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

We consider the orthogonal polynomials, $\{P_n(z){\}}_{n=0,1,\cdots }$, with respect to the measure $$\mathrm{\mid }z-a{\mathrm{\mid }}^{2c}{e}^{-N\mathrm{\mid }z{\mathrm{\mid }}^2}dA(z)$$ 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, $n\mathrm{/}N$, 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 $\eta \in [0,\mathrm{\infty })\mathrm{.}$