤ i < j ≤ N j � ( x i − x j ) 2 } exp {− γ N M 2 ( x ) } exp { γ M 1 ( x ) 2 } , exp { γ = where i < j F. Götze (Bielefeld) Local/Global Universality October 9, 2012 11 / 25

  33. Proof: Simple Example h ( x ) := − x 2 and γ > 0 � Z − 1 N ,α,γ P GUE ( x i − x j ) 2 } , P γ N ,α ( x ) := N ,α ( x ) exp { γ i < j � � � � 1 � 2 exp {− α N � x i − x j P GUE x 2 N ,α ( x ) := j } , Z N ,α 1 ≤ i < j ≤ N j � ( x i − x j ) 2 } exp {− γ N M 2 ( x ) } exp { γ M 1 ( x ) 2 } , exp { γ = where i < j N � x p M p ( x ) := j , p = 1 , 2 , j = 1 F. Götze (Bielefeld) Local/Global Universality October 9, 2012 11 / 25

  34. Proof: Simple Example h ( x ) := − x 2 and γ > 0 � Z − 1 N ,α,γ P GUE ( x i − x j ) 2 } , P γ N ,α ( x ) := N ,α ( x ) exp { γ i < j � � � � 1 � 2 exp {− α N � x i − x j P GUE x 2 N ,α ( x ) := j } , Z N ,α 1 ≤ i < j ≤ N j � ( x i − x j ) 2 } exp {− γ N M 2 ( x ) } exp { γ M 1 ( x ) 2 } , exp { γ = where i < j N � x p M p ( x ) := j , p = 1 , 2 , j = 1 � exp { ε √ γ M 1 ( x ) } exp {− ε 2 / 4 } d ε, exp { γ M 1 ( x ) 2 } = c R F. Götze (Bielefeld) Local/Global Universality October 9, 2012 11 / 25

  35. Orthogonal Polynomials � Z N ,ε P γ c ′ P N ,ε ( x ) exp {− ε 2 / 4 } d ε, N ,α ( x ) = Z N R F. Götze (Bielefeld) Local/Global Universality October 9, 2012 12 / 25

  36. Orthogonal Polynomials � Z N ,ε P γ c ′ P N ,ε ( x ) exp {− ε 2 / 4 } d ε, N ,α ( x ) = Z N R N � ∆( x ) 2 j + √ γε x j ) } ( N ( α + γ ) x 2 P N ,ε ( x ) := exp {− Z N ,ε j = 1 F. Götze (Bielefeld) Local/Global Universality October 9, 2012 12 / 25

  37. Orthogonal Polynomials � Z N ,ε P γ c ′ P N ,ε ( x ) exp {− ε 2 / 4 } d ε, N ,α ( x ) = Z N R N � ∆( x ) 2 j + √ γε x j ) } ( N ( α + γ ) x 2 P N ,ε ( x ) := exp {− Z N ,ε j = 1 � � � γε 2 1 + γ Z N ,ε / Z N = α exp . 4 ( α + γ ) F. Götze (Bielefeld) Local/Global Universality October 9, 2012 12 / 25

  38. Orthogonal Polynomials � Z N ,ε P γ c ′ P N ,ε ( x ) exp {− ε 2 / 4 } d ε, N ,α ( x ) = Z N R N � ∆( x ) 2 j + √ γε x j ) } ( N ( α + γ ) x 2 P N ,ε ( x ) := exp {− Z N ,ε j = 1 � � � γε 2 1 + γ Z N ,ε / Z N = α exp . 4 ( α + γ ) exp {− N ( α + γ ) t 2 + ε √ γ t } Orthogonal polynomials w.r.t. the kernel are shifted Hermite polynomials. F. Götze (Bielefeld) Local/Global Universality October 9, 2012 12 / 25

  39. Orthogonal Polynomials � Z N ,ε P γ c ′ P N ,ε ( x ) exp {− ε 2 / 4 } d ε, N ,α ( x ) = Z N R N � ∆( x ) 2 j + √ γε x j ) } ( N ( α + γ ) x 2 P N ,ε ( x ) := exp {− Z N ,ε j = 1 � � � γε 2 1 + γ Z N ,ε / Z N = α exp . 4 ( α + γ ) exp {− N ( α + γ ) t 2 + ε √ γ t } Orthogonal polynomials w.r.t. the kernel are shifted Hermite polynomials. The ensemble P ε N is determinantal with kernel: F. Götze (Bielefeld) Local/Global Universality October 9, 2012 12 / 25

  40. Orthogonal Polynomials � Z N ,ε P γ c ′ P N ,ε ( x ) exp {− ε 2 / 4 } d ε, N ,α ( x ) = Z N R N � ∆( x ) 2 j + √ γε x j ) } ( N ( α + γ ) x 2 P N ,ε ( x ) := exp {− Z N ,ε j = 1 � � � γε 2 1 + γ Z N ,ε / Z N = α exp . 4 ( α + γ ) exp {− N ( α + γ ) t 2 + ε √ γ t } Orthogonal polynomials w.r.t. the kernel are shifted Hermite polynomials. The ensemble P ε N is determinantal with kernel: √ γ N ( t , s ) = exp { ω ′ 2 ε 2 ω ′ := 4 N } K N ( t − ω ′ ε 2 N , s − ω ′ ε K ∗ 2 N ) , α + γ F. Götze (Bielefeld) Local/Global Universality October 9, 2012 12 / 25

  41. Orthogonal Polynomials � Z N ,ε P γ c ′ P N ,ε ( x ) exp {− ε 2 / 4 } d ε, N ,α ( x ) = Z N R N � ∆( x ) 2 j + √ γε x j ) } ( N ( α + γ ) x 2 P N ,ε ( x ) := exp {− Z N ,ε j = 1 � � � γε 2 1 + γ Z N ,ε / Z N = α exp . 4 ( α + γ ) exp {− N ( α + γ ) t 2 + ε √ γ t } Orthogonal polynomials w.r.t. the kernel are shifted Hermite polynomials. The ensemble P ε N is determinantal with kernel: √ γ N ( t , s ) = exp { ω ′ 2 ε 2 ω ′ := 4 N } K N ( t − ω ′ ε 2 N , s − ω ′ ε K ∗ 2 N ) , α + γ where K N is the kernel of rescaled GUE ω . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 12 / 25

  42. Universality ρ ε, k k -th correlation function of P ε N : N , F. Götze (Bielefeld) Local/Global Universality October 9, 2012 13 / 25

  43. Universality ρ ε, k k -th correlation function of P ε N : N , ω := ( α + γ ) − 1 / 2 , σ Wigner density on [ − ω, ω ] , F. Götze (Bielefeld) Local/Global Universality October 9, 2012 13 / 25

  44. Universality ρ ε, k k -th correlation function of P ε N : N , ω := ( α + γ ) − 1 / 2 , σ Wigner density on [ − ω, ω ] , for all ε ∈ R : F. Götze (Bielefeld) Local/Global Universality October 9, 2012 13 / 25

  45. Universality ρ ε, k k -th correlation function of P ε N : N , ω := ( α + γ ) − 1 / 2 , σ Wigner density on [ − ω, ω ] , ρ 1 ,ε for all ε ∈ R : = ⇒ σ and N F. Götze (Bielefeld) Local/Global Universality October 9, 2012 13 / 25

  46. Universality ρ ε, k k -th correlation function of P ε N : N , ω := ( α + γ ) − 1 / 2 , σ Wigner density on [ − ω, ω ] , ρ 1 ,ε for all ε ∈ R : = ⇒ σ and N � � � sin ( π ( t i − t j ) � 1 t 1 t k σ ( a ) k ρ ε, k lim a + N σ ( a ) , . . . , a + = det , N N σ ( a ) π ( t i − t j ) N →∞ 1 ≤ i , j ≤ k F. Götze (Bielefeld) Local/Global Universality October 9, 2012 13 / 25

  47. Universality ρ ε, k k -th correlation function of P ε N : N , ω := ( α + γ ) − 1 / 2 , σ Wigner density on [ − ω, ω ] , ρ 1 ,ε for all ε ∈ R : = ⇒ σ and N � � � sin ( π ( t i − t j ) � 1 t 1 t k σ ( a ) k ρ ε, k lim a + N σ ( a ) , . . . , a + = det , N N σ ( a ) π ( t i − t j ) N →∞ 1 ≤ i , j ≤ k locally uniformly in t 1 , . . . t k , and a in compact subsets of ( − ω, ω ) . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 13 / 25

  48. Universality ρ ε, k k -th correlation function of P ε N : N , ω := ( α + γ ) − 1 / 2 , σ Wigner density on [ − ω, ω ] , ρ 1 ,ε for all ε ∈ R : = ⇒ σ and N � � � sin ( π ( t i − t j ) � 1 t 1 t k σ ( a ) k ρ ε, k lim a + N σ ( a ) , . . . , a + = det , N N σ ( a ) π ( t i − t j ) N →∞ 1 ≤ i , j ≤ k locally uniformly in t 1 , . . . t k , and a in compact subsets of ( − ω, ω ) . Thm (Venker ’11) ρ k ,γ N ,α , k th correlation function of P γ N ,α : F. Götze (Bielefeld) Local/Global Universality October 9, 2012 13 / 25

  49. Universality ρ ε, k k -th correlation function of P ε N : N , ω := ( α + γ ) − 1 / 2 , σ Wigner density on [ − ω, ω ] , ρ 1 ,ε for all ε ∈ R : = ⇒ σ and N � � � sin ( π ( t i − t j ) � 1 t 1 t k σ ( a ) k ρ ε, k lim a + N σ ( a ) , . . . , a + = det , N N σ ( a ) π ( t i − t j ) N →∞ 1 ≤ i , j ≤ k locally uniformly in t 1 , . . . t k , and a in compact subsets of ( − ω, ω ) . Thm (Venker ’11) ρ k ,γ N ,α , k th correlation function of P γ N ,α : � � � � 1 t 1 t k σ ( a ) k ρ γ, k sin ( π ( t i − t j ) lim a + N σ ( a ) , . . . , a + = det N ,α π ( t i − t j ) N σ ( a ) N →∞ 1 ≤ i , j ≤ k F. Götze (Bielefeld) Local/Global Universality October 9, 2012 13 / 25

  50. Sketch of Proof: Recentering Hoeffding type decomposition of interaction � i < j h ( x i − x j ) = � h ( x i − x j ) + N � � i < j ˜ h ( x i − s ) d µ h Q ( s ) + const . i F. Götze (Bielefeld) Local/Global Universality October 9, 2012 14 / 25

  51. Sketch of Proof: Recentering Hoeffding type decomposition of interaction � i < j h ( x i − x j ) = � h ( x i − x j ) + N � � i < j ˜ h ( x i − s ) d µ h Q ( s ) + const . i into centered fluctuation (w.r.t to µ h Q ) and additional potential h ∗ µ h Q F. Götze (Bielefeld) Local/Global Universality October 9, 2012 14 / 25

  52. Sketch of Proof: Recentering Hoeffding type decomposition of interaction � i < j h ( x i − x j ) = � h ( x i − x j ) + N � � i < j ˜ h ( x i − s ) d µ h Q ( s ) + const . i into centered fluctuation (w.r.t to µ h Q ) and additional potential h ∗ µ h Q Ensemble P h N , Q : N � � 1 ∆( x ) 2 exp {− N P h N , Q ( x ) := Q ( x j ) } exp {− h ( x k − x j ) } Z h N , Q j = 1 j < k F. Götze (Bielefeld) Local/Global Universality October 9, 2012 14 / 25

  53. Sketch of Proof: Recentering Hoeffding type decomposition of interaction � i < j h ( x i − x j ) = � h ( x i − x j ) + N � � i < j ˜ h ( x i − s ) d µ h Q ( s ) + const . i into centered fluctuation (w.r.t to µ h Q ) and additional potential h ∗ µ h Q Ensemble P h N , Q : N � � 1 ∆( x ) 2 exp {− N P h N , Q ( x ) := Q ( x j ) } exp {− h ( x k − x j ) } Z h N , Q j = 1 j < k N � � 1 ∆( x ) 2 exp {− N ˜ = V ( x j ) } exp {− h ( x k − x j ) } , ¯ Z h N , Q j = 1 j < k F. Götze (Bielefeld) Local/Global Universality October 9, 2012 14 / 25

  54. Sketch of Proof: Recentering Hoeffding type decomposition of interaction � i < j h ( x i − x j ) = � h ( x i − x j ) + N � � i < j ˜ h ( x i − s ) d µ h Q ( s ) + const . i into centered fluctuation (w.r.t to µ h Q ) and additional potential h ∗ µ h Q Ensemble P h N , Q : N � � 1 ∆( x ) 2 exp {− N P h N , Q ( x ) := Q ( x j ) } exp {− h ( x k − x j ) } Z h N , Q j = 1 j < k N � � 1 ∆( x ) 2 exp {− N ˜ = V ( x j ) } exp {− h ( x k − x j ) } , ¯ Z h N , Q j = 1 j < k � h ( t − s ) d µ h V ( t ) := Q ( t ) + Q ( s ) Correlation-Fct. of P h Claim: N , Q equivalent to P N , V as N → ∞ , where N � 1 ∆( x ) 2 exp {− N P N , V ( x ) := V ( x j ) } Z N , V j = 1 F. Götze (Bielefeld) Local/Global Universality October 9, 2012 14 / 25

  55. Equilibrium Measure For ν ∈ M 1 ( R ) ( Q , h as above) consider potential F. Götze (Bielefeld) Local/Global Universality October 9, 2012 15 / 25

  56. Equilibrium Measure For ν ∈ M 1 ( R ) ( Q , h as above) consider potential � V ν, Q ( t ) := Q ( t ) + h ( t − s ) d ν ( s ) . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 15 / 25

  57. Equilibrium Measure For ν ∈ M 1 ( R ) ( Q , h as above) consider potential � V ν, Q ( t ) := Q ( t ) + h ( t − s ) d ν ( s ) . Equilibrium measure for potential V (like V ν, Q above) is the unique solution, say µ = T ( V ) , F. Götze (Bielefeld) Local/Global Universality October 9, 2012 15 / 25

  58. Equilibrium Measure For ν ∈ M 1 ( R ) ( Q , h as above) consider potential � V ν, Q ( t ) := Q ( t ) + h ( t − s ) d ν ( s ) . Equilibrium measure for potential V (like V ν, Q above) is the unique solution, say µ = T ( V ) , to the minimization problem � � � log | t − s | − 1 d µ ( t ) d µ ( s ) . min V ( t ) d µ ( t ) + µ ∈M 1 ( R ) F. Götze (Bielefeld) Local/Global Universality October 9, 2012 15 / 25

  59. Equilibrium Measure For ν ∈ M 1 ( R ) ( Q , h as above) consider potential � V ν, Q ( t ) := Q ( t ) + h ( t − s ) d ν ( s ) . Equilibrium measure for potential V (like V ν, Q above) is the unique solution, say µ = T ( V ) , to the minimization problem � � � log | t − s | − 1 d µ ( t ) d µ ( s ) . min V ( t ) d µ ( t ) + µ ∈M 1 ( R ) By Schauder let µ = µ h Q be a fixed point of ν → T ( V ν, Q ) , i.e. F. Götze (Bielefeld) Local/Global Universality October 9, 2012 15 / 25

  60. Equilibrium Measure For ν ∈ M 1 ( R ) ( Q , h as above) consider potential � V ν, Q ( t ) := Q ( t ) + h ( t − s ) d ν ( s ) . Equilibrium measure for potential V (like V ν, Q above) is the unique solution, say µ = T ( V ) , to the minimization problem � � � log | t − s | − 1 d µ ( t ) d µ ( s ) . min V ( t ) d µ ( t ) + µ ∈M 1 ( R ) By Schauder let µ = µ h Q be a fixed point of ν → T ( V ν, Q ) , i.e. Selfconsistency: µ = T ( V µ, Q ) , F. Götze (Bielefeld) Local/Global Universality October 9, 2012 15 / 25

  61. Equilibrium Measure For ν ∈ M 1 ( R ) ( Q , h as above) consider potential � V ν, Q ( t ) := Q ( t ) + h ( t − s ) d ν ( s ) . Equilibrium measure for potential V (like V ν, Q above) is the unique solution, say µ = T ( V ) , to the minimization problem � � � log | t − s | − 1 d µ ( t ) d µ ( s ) . min V ( t ) d µ ( t ) + µ ∈M 1 ( R ) By Schauder let µ = µ h Q be a fixed point of ν → T ( V ν, Q ) , i.e. Selfconsistency: µ = T ( V µ, Q ) , with continuous density µ h Q ( x ) and compact support. F. Götze (Bielefeld) Local/Global Universality October 9, 2012 15 / 25

  62. Fourier Representation of ˜ h � µ = µ h Fourier representation, h real, Q , � � � � � 2 � � e i x j t − � e i x j · � µ ˜ � − h ( x l − x k ) = − h ( t ) dt , � � l � = k j F. Götze (Bielefeld) Local/Global Universality October 9, 2012 16 / 25

  63. Fourier Representation of ˜ h � µ = µ h Fourier representation, h real, Q , � � � � � 2 � � e i x j t − � e i x j · � µ ˜ � − h ( x l − x k ) = − h ( t ) dt , � � l � = k j Let e.g. S ( t ) = � j sin ( t x j ) . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 16 / 25

  64. Fourier Representation of ˜ h � µ = µ h Fourier representation, h real, Q , � � � � � 2 � � e i x j t − � e i x j · � µ ˜ � − h ( x l − x k ) = − h ( t ) dt , � � l � = k j Let e.g. S ( t ) = � If g ( t ) = − � j sin ( t x j ) . h ≥ 0 F. Götze (Bielefeld) Local/Global Universality October 9, 2012 16 / 25

  65. Fourier Representation of ˜ h � µ = µ h Fourier representation, h real, Q , � � � � � 2 � � e i x j t − � e i x j · � µ ˜ � − h ( x l − x k ) = − h ( t ) dt , � � l � = k j Let e.g. S ( t ) = � If g ( t ) = − � j sin ( t x j ) . h ≥ 0 � ∞ � ∞ � 1 � g ( t ) S ( t ) 2 dt g 1 / 2 ( t ) S ( t ) dB t } exp = E exp { 2 0 0 � =: E ω exp { f ( x j , ω ) } , j F. Götze (Bielefeld) Local/Global Universality October 9, 2012 16 / 25

  66. Fourier Representation of ˜ h � µ = µ h Fourier representation, h real, Q , � � � � � 2 � � e i x j t − � e i x j · � µ ˜ � − h ( x l − x k ) = − h ( t ) dt , � � l � = k j Let e.g. S ( t ) = � If g ( t ) = − � j sin ( t x j ) . h ≥ 0 � ∞ � ∞ � 1 � g ( t ) S ( t ) 2 dt g 1 / 2 ( t ) S ( t ) dB t } exp = E exp { 2 0 0 � =: E ω exp { f ( x j , ω ) } , j − � l � = k ˜ and one may linearize h ( x l − x k ) . F. Götze (Bielefeld) Local/Global Universality October 9, 2012 16 / 25

  67. Fourier Representation of ˜ h � µ = µ h Fourier representation, h real, Q , � � � � � 2 � � e i x j t − � e i x j · � µ ˜ � − h ( x l − x k ) = − h ( t ) dt , � � l � = k j Let e.g. S ( t ) = � If g ( t ) = − � j sin ( t x j ) . h ≥ 0 � ∞ � ∞ � 1 � g ( t ) S ( t ) 2 dt g 1 / 2 ( t ) S ( t ) dB t } exp = E exp { 2 0 0 � =: E ω exp { f ( x j , ω ) } , j − � l � = k ˜ and one may linearize h ( x l − x k ) . g ( t ) = − � Need real f : extend limit results to h ( t ) < 0: F. Götze (Bielefeld) Local/Global Universality October 9, 2012 16 / 25

  68. Fourier Representation of ˜ h � µ = µ h Fourier representation, h real, Q , � � � � � 2 � � e i x j t − � e i x j · � µ ˜ � − h ( x l − x k ) = − h ( t ) dt , � � l � = k j Let e.g. S ( t ) = � If g ( t ) = − � j sin ( t x j ) . h ≥ 0 � ∞ � ∞ � 1 � g ( t ) S ( t ) 2 dt g 1 / 2 ( t ) S ( t ) dB t } exp = E exp { 2 0 0 � =: E ω exp { f ( x j , ω ) } , j − � l � = k ˜ and one may linearize h ( x l − x k ) . g ( t ) = − � Need real f : extend limit results to h ( t ) < 0: to family: g z := g + + zg − ≥ 0 , z ≥ 0 F. Götze (Bielefeld) Local/Global Universality October 9, 2012 16 / 25

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