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Seminar
FIELD Mathematics
DATE November 12 (Thu), 2020
TIME 17:00-18:00
PLACE Online
SPEAKER Yang, Meng
HOST Seo, Seong-Mi
INSTITUTE University of Copenhagen
TITLE Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Point Charges
ABSTRACT We consider the planar orthogonal polynomials $\{p_n(z)\}$ with respect to the measure supported on the complex plane
$$e^{-N|z|^2} \prod_{j=1}^{\nu} |z - a_j |^{2 c_j} dA(z) $$
where $dA$ is the Lebesgue measure of the plane, $N$ is a positive constant, $\{c_1, \cdots , c_\nu\}$ are nonzero real numbers greater than $-1$ and $\{a_1, \cdots , a_\nu \} \subset \mathbb{D} \setminus \{0\}$ are distinct points inside the unit disk. The orthogonal polynomials are related to the interacting Coulomb particles with charge $+1$ for each, in the presence of extra particles with charge $+c_j$ at $a_j$. For fixed $c_j$, these can be considered as small perturbations of the Gaussian weight. When $\nu = 1$, in the scaling limit $n/N = 1$ and $n \to \infty$, we obtain strong asymptotics of $p_n(z)$ via a matrix Riemann?Hilbert problem. From the asymptotic behavior of $p_n(z)$, we find that, as we vary $c_1$, the limiting distribution of zeros behaves discontinuously at $c_1 = 0$. We observe that the generalized Szeg\H{o} curve (a kind of potential theoretic skeleton) also behaves discontinuously at $c_1 = 0$. We also derive the strong asymptotics of $p_n(z)$ for the case of $\nu > 1$ by applying the nonlinear steepest descent method on the matrix Riemann-Hilbert problem of size $(\nu + 1) \times (\nu + 1)$ This talk is based on joint work with Seung-Yeop Lee.
FILE 484581604322359893_1.pdf
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