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arXiv · 1607.02821

Discontinuity in the asymptotic behavior of planar orthogonal polynomials under a perturbation of the Gaussian weight

Seung-Yeop Lee·Meng Yang
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Abstract

We consider the orthogonal polynomials, $\{P_n(z)\}_{n=0,1,\cdots}$, with respect to the measure $$|z-a|^{2c} e^{-N|z|^2}dA(z)$$ supported over the whole complex plane, where $a>0$, $N>0$ and $c>-1$. We look at the scaling limit where $n$ and $N$ tend to infinity while keeping their ratio, $n/N$, fixed. The support of the limiting zero distribution is given in terms of certain "limiting potential-theoretic skeleton" of the unit disk. We show that, as we vary $c$, both the skeleton and the asymptotic distribution of the zeros behave discontinuously at $c=0$. The smooth interpolation of the discontinuity is obtained by the further scaling of $c=e^{-\eta N}$ in terms of the parameter $\eta\in[0,\infty).$

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Seung-Yeop Lee, Meng Yang. 2016-07-11. Discontinuity in the asymptotic behavior of planar orthogonal polynomials under a perturbation of the Gaussian weight. https://doi.org/10.1007/s00220-017-2888-8

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