Discontinuity in the Asymptotic Behavior of Planar Orthogonal Polynomials Under a Perturbation of the Gaussian Weight

Abstract

We consider the orthogonal polynomials, \({\{P_n(z)\}_{n=0,1,\ldots}}\) , 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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