Abstract
We consider the orthogonal polynomials, \({\{P_n(z)\}_{n=0,1,\ldots}}\), with respect to the measure
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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Lee, SY., Yang, M. Discontinuity in the Asymptotic Behavior of Planar Orthogonal Polynomials Under a Perturbation of the Gaussian Weight. Commun. Math. Phys. 355, 303–338 (2017). https://doi.org/10.1007/s00220-017-2888-8
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DOI: https://doi.org/10.1007/s00220-017-2888-8