local universality of repulsive particle systems and
play

Local Universality of Repulsive Particle Systems and Random Matrices - PowerPoint PPT Presentation

Nov 09, 2023 •213 likes •1.64k views

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

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Universality in the equilibration of Universality in the equilibration of isolated systems after

Universality in the equilibration of Universality in the equilibration of isolated systems after

Universality in the equilibration of Universality in the equilibration of isolated systems after a small quench isolated systems after a small quench Lorenzo Campos Venuti University of Southern California, Los Angeles Paolo Zanardi (USC)

222 views • 18 slides
Universality M. Hairer University of Warwick New fellows seminar Universality Experimental /

Universality M. Hairer University of Warwick New fellows seminar Universality Experimental /

Universality M. Hairer University of Warwick New fellows seminar Universality Experimental / numerical fact: large-scale behaviour of systems displaying random behaviour self-similar and independent of microscopic description. Universality

731 views • 19 slides
20 Particle Systems Steve Marschner Eston Schweickart CS4620 Spring 2017 Examples of Particle

20 Particle Systems Steve Marschner Eston Schweickart CS4620 Spring 2017 Examples of Particle

20 Particle Systems Steve Marschner Eston Schweickart CS4620 Spring 2017 Examples of Particle Systems Particle Dreams [Karl Sims, 1988] Examples of Particle Systems Balloon Burst [Macklin et al., SIGGRAPH 2015] Examples of Particle Systems

603 views • 32 slides
Universality of local spectral statistics of random matrices L aszl o Erd os

Universality of local spectral statistics of random matrices L aszl o Erd os

Universality of local spectral statistics of random matrices L aszl o Erd os Ludwig-Maximilians-Universit at, Munich, Germany Abel Symposium, Oslo, Aug 21 2012 With P. Bourgade, A. Knowles, B. Schlein, H.T. Yau, and J. Yin 1

524 views • 31 slides
From Wigner-Dyson to Pearcey: Universality of the Local Eigenvalue Statistics of Random Matrices

From Wigner-Dyson to Pearcey: Universality of the Local Eigenvalue Statistics of Random Matrices

From Wigner-Dyson to Pearcey: Universality of the Local Eigenvalue Statistics of Random Matrices at the Cusp Lszl Erds IST Austria (supported by ERC Advanced Grant RANMAT) Workshop on Statistical Mechanics Les Diablerets, Switzerland

831 views • 55 slides
Cusp Local Law and Complex Hermitian Universality joint work with Johannes Alt, Lszl Erds and

Cusp Local Law and Complex Hermitian Universality joint work with Johannes Alt, Lszl Erds and

Cusp Local Law and Complex Hermitian Universality joint work with Johannes Alt, Lszl Erds and Torben Krger Feburary 19, 2019 Workshop on Statistical Mechanics, Les Diablerets Dominik Schrder Partially supported by ERC Advanced

746 views • 36 slides
Variational mapping particle filter combining particle filters with local optimal transport Manuel

Variational mapping particle filter combining particle filters with local optimal transport Manuel

Variational mapping particle filter combining particle filters with local optimal transport Manuel Pulido 1 , 2 and Peter Jan vanLeeuwen 1 , 3 1 Data Assimilation Research Centre, University of Reading, UK 2 Department of Physics, Universidad

388 views • 19 slides
Two-particle systems (Ch. 5) Generalization to two-particles is very natural. A two-particle state

Two-particle systems (Ch. 5) Generalization to two-particles is very natural. A two-particle state

EE201/MSE207 Lecture 12 Two-particle systems (Ch. 5) Generalization to two-particles is very natural. A two-particle state is characterized by wavefunction 1 , 2 , (for a moment forget about spin) = 2 2 2

159 views • 12 slides
Reproducibility, trust, and proof checking From Universality of Facts to Universality of Proofs

Reproducibility, trust, and proof checking From Universality of Facts to Universality of Proofs

Reproducibility, trust, and proof checking From Universality of Facts to Universality of Proofs Dale Miller Inria Saclay &amp; LIX, Ecole Polytechnique Palaiseau, France 18 October 2016 Reference:Communicating and trusting proofs: The

165 views • 13 slides
Universality classes of Quantum Gravity Frank Saueressig Research Institute for Mathematics,

Universality classes of Quantum Gravity Frank Saueressig Research Institute for Mathematics,

Universality classes of Quantum Gravity Frank Saueressig Research Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen A. Contillo, G. DOdorico, E. Manrique, S. Rechenberger, M. Schutten arXiv:1102.5012,

567 views • 46 slides
Digital Circuits and Systems Universality, Rearranging Truth Tables Shankar Balachandran*

Digital Circuits and Systems Universality, Rearranging Truth Tables Shankar Balachandran*

Spring 2015 Week 2 Module 7 Digital Circuits and Systems Universality, Rearranging Truth Tables Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras *Currently a Visiting Professor at IIT Bombay

484 views • 14 slides
Time-Development of Explosions and a Path-Space Measure for Diffusion Process with Repulsive

Time-Development of Explosions and a Path-Space Measure for Diffusion Process with Repulsive

Time-Development of Explosions and a Path-Space Measure for Diffusion Process with Repulsive Higer Order Drift Hiroshi Ezawa Keiji Watanabe and Toru Nakamura 1. Time-development of explosion 1. Explosion Stochastic differential equation

295 views • 13 slides
Universality of step bunching behavior in systems with non-conserved dynamics Joachim Krug

Universality of step bunching behavior in systems with non-conserved dynamics Joachim Krug

Universality of step bunching behavior in systems with non-conserved dynamics Joachim Krug Institute for Theoretical Physics, University of Cologne Electromigration-induced step bunching on Si(111) [Yang, Fu &amp; Williams 1996] M. Ivanov, JK,

718 views • 26 slides
Probing lepton flavour universality in B (0) K (0 ) + decays at CMS

Probing lepton flavour universality in B (0) K (0 ) + decays at CMS

Probing lepton flavour universality in B (0) K (0 ) + decays at CMS Author: Julia-Suzana Dancu Supervisor: Dr Oliver Buchm uller Post-doc: Dr Matthias Komm 1/11 Lepton Universality Tests at CMS February 28, 2019

580 views • 19 slides
Why Complexify? Universality Principles of Complex Systems Symmetry CSYS/MATH 300, Spring, 2013

Why Complexify? Universality Principles of Complex Systems Symmetry CSYS/MATH 300, Spring, 2013

Why Complexify? Why Complexify? Universality Principles of Complex Systems Symmetry CSYS/MATH 300, Spring, 2013 | #SpringPoCS2013 Breaking The Big Theory Final words Prof. Peter Dodds For your consideration @peterdodds References

963 views • 67 slides
COMP30019 Graphics and Interaction Particle Systems Adrian Pearce Department of Computer Science

COMP30019 Graphics and Interaction Particle Systems Adrian Pearce Department of Computer Science

Introduction Particle creation Geometry Shaders Sprites Voxels Unity3D Particle System COMP30019 Graphics and Interaction Particle Systems Adrian Pearce Department of Computer Science and Software Engineering University of Melbourne The

256 views • 23 slides
Interpolating Curves Intro to curve interpolation &amp; approximation Polynomial

Interpolating Curves Intro to curve interpolation &amp; approximation Polynomial

Interpolating Curves Intro to curve interpolation &amp; approximation Polynomial interpolation Bzier curves Showtime: Logistics Assignment 2 is available For assignment questions use the bulletin board or email:

740 views • 47 slides
Numerical Analysis I Interpolation and Polynomial Approximation Instructor: Wei-Cheng Wang 1

Numerical Analysis I Interpolation and Polynomial Approximation Instructor: Wei-Cheng Wang 1

Numerical Analysis I Interpolation and Polynomial Approximation Instructor: Wei-Cheng Wang 1 Department of Mathematics National TsingHua University Fall 2010 1These slides are based on Prof. Tsung-Ming Huang(NTNU)s original slides Wei-Cheng

1.56k views • 64 slides
Reducing number field defining polynomials: An application to class group computations Alexandre

Reducing number field defining polynomials: An application to class group computations Alexandre

Overview on class groups Reducing the defining polynomial Reducing number field defining polynomials: An application to class group computations Alexandre Glin Laboratoire dInformatique de Paris 6, UPMC, Sorbonne Universits 09/11/2016

575 views • 42 slides
On the interface between Hermitian and normal random matrices Yacin Ameur. Centre for

On the interface between Hermitian and normal random matrices Yacin Ameur. Centre for

On the interface between Hermitian and normal random matrices Yacin Ameur. Centre for Mathematical Sciences Lund University, Sweden Yacin.Ameur@maths.lth.se Random Matrices and Related Topics, KIAS, May 7, 2019 1 / 41 External potential Let

837 views • 48 slides
Elements of differential calculus and optimization. Joan Alexis Glaun` es October 24, 2019 1/29

Elements of differential calculus and optimization. Joan Alexis Glaun` es October 24, 2019 1/29

Elements of differential calculus and optimization. Joan Alexis Glaun` es October 24, 2019 1/29 Differential Calculus in R n partial derivatives Partial derivatives of a real-valued function defined on R n : f : R n R . example : f : R

476 views • 29 slides
Inverse Kinematics (part 2) CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter

Inverse Kinematics (part 2) CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter

Inverse Kinematics (part 2) CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2017 Forward Kinematics We will use the vector: ... 1 2 M to represent the array of M joint DOF values

1.04k views • 87 slides
Structure of the Hessian Graham C. Goodwin September 2004 Centre for Complex Dynamic Systems

Structure of the Hessian Graham C. Goodwin September 2004 Centre for Complex Dynamic Systems

Structure of the Hessian Graham C. Goodwin September 2004 Centre for Complex Dynamic Systems and Control 1. Introduction We saw in earlier lectures that a core ingredient in quadratic constrained optimisation problems is the Hessian matrix H .

628 views • 42 slides
Numerical Computation Sargur N. Srihari srihari@cedar.buffalo.edu This is part of lecture slides

Numerical Computation Sargur N. Srihari srihari@cedar.buffalo.edu This is part of lecture slides

Srihari Deep Learning Numerical Computation Sargur N. Srihari srihari@cedar.buffalo.edu This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/CSE676 1 Srihari Deep Learning Topics Overflow and Underflow

694 views • 40 slides

More recommend