Multiple Orthogonal Polynomials and the Normal Matrix Model Arno - PowerPoint PPT Presentation

Multiple Orthogonal Polynomials and the Normal Matrix Model Arno Kuijlaars Department of Mathematics KU Leuven, Belgium Joint work with Pavel Bleher Adv. Math. 2012 Random Matrices and their Applications T´ el´ ecom ParisTech, Paris, 8 October 2012

1. Orthogonal and multiple orthogonal polynomials Orthogonal polynomial P n ( x ) = x n + · · · satisfies � ∞ P n ( x ) x k w ( x ) dx = 0 , k = 0 , 1 , . . . , n − 1 , −∞ OPs have many nice properties including a three term recurrence relation xP n ( x ) = P n +1 ( x ) + b n P n ( x ) + a n P n − 1 ( x )

1. Orthogonal and multiple orthogonal polynomials Orthogonal polynomial P n ( x ) = x n + · · · satisfies � ∞ P n ( x ) x k w ( x ) dx = 0 , k = 0 , 1 , . . . , n − 1 , −∞ OPs have many nice properties including a three term recurrence relation xP n ( x ) = P n +1 ( x ) + b n P n ( x ) + a n P n − 1 ( x ) and a Riemann-Hilbert problem

Riemann Hilbert problem Fokas-Its-Kitaev (1992) characterized OPs by means of 2 × 2 matrix valued Riemann-Hilbert problem (1) Y : C \ R → C 2 × 2 is analytic, � 1 � w (2) Y + = Y − on R , 0 1 � � z n 0 (3) Y ( z ) = ( I 2 + O (1 / z )) as z → ∞ . z − n 0

Riemann Hilbert problem Fokas-Its-Kitaev (1992) characterized OPs by means of 2 × 2 matrix valued Riemann-Hilbert problem (1) Y : C \ R → C 2 × 2 is analytic, � 1 � w (2) Y + = Y − on R , 0 1 � � z n 0 (3) Y ( z ) = ( I 2 + O (1 / z )) as z → ∞ . z − n 0 Unique solution  � ∞  1 P n ( s ) w ( s ) P n ( z ) ds   2 π i s − z   � ∞ −∞ Y ( z ) =   P n − 1 ( s ) w ( s ) − 2 π i γ − 1 − γ − 1 n − 1 P n − 1 ( z ) ds n − 1 s − z −∞ � ∞ P n − 1 ( x ) x n − 1 w ( x ) dx > 0 . where γ n − 1 = −∞

Multiple orthogonal polynomials Multiple orthogonal polynomial (MOP) is a monic polynomial of degree n 1 + n 2 P n 1 , n 2 ( x ) = x n 1 + n 2 + · · · characterized by � ∞ P n 1 , n 2 ( x ) x k w 1 ( x ) dx = 0 , k = 0 , 1 , . . . , n 1 − 1 , −∞ � ∞ P n 1 , n 2 ( x ) x k w 2 ( x ) dx = 0 , k = 0 , 1 , . . . , n 2 − 1 . −∞ Immediate extension to r weights w 1 , . . . , w r and ( n 1 , . . . , n r ) ∈ N r .

MOP in random matrix theory MOPs appear in random matrix theory and related stochastic processes (a) Random matrices with external source (b) Non-intersecting Brownian motions (c) Non-intersecting squared Bessel paths (d) Coupled random matrices - two matrix model - Cauchy matrix model

Properties of MOPS 1: short recurrence MOPs P n 1 , n 2 with two weight functions The polynomials Q n defined by Q 2 k = P k , k , Q 2 k +1 = P k +1 , k have a four term recurrence xQ n ( x ) = Q n +1 ( x ) + a n Q n ( x ) + b n Q n − 1 ( x ) + c n Q n − 2 ( x )

Properties of MOPS 1: short recurrence MOPs P n 1 , n 2 with two weight functions The polynomials Q n defined by Q 2 k = P k , k , Q 2 k +1 = P k +1 , k have a four term recurrence xQ n ( x ) = Q n +1 ( x ) + a n Q n ( x ) + b n Q n − 1 ( x ) + c n Q n − 2 ( x ) MOPs with r weight functions and near-diagonal multi-indices satisfy an r + 2 -term recurrence.

Properties of MOPS 2: RH problem MOPs with two weight functions have a Riemann-Hilbert problem of size 3 × 3 (1) Y : C \ R → C 3 × 3 is analytic,   1 w 1 w 2   on R , (2) Y + = Y − 0 1 0 0 0 1   z n 1 + n 2 0 0   as z → ∞ . z − n 1 (3) Y ( z ) = ( I 3 + O (1 / z )) 0 0 z − n 2 0 0 Van Assche-Geronimo-K (2001)

Properties of MOPS 2: RH problem MOPs with two weight functions have a Riemann-Hilbert problem of size 3 × 3 (1) Y : C \ R → C 3 × 3 is analytic,   1 w 1 w 2   on R , (2) Y + = Y − 0 1 0 0 0 1   z n 1 + n 2 0 0   as z → ∞ . z − n 1 (3) Y ( z ) = ( I 3 + O (1 / z )) 0 0 z − n 2 0 0 Van Assche-Geronimo-K (2001) RH problem has a unique solution if and only if the MOP P n 1 , n 2 uniquely exists and in that case Y 11 ( z ) = P n 1 , n 2 ( z ) MOPs with r weight functions have a RH problem of size ( r + 1) × ( r + 1) .

2. Normal matrix model Probability measure on n × n complex matrices 1 e − n t 0 Tr( MM ∗ − V ( M ) − V ( M ∗ )) dM , t 0 > 0 , Z n with ∞ � t k k M k V ( M ) = k =1

2. Normal matrix model Probability measure on n × n complex matrices 1 e − n t 0 Tr( MM ∗ − V ( M ) − V ( M ∗ )) dM , t 0 > 0 , Z n with ∞ � t k k M k V ( M ) = k =1 Model depends on parameters t 0 > 0 , t 1 , t 2 , . . . , t k , . . . . For t 1 = t 2 = · · · = 0 this is the Ginibre ensemble. Ginibre (1965)

Ginibre ensemble Eigenvalues in the Ginibre ensemble have a limiting distribution as n → ∞ that is uniform in a disk around 0 with radius √ t 0 . 1.5 1 0.5 0 −0.5 −1 −1.5 −1.5 −1 −0.5 0 0.5 1 1.5

Laplacian growth For general t 1 , t 2 , . . . , and t 0 sufficiently small, the eigenvalues of M fill out a two-dimensional domain Ω = Ω( t 0 , t 1 , . . . ) Ω is characterized by �� t 0 = 1 t k = − 1 dA ( z ) π area(Ω) , , k ≥ 1 z k π C \ Ω

Laplacian growth For general t 1 , t 2 , . . . , and t 0 sufficiently small, the eigenvalues of M fill out a two-dimensional domain Ω = Ω( t 0 , t 1 , . . . ) Ω is characterized by �� t 0 = 1 t k = − 1 dA ( z ) π area(Ω) , , k ≥ 1 z k π C \ Ω As a function of t 0 , the boundary of Ω evolves according to the model of Laplacian growth. Laplacian growth is unstable. Singularities develop in finite time. Wiegmann-Zabrodin (2000) Teoderescu-Bettelheim-Agam-Zabrodin-Wiegmann (2005)

Cubic case V ( z ) = t 3 3 z 3 2 , 0 –2 0 2

Cubic case 2 , 0 –2 0 2

Cubic case 2 , 0 –2 0 2

Cubic case 2 , 0 –2 0 2

Cubic case 2 , 0 –2 0 2

Cubic case 2 , 0 –2 0 2

Cubic case 2 , 0 –2 0 2

Cubic case 2 , 0 –2 0 2

3. Mathematical problem Normal matrix model 1 e − n t 0 Tr( MM ∗ − V ( M ) − V ( M ∗ )) dM , t 0 > 0 , Z n is not well-defined if V is a polynomial of degree ≥ 3

3. Mathematical problem Normal matrix model 1 e − n t 0 Tr( MM ∗ − V ( M ) − V ( M ∗ )) dM , t 0 > 0 , Z n is not well-defined if V is a polynomial of degree ≥ 3 The normalization constant (partition function) � e − n t 0 Tr( MM ∗ − V ( M ) − V ( M ∗ )) dM = + ∞ . Z n = is divergent.

Elbau-Felder approach Elbau and Felder use a cut-off. They restrict to matrices with eigenvalues in a well-chosen bounded domain D .

Elbau-Felder approach Elbau and Felder use a cut-off. They restrict to matrices with eigenvalues in a well-chosen bounded domain D . Then the induced probability measure on eigenvalues is a determinantal point process on D . Eigenvalues fill out a domain Ω that evolves according to Laplacian growth provided t 0 is small enough. Elbau-Felder (2005)

Orthogonal polynomials Average characteristic polynomial P n ( z ) = E [ zI n − M ] in the cut-off model is an orthogonal polynomial for scalar product �� f ( z ) g ( z ) e − n t 0 ( | z | 2 − V ( z ) − V ( z )) dA ( z ) � f , g � = D Elbau (ETH thesis, arXiv 2007) Orthogonality does not make sense if D = C , since integrals would diverge if f and g are polynomials

Recurrence relation OPs in the cut-off model satisfy a recurrence relation zP n ( z ) = P n +1 ( z ) + a (1) n P n ( z ) + · · · + a ( r ) n P n − r ( z ) + “remainder term”

Recurrence relation OPs in the cut-off model satisfy a recurrence relation zP n ( z ) = P n +1 ( z ) + a (1) n P n ( z ) + · · · + a ( r ) n P n − r ( z ) + “remainder term” Remainder term comes from boundary integrals that are due to the cut-off. Remainder term is exponentially small for t 0 > 0 sufficiently small.

Zeros of OPs Conjecture: The zeros of P n do not fill out the twodimensional domain Ω as n → ∞ , but instead accumulate along a contour Σ 1 inside Ω . Singularities appear when Σ 1 meets the boundary of Ω .

Zeros of OPs Conjecture: The zeros of P n do not fill out the twodimensional domain Ω as n → ∞ , but instead accumulate along a contour Σ 1 inside Ω . Singularities appear when Σ 1 meets the boundary of Ω . In the cubic case V ( z ) = t 3 3 z 3 , t 3 > 0 , the contour is a three-star Σ 1 = [0 , x ∗ ] ∪ [0 , e 2 π i / 3 x ∗ ] ∪ [0 , e − 2 π i / 3 x ∗ ] . Elbau (ETH thesis, arXiv 2007)

Cubic case 2 , 0 –2 0 2

Cubic case 2 , 0 –2 0 2

4. Different approach Scalar product in the cut-off model �� f ( z ) g ( z ) e − n t 0 ( | z | 2 − V ( z ) − V ( z )) dA ( z ) � f , g � = D satisfies (due to Green’s theorem) n � zf , g � = t 0 � f , g ′ � + n � f , V ′ g � � − t 0 f ( z ) g ( z ) e − n t 0 ( | z | 2 − V ( z ) − V ( z ) ) dz 2 i ∂ D Our idea: drop the boundary term

Hermitian form Consider an a priori abstract sesquilinear form on the space of polynomials satisfying n � zf , g � = t 0 � f , g ′ � + n � f , V ′ g �

Hermitian form Consider an a priori abstract sesquilinear form on the space of polynomials satisfying n � zf , g � = t 0 � f , g ′ � + n � f , V ′ g � We also want to keep the Hermitian form condition � g , f � = � f , g �