Universality for zeros of random polynomials Turgay Bayraktar Universality for zeros of random polynomials Motivation Random polynomials Turgay Bayraktar Random Holomorphic Sections Syracuse University Further Study Asymptotic Normality MWAA Bloomington October 11, 2015 Turgay Bayraktar Universality for zeros of random polynomials
Kac polynomials Universality A Kac polynomial on the complex plane is of the form for zeros of random polynomials N � a j z j f N ( z ) = Turgay Bayraktar j =0 Motivation We assume that a j ’s are real or complex identically distributed Random independent i.i.d. random variables and let P denote their polynomials distribution law. Identifying Random Holomorphic Sections Poly N → C N +1 Further Study Asymptotic f N → ( a j ) N Normality j =0 we obtain the probability space ( Poly N , Prob N ) where Prob N is the ( N + 1)-fold product probability measure induced from P . Then we from the product probability space � ∞ N =1 ( Poly N , Prob N ) whose elements are sequences of random polynomials. Turgay Bayraktar Universality for zeros of random polynomials
Kac polynomials Universality A Kac polynomial on the complex plane is of the form for zeros of random polynomials N � a j z j f N ( z ) = Turgay Bayraktar j =0 Motivation We assume that a j ’s are real or complex identically distributed Random independent i.i.d. random variables and let P denote their polynomials distribution law. Identifying Random Holomorphic Sections Poly N → C N +1 Further Study Asymptotic f N → ( a j ) N Normality j =0 we obtain the probability space ( Poly N , Prob N ) where Prob N is the ( N + 1)-fold product probability measure induced from P . Then we from the product probability space � ∞ N =1 ( Poly N , Prob N ) whose elements are sequences of random polynomials. Turgay Bayraktar Universality for zeros of random polynomials
Empirical measure of zeros Writing Universality for zeros of N N random a j z j = a N � � f N ( z ) = ( z − ζ j ) polynomials j =0 j =1 Turgay Bayraktar where ζ j ’s are the roots of f N . We may define a random variable Motivation Poly N → M ( C ) Random polynomials N Random � Holomorphic f N → [ Z f N ] := δ ζ j . Sections j =1 Further Study Asymptotic Normality Turgay Bayraktar Universality for zeros of random polynomials
Empirical measure of zeros Writing Universality for zeros of N N random a j z j = a N � � f N ( z ) = ( z − ζ j ) polynomials j =0 j =1 Turgay Bayraktar where ζ j ’s are the roots of f N . We may define a random variable Motivation Poly N → M ( C ) Random polynomials N Random � Holomorphic f N → [ Z f N ] := δ ζ j . Sections j =1 Further Study Asymptotic and we define the expected zero measure Normality N � � � E [ Z f N ] , ϕ � := ϕ ( ζ j ) dProb N Poly N j =1 for continuous function ϕ ∈ C c ( C ) . Turgay Bayraktar Universality for zeros of random polynomials
Kac Ensemble: Gaussian Case Universality Theorem (Kac-Hammersley-Shepp-Vanderbei) for zeros of random Assume that a j are i.i.d. complex (or real) valued Gaussian random polynomials variables of mean zero and variance one. Then Turgay Bayraktar 1 1 N E [ Z f N ] → 2 π d θ weakly as N → ∞ . Motivation Almost surely 1 1 N Z f N → 2 π d θ weakly as N → ∞ . Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality Turgay Bayraktar Universality for zeros of random polynomials
Kac Ensemble: Gaussian Case Universality Theorem (Kac-Hammersley-Shepp-Vanderbei) for zeros of random Assume that a j are i.i.d. complex (or real) valued Gaussian random polynomials variables of mean zero and variance one. Then Turgay Bayraktar 1 1 N E [ Z f N ] → 2 π d θ weakly as N → ∞ . Motivation Almost surely 1 1 N Z f N → 2 π d θ weakly as N → ∞ . Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality f 10 . Turgay Bayraktar Universality for zeros of random polynomials
Kac Ensemble: Gaussian Case Universality Theorem (Kac-Hammersley-Shepp-Vanderbei) for zeros of random Assume that a j are i.i.d. complex (or real) valued Gaussian random polynomials variables of mean zero and variance one. Then Turgay Bayraktar 1 1 N E [ Z f N ] → 2 π d θ weakly as N → ∞ . Motivation Almost surely 1 1 N Z f N → 2 π d θ weakly as N → ∞ . Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality f 10 , f 50 Turgay Bayraktar Universality for zeros of random polynomials
Kac Ensemble: Gaussian Case Universality Theorem (Kac-Hammersley-Shepp-Vanderbei) for zeros of random Assume that a j are i.i.d. complex (or real) valued Gaussian random polynomials variables of mean zero and variance one. Then Turgay Bayraktar 1 1 N E [ Z f N ] → 2 π d θ weakly as N → ∞ . Motivation Almost surely 1 1 N Z f N → 2 π d θ weakly as N → ∞ . Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality f 10 , f 50 , f 100 Turgay Bayraktar Universality for zeros of random polynomials
Kac Ensemble: Gaussian Case Universality Theorem (Kac-Hammersley-Shepp-Vanderbei) for zeros of random Assume that a j are i.i.d. complex (or real) valued Gaussian random polynomials variables of mean zero and variance one. Then Turgay Bayraktar 1 1 N E [ Z f N ] → 2 π d θ weakly as N → ∞ . Motivation Almost surely 1 1 N Z f N → 2 π d θ weakly as N → ∞ . Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality f 10 , f 50 , f 100 , f 250 Turgay Bayraktar Universality for zeros of random polynomials
Kac Ensemble: Gaussian Case Universality Theorem (Kac-Hammersley-Shepp-Vanderbei) for zeros of random Assume that a j are i.i.d. complex (or real) valued Gaussian random polynomials variables of mean zero and variance one. Then Turgay Bayraktar 1 1 N E [ Z f N ] → 2 π d θ weakly as N → ∞ . Motivation Almost surely 1 1 N Z f N → 2 π d θ weakly as N → ∞ . Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality f 10 , f 50 , f 100 , f 250 , f 500 Turgay Bayraktar Universality for zeros of random polynomials
Kac Ensemble: Gaussian Case Universality Theorem (Kac-Hammersley-Shepp-Vanderbei) for zeros of random Assume that a j are i.i.d. complex (or real) valued Gaussian random polynomials variables of mean zero and variance one. Then Turgay Bayraktar 1 1 N E [ Z f N ] → 2 π d θ weakly as N → ∞ . Motivation Almost surely 1 1 N Z f N → 2 π d θ weakly as N → ∞ . Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality f 10 , f 50 , f 100 , f 250 , f 500 , f 1000 Turgay Bayraktar Universality for zeros of random polynomials
Kac Ensemble: Gaussian Case Universality Theorem (Kac-Hammersley-Shepp-Vanderbei) for zeros of random Assume that a j are i.i.d. complex (or real) valued Gaussian random polynomials variables of mean zero and variance one. Then Turgay Bayraktar 1 1 N E [ Z f N ] → 2 π d θ weakly as N → ∞ . Motivation Almost surely 1 1 N Z f N → 2 π d θ weakly as N → ∞ . Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality f 10 , f 50 , f 100 , f 250 , f 500 , f 1000 , f 2000 Turgay Bayraktar Universality for zeros of random polynomials
Why? Universality Monomials z j for an ONB for Poly N relative to for zeros of random polynomials � 2 π � f , g � := 1 f ( z ) g ( z ) d θ Turgay 2 π Bayraktar 0 and Bergman kernel Motivation Random polynomials N � z j w j K N ( z , w ) = Random Holomorphic j =0 Sections Further Study is reproducing kernel of point evaluation at z that is Asymptotic Normality � S 1 f ( w ) K N ( z , w ) d θ f ( z ) = 2 π and K ( z , z ) = 1 −| z | 2 n +2 1 −| z | 2 . Moreover, K N ( e i θ , e i θ ) = N + 1 . Turgay Bayraktar Universality for zeros of random polynomials
Proof of Gaussian case Universality This implies that for zeros of random polynomials 1 2 N log K N ( z , z ) → log + | z | := max(log | z | , 0) Turgay Bayraktar locally uniformly on C . Therefore ∆( 1 2 N log K N ( z , z )) → d θ Motivation 2 π Random polynomials Random Holomorphic Sections Further Study Asymptotic Normality Turgay Bayraktar Universality for zeros of random polynomials
Proof of Gaussian case Universality This implies that for zeros of random polynomials 1 2 N log K N ( z , z ) → log + | z | := max(log | z | , 0) Turgay Bayraktar locally uniformly on C . Therefore ∆( 1 2 N log K N ( z , z )) → d θ Motivation 2 π Now, Random polynomials N log | f N ( z ) | = 1 1 N log |� a N , u N ( z ) �| + 1 Random 2 N log K N ( z , z ) Holomorphic Sections Further Study where Asymptotic z j Normality � � a N , u N ( z ) � = a j � K N ( z , z ) j K N ( z , z ) ) ∈ C N +1 is a unit z N and u N ( z ) = ( √ 1 √ z √ K N ( z , z ) , K N ( z , z ) , . . . , vector. Turgay Bayraktar Universality for zeros of random polynomials
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