Randomness in C 2 and Pluripotential Theory Randomness in C 2 and - PowerPoint PPT Presentation

Randomness in C 2 and Pluripotential Theory Randomness in C 2 and Pluripotential Theory

Outline 1 Zeros of univariate random polynomials p : C → C and potential theory; recent results of Bloom-Dauvergne 2 Random polynomials p : C 2 → C and random polynomial mappings F = ( p , q ) : C 2 → C 2 and pluripotential theory; recent results of Bayraktar 3 Generalizations/modifications and open questions Randomness in C 2 and Pluripotential Theory

Kac-Hammersley polynomials Consider random polynomials p n ( z ) = � n j =0 a j z j where the coefficients a 0 , ..., a n are i.i.d. complex Gaussian random variables with E ( a j ) = E ( a j a k ) = 0 and E ( a j ¯ a k ) = δ jk . Thus we get a probability measure Prob n on P n , the polynomials of degree at most n , identified with C n +1 , where, for G ⊂ C n +1 , � 1 e − � n j =0 | a j | 2 dm ( a 0 ) · · · dm ( a n ) Prob n ( G ) = π n +1 G where dm =Lebesgue measure on C . Randomness in C 2 and Pluripotential Theory

Asymptotic expectation � n � n j =1 ( z − ζ j ) and call � Z p n := 1 Write p n ( z ) = a n j =1 δ ζ j the n normalized zero measure of p n . Note Z p n = ∆1 � n log | p n | where (ignore 2 π ) ∆ log | z | = δ 0 . What can we say about asymptotics of E ( � Z p n ) as n → ∞ ? Here, E ( � Z p n ) is a measure defined, for ψ ∈ C c ( C ), as � � � E ( � C n +1 ( � Z p n , ψ ) C dProb n ( a ( n ) ) Z p n ) , ψ C := � n where a ( n ) = ( a 0 , ..., a n ) and ( � Z p n , ψ ) C = 1 j =1 ψ ( ζ j ). n Randomness in C 2 and Pluripotential Theory

Key idea: Reproducing kernel and monomials Note that { z j } j =0 ,..., n := { b j ( z ) } j =0 ,..., n form an orthonormal basis 2 π d θ on S 1 = { z : | z | = 1 } . for P n in L 2 ( µ S 1 ) where µ S 1 = 1 Proposition. lim n →∞ E ( � Z p n ) = µ S 1 . n n � � z j ¯ w j S n ( z , w ) := b j ( z ) b j ( w )= j =0 j =0 is the reproducing kernel for point evaluation at z on P n . On the diagonal w = z , we have S n ( e i θ , e i θ ) = n + 1 and n � | z | 2 j = 1 − | z | 2 n +2 K n ( z ) := S n ( z , z ) = Thus: 1 − | z | 2 j =0 Randomness in C 2 and Pluripotential Theory

2 n log 1 − | z | 2 n +2 2 n log K n ( z ) = 1 1 → log + | z | = max[0 , log | z | ] 1 − | z | 2 locally uniformly on C . Note that ∆ log + | z | = µ S 1 ; thus � 1 � ∆ 2 n log K n ( z ) → µ S 1 . Write | p n ( z ) | = | � n j =0 a j b j ( z ) | =: | < a ( n ) , b ( n ) ( z ) > C n +1 | = K n ( z ) 1 / 2 | < a ( n ) , u ( n ) ( z ) > C n +1 | where || b ( n ) ( z ) || = b ( n ) ( z ) b ( n ) ( z ) u ( n ) ( z ) := K n ( z ) 1 / 2 . Randomness in C 2 and Pluripotential Theory

Use | p n ( z ) | = K n ( z ) 1 / 2 | < a ( n ) , u ( n ) ( z ) > C n +1 | : For ψ ∈ C c ( C ) (recall � Z p n = ∆ 1 n log | p n | ) � � � � � ∆1 E ( � C dProb n ( a ( n ) ) Z p n ) , ψ C = n log | p n ( z ) | , ψ ( z ) C n +1 � � � ∆ 1 C dProb n ( a ( n ) ) = 2 n log K n ( z ) , ψ ( z ) C n +1 � � � ∆1 n log | < a ( n ) , u ( n ) ( z ) > C n +1 | , ψ ( z ) C dProb n ( a ( n ) ) . + C n +1 � The first term (deterministic) goes to S 1 ψ d µ S 1 as n → ∞ and the second term can be rewritten: � � 1 � n log | < a ( n ) , u ( n ) ( z ) > C n +1 | , ∆ ψ ( z ) C dProb n ( a ( n ) ) C n +1 Randomness in C 2 and Pluripotential Theory

� � � 1 � C n +1 log | < a ( n ) , u ( n ) ( z ) > C n +1 | dProb n ( a ( n ) ) = ∆ ψ ( z ) dm ( z ) n C (Fubini). By unitary invariance of dProb n ( a ( n ) ), � I n ( u ( n ) ( z )) := C n +1 log | < a ( n ) , u ( n ) ( z ) > C n +1 | dProb n ( a ( n ) ) � 1 π n +1 log | < a ( n ) , u ( n ) ( z ) > C n +1 | e − � n j =0 | a j | 2 dm ( a 0 ) · · · dm ( a n ) = C n +1 � = 1 log | a 0 | e −| a 0 | 2 dm ( a 0 ) = E (log | a 0 | ) (let u ( n ) ( z ) → (1 , 0 , ..., 0)) π C is a constant for unit vectors u ( n ) ( z ), independent of n (and z ). � � E ( � Thus the second term in Z p n ) , ψ C is 0(1 / n ) and n →∞ E ( � lim Z p n ) = µ S 1 . Randomness in C 2 and Pluripotential Theory

Remarks 1 Clearly “wiggle room” for improvement: more general random coefficients than normalized complex Gaussian 2 Generalizations to random polynomials � n j =0 a j b j ( z ) 3 “Harder” probabilistic results involve analyzing n � | b j ( z ) | 2 K n ( z ) = S n ( z , z ) = j =0 and off-diagonal asymptotics of S n ( z , w ) 4 Sequences vs. arrays of i.i.d. random variables n n � � a ( n ) a j b j ( z ) vs. b j ( z ) . j j =0 j =0 5 Weighted case: � n j =0 a ( n ) b ( n ) ( z ) j j Randomness in C 2 and Pluripotential Theory

General univariate setting: Extremal functions For K ⊂ C compact, we define V K ( z ) := sup { u ( z ) : u ∈ L ( C ) , u ≤ 0 on K } 1 = sup { deg ( p ) log | p ( z ) | : p ∈ ∪ n P n , || p || K ≤ 1 } where L ( C ) = { u ∈ SH ( C ) : u ( z ) − log | z | = 0(1) , | z | → ∞} . For K = S 1 , V S 1 ( z ) = log + | z | . If V K is continuous, defining φ n ( z ) := sup {| p ( z ) | : p ∈ P n , || p || K ≤ 1 } , we have 1 n log φ n ( z ) → V K ( z ) locally uniformly on C . Let µ K := ∆ V K . Randomness in C 2 and Pluripotential Theory

General univariate setting: Potential theory � 1 Let p µ K ( z ) := K log | z − ζ | d µ K ( ζ ) so ∆ p µ K = − µ K and � I ( µ K ) = p µ K ( z ) d µ K ( z ) = µ ∈M ( K ) I ( µ ) inf K � � 1 where I ( µ ) = K log | z − ζ | d µ ( z ) d µ ( ζ ). Then K V K ( z ) = I ( µ K ) − p µ K ( z ) so ∆ V K = µ K . We can recover V K and µ K via L 2 − methods. Note if τ is a measure on K such that || p || K ≤ M n || p || L 2 ( τ ) for all p ∈ P n , then (exercise!) the best constant is given by n � z ∈ K K n ( z ) 1 / 2 = max | b j ( z ) | 2 ) 1 / 2 M n = max z ∈ K ( j =0 j =0 form an orthonormal basis for P n in L 2 ( τ ). where { b j } n Randomness in C 2 and Pluripotential Theory

n +1 ≤ K n ( z ) 1 φ n ( z ) 2 ≤ M 2 Relate K n , φ n : n ( n + 1) The right-hand inequality is from || p || K ≤ M n || p || L 2 ( τ ) ; the left-hand inequality uses the reproducing property of S n ( z , w ). If ( K , τ ) is (BM) i.e., M 1 / n → 1, this shows n 2 n log K n ( z ) ≍ 1 1 n log φ n ( z ) ≍ V K ( z ) . Indeed: If V K is continuous, then (BM) for ( K , τ ) is equivalent to 1 lim 2 n log K n ( z ) = V K ( z ) locally uniformly on C . n →∞ Hence ∆ 1 2 n log K n ( z ) → µ K . Randomness in C 2 and Pluripotential Theory

Summary Thus, what we have really proved is the following: Theorem Let τ be a (BM) measure on a compact set K with V K continuous. Consider random polynomials of the form p n ( z ) = � n j =0 a j b j ( z ) where { b j ( z ) } j =0 ,..., n form an orthonormal basis for P n in L 2 ( τ ) and a 0 , ..., a n are i.i.d. complex Gaussian random variables with E ( a j ) = E ( a j a k ) = 0 and E ( a j ¯ a k ) = δ jk . Then n →∞ E ( � lim Z p n ) = µ K . Note any (BM) measure yields the same limit measure µ K (this is a type of “universality”). “Same” result in weighted case ( b ( n ) j change with n ); limit µ K , Q . Conclusion: limit depends on basis . Randomness in C 2 and Pluripotential Theory

Further questions on random polynomials The method above was used (and generalized) by Bloom, Shiffman, Zelditch (and others). We briefly address the following questions: 1 What can we say about generic convergence of the (random) sequence of subharmonic functions { 1 n log | p n |} ? 2 Can we allow more general coefficients than i.i.d. complex Gaussian? We write P for the space of sequences of random polynomials; note if we consider random polynomials p n ∈ P n as n � a ( n ) b j ( z ) , a ( n ) p n ( z ) = i.i.d j j j =0 then P := ⊗ ∞ n =1 ( P n , Prob n ) = ⊗ ∞ n =1 ( C n +1 , Prob n ) . Also (relevant for weighted case) can have b ( n ) ( z ). j Randomness in C 2 and Pluripotential Theory

General coefficients: The following is due to Ibragimov/Zaporozhets (2013): Theorem For random Kac polynomials of the form p n ( z ) = � n j =0 a j z j with a j i.i.d., E (log (1 + | a j | )) < ∞ is a necessary and sufficient condition for Z p n = ∆(1 n log | p n | ) → 1 � 2 π d θ amost surely in P . Kabluchko/Zaporozhets (2014) considered p. s. of random analytic functions of the form G n ( z ) = � n j =0 a j f n , j z j with deterministic coefficients { f n , j } satisfying certain hypotheses to get conv. in prob. to a target measure. We discuss recent generalizations by Tom BLOOM and Duncan DAUVERGNE (2018). Randomness in C 2 and Pluripotential Theory

Conv. in prob. vs. a.s. conv. Let a j be i.i.d. complex random variables defined on a probability space (Ω , F , P ). For ǫ > 0, n ∈ Z + , let Ω n ,ǫ := { ω ∈ Ω : | a j ( ω ) | ≤ e ǫ n , j = 0 , ..., n } . ∞ � P (Ω c E (log (1 + | a j | )) < ∞ ⇐ ⇒ ∀ ǫ, n ,ǫ ) < ∞ . n =0 P ( | a j | > e | z | ) = o (1 / | z | ) ⇒ ∀ ǫ, n →∞ P (Ω c lim n ,ǫ ) = 0 . When does � Z p n → µ K a.s.? In probability? This latter means for any open set U in the space of prob. measures on C with µ K ∈ U , we have P ( � Z p n ∈ U ) → 0 as n → ∞ . Randomness in C 2 and Pluripotential Theory