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Documents  33C45 | enregistrements trouvés : 4

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I will discuss polynomials $P_{N}$ of degree $N$ that satisfy non-Hermitian orthogonality conditions with respect to the weight $\frac{\left ( z+1 \right )^{N}\left ( z+a \right )^{N}}{z^{2N}}$ on a contour in the complex plane going around 0. These polynomials reduce to Jacobi polynomials in case a = 1 and then their zeros cluster along an open arc on the unit circle as the degree tends to infinity.
For general a, the polynomials are analyzed by a Riemann-Hilbert problem. It follows that the zeros exhibit an interesting transition for the value of a = 1/9, when the open arc closes to form a closed curve with a density that vanishes quadratically. The transition is described by a Painlevé II transcendent.
The polynomials arise in a lozenge tiling problem of a hexagon with a periodic weighting. The transition in the behavior of zeros corresponds to a tacnode in the tiling problem.
This is joint work in progress with Christophe Charlier, Maurice Duits and Jonatan Lenells and we use ideas that were developed in [2] for matrix valued orthogonal polynomials in connection with a domino tiling problem for the Aztec diamond.
I will discuss polynomials $P_{N}$ of degree $N$ that satisfy non-Hermitian orthogonality conditions with respect to the weight $\frac{\left ( z+1 \right )^{N}\left ( z+a \right )^{N}}{z^{2N}}$ on a contour in the complex plane going around 0. These polynomials reduce to Jacobi polynomials in case a = 1 and then their zeros cluster along an open arc on the unit circle as the degree tends to infinity.
For general a, the polynomials are analyzed ...

05B45 ; 52C20 ; 33C45 ; 60B20

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We establish a new connection between moments of n×n random matrices $X_{n}$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s\in\mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order n→∞ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials. This is work in collaboration with Fabio Cunden, Neil O' Connell and Nick Simm.
We establish a new connection between moments of n×n random matrices $X_{n}$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s\in\mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and ...

15B52 ; 05E05 ; 33C45

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Multi angle  Le diamant aztèque - Cours 2
Corteel, Sylvie (Auteur de la Conférence) | CIRM (Editeur )

Le but du mini-cours sera de faire un cours introductif à différentes méthodes énumeratives à travers l’exemple des pavages par dominos du diamant aztèque. On essaiera de voir les fonctions (super)-symétriques, les moments de polynômes bi-orthogonaux, les évaluations de determinants, les algorithmes de génération...

05A15 ; 33C45

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Multi angle  Le diamant aztèque - Cours 1
Corteel, Sylvie (Auteur de la Conférence) | CIRM (Editeur )

Le but du mini-cours sera de faire un cours introductif à différentes méthodes énumeratives à travers l’exemple des pavages par dominos du diamant aztèque. On essaiera de voir les fonctions (super)-symétriques, les moments de polynômes bi-orthogonaux, les évaluations de determinants, les algorithmes de génération...

05A15 ; 33C45

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