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

TL;DR

This paper investigates the asymptotic zero distribution of planar orthogonal polynomials with a perturbed Gaussian weight, revealing a discontinuity at a specific parameter value and providing a smooth interpolation via a secondary scaling.

Contribution

It uncovers a discontinuous change in the zero distribution and potential-theoretic skeleton of orthogonal polynomials under weight perturbation, with a new scaling approach for smooth transition.

Findings

Discontinuity in zero distribution at c=0

Support characterized by potential-theoretic skeleton

Smooth interpolation achieved through secondary scaling

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 $\eta\in[0,\infty).$