Random Orthogonal Polynomials: From matrices to point processes Diane Holcomb, KTH Integrability and Randomness in Math Physics CIRM, April 2019 Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Outline OPUCs and matrices 1 Random Orthogonal Polynomials and β -ensembles 2 Counting functions and a nice CLT (Killip) 3 The Sine β limit process via it’s counting function 4 Results for Sine β 5 OPUCs and Dirac Operators (if there’s time) 6 Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
OPUCs (part I) For any measure on the unit circle ( ∂ D ), we can associate a family of orthogonal polynomials, Φ 0 ( z ) , Φ 1 ( z ) , Φ 2 ( z ) , ... . Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
OPUCs (part I) For any measure on the unit circle ( ∂ D ), we can associate a family of orthogonal polynomials, Φ 0 ( z ) , Φ 1 ( z ) , Φ 2 ( z ) , ... . There exists a bijection between measures on the unit circle and sequences of Verblunsky coefficients. µ ↔ { α k } ∞ k =0 where the α k ’s give recurrence coefficients that may be used to build the OPUCs that are orthogonal with respect to the measure µ . Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
OPUCs (part I) For any measure on the unit circle ( ∂ D ), we can associate a family of orthogonal polynomials, Φ 0 ( z ) , Φ 1 ( z ) , Φ 2 ( z ) , ... . There exists a bijection between measures on the unit circle and sequences of Verblunsky coefficients. µ ↔ { α k } ∞ k =0 where the α k ’s give recurrence coefficients that may be used to build the OPUCs that are orthogonal with respect to the measure µ . Particularly in the case where µ has finite support we may study the orthogonal polynomials to obtain information about the measure. If the measure is random this can be more useful that studying the measure directly. Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
OPUCs (part II): The Szeg¨ o Recursion Suppose that Φ 0 ( z ) , Φ 1 ( z ) , ... are a family of OPUCs associated to a measure µ on ∂ D . Define: Φ ∗ k ( z ) = z k Φ k ( 1 z ). Then: α k Φ ∗ Φ k +1 ( z ) = z Φ k ( z ) − ¯ k ( z ) Φ ∗ k +1 ( z ) = Φ ∗ k ( z ) − α k z Φ k ( z ) � Φ k +1 ( z ) � � z − ¯ α k � � Φ k ( z ) � � Φ k ( z ) � = = T k Φ ∗ Φ ∗ Φ ∗ k +1 ( z ) − α k z 1 k ( z ) k ( z ) Using this notation we can write � Φ k +1 ( z ) � 1 � � = T k · · · T 0 Φ ∗ k +1 ( z ) 1 Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
OPUCs and Matrices Suppose that U n is an n × n unitary matrix. We can define a spectral measure µ n by � f ( z ) d µ n ( z ) = � f ( U n ) e 1 , e 1 � ∂ D Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
OPUCs and Matrices Suppose that U n is an n × n unitary matrix. We can define a spectral measure µ n by � f ( z ) d µ n ( z ) = � f ( U n ) e 1 , e 1 � ∂ D In this case we have that If the measure µ n = � n k =1 q k δ z k and there exists a bijection k =1 , { q k } n − 1 k =1 ) ↔ { α k } n − 1 ( { z k } n k =0 with α k ∈ D for k ≤ n − 1 and | α n − 1 | = 1. Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
OPUCs and Matrices Suppose that U n is an n × n unitary matrix. We can define a spectral measure µ n by � f ( z ) d µ n ( z ) = � f ( U n ) e 1 , e 1 � ∂ D In this case we have that If the measure µ n = � n k =1 q k δ z k and there exists a bijection k =1 , { q k } n − 1 k =1 ) ↔ { α k } n − 1 ( { z k } n k =0 with α k ∈ D for k ≤ n − 1 and | α n − 1 | = 1. The associated Verblunsky coefficients { α k } n − 1 k =0 allow us to generate Φ 0 ( z ) , Φ 1 ( z ) , ..., Φ n ( z ) = det( U n − zI ) Notice that Φ n ( z ) is not actually orthogonal to the previous polynomials with respect to µ n . Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Random Matrix Ensembles Recall that if we choose O n and U n according to Haar measure on the orthogonal and unitary groups respectively, then the eigenvalues of O n or U n have joint distribution given by 1 | e i θ j − e i θ k | β . � f ( θ 1 , ..., θ n ) = (0.1) Z n ,β j < k for β = 1 , 2. Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Random Matrix Ensembles Recall that if we choose O n and U n according to Haar measure on the orthogonal and unitary groups respectively, then the eigenvalues of O n or U n have joint distribution given by 1 | e i θ j − e i θ k | β . � f ( θ 1 , ..., θ n ) = (0.1) Z n ,β j < k for β = 1 , 2. If we study µ n defined as the spectral measure at e 1 then n � � µ n = q k δ e i θ k , where q k = 1 . k =1 and the weights { q k } are independent from the { θ k } with ( q 1 , ..., q n ) ∼ Dirichlet( β 2 , ..., β 2 ) Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Verblunsky’s for the β -circular ensemble The joint density on the previous slide defines an n -point measure on the unit circle (or [ − π, π ]) for any β > 0. A set of angles with joint density 1 | e i θ j − e i θ k | β . � f ( θ 1 , ..., θ n ) = (0.2) Z n ,β j < k is called the β -circular ensemble Theorem (Killip-Nenciu) Let µ n = � n k =1 q k δ e i θ k with { θ k } having β -circular distribution and ( q 1 , ..., q n ) ∼ Dirichlet ( β 2 , ..., β 2 ) . then the associated Verblunsky coefficients will be independent with rotationally invariant distribution and � Beta (1 , β 2 ( n − k − 1)) k < n − 1 | α k | 2 ∼ 1 k = n − 1 Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Finding a counting function For a measure µ supported on n points we can use the Szeg¨ o recursion to define the function Φ n ( z ) (not an OPUC) which is 0 on the support of µ . e ix ∈ supp µ ⇐ ⇒ Φ n ( e ix ) = 0 ⇒ e ix Φ n − 1 ( e ix ) = α n − 1 Φ ∗ n − 1 ( e ix ) ⇐ On ∂ D the definition of Φ ∗ k becomes Φ ∗ k ( e ix ) = e ixk Φ k ( e ix ): e ix ∈ supp µ ⇐ ⇒ arg α n − 1 = 2 arg(Φ n − 1 ( e ix )) − x ( n − 2) . Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Finding a counting function For a measure µ supported on n points we can use the Szeg¨ o recursion to define the function Φ n ( z ) (not an OPUC) which is 0 on the support of µ . e ix ∈ supp µ ⇐ ⇒ Φ n ( e ix ) = 0 ⇒ e ix Φ n − 1 ( e ix ) = α n − 1 Φ ∗ n − 1 ( e ix ) ⇐ On ∂ D the definition of Φ ∗ k becomes Φ ∗ k ( e ix ) = e ixk Φ k ( e ix ): e ix ∈ supp µ ⇐ ⇒ arg α n − 1 = 2 arg(Φ n − 1 ( e ix )) − x ( n − 2) . More generally define ω k ( x ) = 2 arg(Φ k ( e ix )) − x ( k − 1) , then... � ω n − 1 ( x ) − arg α n − 1 � N ([0 , x ]) = 2 π Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
The counting function from ω n − 1 ( x ) for Circular β P 350 ( x ) for n = 1000 ω n − 1 ( x ) for n = 12, β = 4. Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Counting functions are useful! Theorem (Killip) Let N n ( a , b ) be the number of points of an n-point β -circular ensemble that lie in the arc between a and b, then � π 2 β N n ( a , b ) − n ( b − a ) � � ⇒ N (0 , 1) . 2 log n 2 π Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Counting functions are useful! Theorem (Killip) Let N n ( a , b ) be the number of points of an n-point β -circular ensemble that lie in the arc between a and b, then � π 2 β N n ( a , b ) − n ( b − a ) � � ⇒ N (0 , 1) . 2 log n 2 π Rotational invariance means we can study [ a , b ] = [0 , x ]. We can compute ω k ( x ) − ω k − 1 ( x ) = 2 arg(1 + ˜ α k ) + x d Where ˜ α k = α k (only for a fixed x ) n − 1 � ω n − 1 ( x ) − nx = 2 arg(1 + ˜ α k ) k =0 Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Counting functions are useful! Theorem (Killip) Let N n ( a , b ) be the number of points of an n-point β -circular ensemble that lie in the arc between a and b, then � π 2 β N n ( a , b ) − n ( b − a ) � � ⇒ N (0 , 1) . 2 log n 2 π Rotational invariance means we can study [ a , b ] = [0 , x ]. We can compute ω k ( x ) − ω k − 1 ( x ) = 2 arg(1 + ˜ α k ) + x d Where ˜ α k = α k (only for a fixed x ) n − 1 � ω n − 1 ( x ) − nx = 2 arg(1 + ˜ α k ) k =0 If we reverse the order of the Verblunsky coefficients we get that ω n − 1 ( x ) − nx is a martinage in n . The martingale central limit theorem will give the theorem. Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Local limits for Circular β What if we want to see the local interaction between eigenvalues? − π x π 0 Λ n 0 n (Λ n − x ) Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Local limits for Circular β What if we want to see the local interaction between eigenvalues? − π x π 0 Λ n 0 n (Λ n − x ) Rotational invariance means that we will see the same type of structure everywhere in the spectrum (on the circle). Diane Holcomb, KTH Random Orthogonal Polynomials: From matrices to point processes
Recommend
More recommend
