The normal matrix model with monomial potential and - PowerPoint PPT Presentation

The normal matrix model with monomial potential and multi-orthogonality on a star A.B.J. Kuijlaars 1 A. López-García 2 1 KU Leuven 2 University of South Alabama A. López-García (U. South Alabama) 1 / 19

Main ideas in the talk There is a natural connection between: 1) The global asymptotic distribution of eigenvalues in the normal matrix model with monomial potential. 2) The limiting zero distribution of a certain sequence of polynomials. The limiting zero distribution of the sequence of polynomials is part of the solution to a (vector) equilibrium problem. A. López-García (U. South Alabama) 2 / 19

Main ideas in the talk There is a natural connection between: 1) The global asymptotic distribution of eigenvalues in the normal matrix model with monomial potential. 2) The limiting zero distribution of a certain sequence of polynomials. The limiting zero distribution of the sequence of polynomials is part of the solution to a (vector) equilibrium problem. The polynomials are multi-orthogonal with respect to a system of weights defined on a star-like set. A. López-García (U. South Alabama) 2 / 19

Main ideas in the talk There is a natural connection between: 1) The global asymptotic distribution of eigenvalues in the normal matrix model with monomial potential. 2) The limiting zero distribution of a certain sequence of polynomials. The limiting zero distribution of the sequence of polynomials is part of the solution to a (vector) equilibrium problem. The polynomials are multi-orthogonal with respect to a system of weights defined on a star-like set. The problem is investigated in a pre-critical regime (for a certain parameter in the model). A. López-García (U. South Alabama) 2 / 19

Normal matrix model D : two-dimensional compact domain in C . V : real-valued continuous function on D . A. López-García (U. South Alabama) 3 / 19

Normal matrix model D : two-dimensional compact domain in C . V : real-valued continuous function on D . NMM: Probability measure on { M ∈ C n × n : M normal , σ ( M ) ⊂ D } , that induces on D n (the space of eigenvalues) the probability distribution � � � n � 1 | z i − z j | 2 d A ( z 1 ) . . . d A ( z n ) , exp − n V ( z i ) Z n i = 1 i < j where d A is area measure on D . A. López-García (U. South Alabama) 3 / 19

Normal matrix model D : two-dimensional compact domain in C . V : real-valued continuous function on D . NMM: Probability measure on { M ∈ C n × n : M normal , σ ( M ) ⊂ D } , that induces on D n (the space of eigenvalues) the probability distribution � � � n � 1 | z i − z j | 2 d A ( z 1 ) . . . d A ( z n ) , exp − n V ( z i ) Z n i = 1 i < j where d A is area measure on D . Wiegmann-Zabrodin ’00 Elbau-Felder ’05 Ameur-Hedenmalm-Makarov ’11 Bleher-Kuijlaars ’12 A. López-García (U. South Alabama) 3 / 19

� � � n � 1 | z i − z j | 2 d A ( z 1 ) . . . d A ( z n ) . exp − n V ( z i ) (1) Z n i = 1 i < j A. López-García (U. South Alabama) 4 / 19

� � � n � 1 | z i − z j | 2 d A ( z 1 ) . . . d A ( z n ) . exp − n V ( z i ) (1) Z n i = 1 i < j NMM is naturally tied with the study of (Bergmann) orthogonal polynomials associated with the inner product �� f ( z ) g ( z ) e − n V ( z ) d A ( z ) . � f , g � D = (2) D A. López-García (U. South Alabama) 4 / 19

� � � n � 1 | z i − z j | 2 d A ( z 1 ) . . . d A ( z n ) . exp − n V ( z i ) (1) Z n i = 1 i < j NMM is naturally tied with the study of (Bergmann) orthogonal polynomials associated with the inner product �� f ( z ) g ( z ) e − n V ( z ) d A ( z ) . � f , g � D = (2) D (1) is a determinantal point process with correlation kernel � � n − 1 � − n K n ( z , w ) = exp 2 ( V ( w ) + V ( z )) q k , n ( z ) q k , n ( w ) , k = 0 where ( q k , n ) ∞ k = 0 are the orthonormal polynomials associated to (2), i.e., q k , n ( z ) = γ k z k + · · · , � q k , n , q l , n � D = δ kl , γ k > 0 . A. López-García (U. South Alabama) 4 / 19

Global asymptotic distribution of eigenvalues V ( z ) := 1 ( | z | 2 − V ( z ) − V ( z )) , t 0 > 0 , t 0 d + 1 � t k k z k , { t k } d + 1 V ( z ) := k = 1 ⊂ C , k = 1 D : compact domain with 0 ∈ int ( D ) . A. López-García (U. South Alabama) 5 / 19

Global asymptotic distribution of eigenvalues V ( z ) := 1 ( | z | 2 − V ( z ) − V ( z )) , t 0 > 0 , t 0 d + 1 � t k k z k , { t k } d + 1 V ( z ) := k = 1 ⊂ C , k = 1 D : compact domain with 0 ∈ int ( D ) . Theorem (Elbau-Felder ’05) Under certain assumptions on V and D, for all t 0 > 0 small enough , 1 ∗ n K n ( z , z ) d A ( z ) − n →∞ λ Ω , − − → λ Ω : normalized area measure on Ω , where Ω ⊂ D is a Jordan domain with 0 ∈ int (Ω) . A. López-García (U. South Alabama) 5 / 19

Global asymptotic distribution of eigenvalues V ( z ) := 1 ( | z | 2 − V ( z ) − V ( z )) , t 0 > 0 , t 0 d + 1 � t k k z k , { t k } d + 1 V ( z ) := k = 1 ⊂ C , k = 1 D : compact domain with 0 ∈ int ( D ) . Theorem (Elbau-Felder ’05) Under certain assumptions on V and D, for all t 0 > 0 small enough , 1 ∗ n K n ( z , z ) d A ( z ) − n →∞ λ Ω , − − → λ Ω : normalized area measure on Ω , where Ω ⊂ D is a Jordan domain with 0 ∈ int (Ω) . Moreover, area (Ω) = π t 0 and � � t k , k ∈ { 1 , . . . , d + 1 } , 1 zz − k d z = 2 π i 0 , k ∈ N \ { 1 , . . . , d + 1 } . ∂ Ω A. López-García (U. South Alabama) 5 / 19

Dynamics of Ω = Ω( t 0 ) : Laplacian growth, Hele-Shaw flow (work of Wiegmann, Zabrodin, Teodorescu, Lee, Bettelheim, Elbau, Ameur, Makarov and others). Problem : Relation between the eigenvalue asymptotic distribution in the NMM and the zero asymptotic distribution of the orthogonal polynomials q n , n . Elbau, 2007: Unless V ( z ) = 0, if σ is a limiting distribution of the zeros of q n , n , then σ is determined by the Schwarz function associated with ∂ Ω . A. López-García (U. South Alabama) 6 / 19

Dynamics of Ω = Ω( t 0 ) : Laplacian growth, Hele-Shaw flow (work of Wiegmann, Zabrodin, Teodorescu, Lee, Bettelheim, Elbau, Ameur, Makarov and others). Problem : Relation between the eigenvalue asymptotic distribution in the NMM and the zero asymptotic distribution of the orthogonal polynomials q n , n . Elbau, 2007: Unless V ( z ) = 0, if σ is a limiting distribution of the zeros of q n , n , then σ is determined by the Schwarz function associated with ∂ Ω . In the case V ( z ) = t d + 1 d + 1 z d + 1 , t d + 1 > 0 , we establish a relation between the eigenvalue asymptotic distribution in the NMM and the zero asymptotic distribution of a sequence of multi-orthogonal polynomials P n , n associated with weights supported on a star-like set. The zero asymptotic distribution solves a (vector) equilibrium problem. This generalizes work of Bleher-Kuijlaars ’12 for d = 2. A. López-García (U. South Alabama) 6 / 19

The approach of Bleher-Kuijlaars to the NMM Inner product �� f ( z ) g ( z ) e − n V ( z ) d A ( z ) , � f , g � D = D V ( z ) = 1 ( | z | 2 − V ( z ) − V ( z )) . t 0 Applying Green’s formula on D , for polynomials p and q , � t 0 � p , q ′ � D − n � zp , q � D + n � p , V ′ q � D = t 0 p ( z ) q ( z ) e − n V ( z ) d z . 2 i ∂ D A. López-García (U. South Alabama) 7 / 19

The approach of Bleher-Kuijlaars to the NMM Inner product �� f ( z ) g ( z ) e − n V ( z ) d A ( z ) , � f , g � D = D V ( z ) = 1 ( | z | 2 − V ( z ) − V ( z )) . t 0 Applying Green’s formula on D , for polynomials p and q , � t 0 � p , q ′ � D − n � zp , q � D + n � p , V ′ q � D = t 0 p ( z ) q ( z ) e − n V ( z ) d z . 2 i ∂ D Bleher-Kuijlaars neglect the boundary term on the right-hand side and this leads to the study of sesquilinear forms �· , ·� satisfying the structure relation t 0 � p , q ′ � − n � zp , q � + n � p , V ′ q � = 0 . A. López-García (U. South Alabama) 7 / 19

t 0 � p , q ′ � − n � zp , q � + n � p , V ′ q � = 0 (3) Bleher-Kuijlaars conjecture : For any polynomial d + 1 � t k k z k , V ( z ) = k = 1 there is a suitable choice of a sesquilinear form �· , ·� satisfying (3) such that, for t 0 small enough, the orthogonal polynomials associated with the sesquilinear form and the Bergmann orthogonal polynomials in the NMM will have the same asymptotic behavior. A. López-García (U. South Alabama) 8 / 19

The monomial case V ( z ) = t d + 1 d + 1 z d + 1 , t d + 1 > 0 D : simply-connected, Jordan domain with 0 in its origin, invariant under � � 2 π i z �→ exp z and z �→ z . d + 1 Σ D Let Σ = { z ∈ D : z d + 1 ∈ R + } , the ( d + 1 ) -star. Green’s theorem applied on the sectors of D gives �� � � t 0 ( | z | 2 − V ( z ) − V ( z )) d A ( z ) = Q ( z ) z j e − n Q ( z ) � 2 i Q ( z ) w j , n ( z ) d z + w j , n ( z ) d z D Σ ∂ D See also Balogh-Bertola-Lee-McLaughlin ’12. A. López-García (U. South Alabama) 9 / 19