Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges

Abstract

We consider the planar orthogonal polynomial $p_{n}(z)$ with respect to the measure supported on the whole complex plane $${\rm e}^{-N|z|^2} \prod_{j=1}^ν|z-a_j|^{2c_j}\,{\rm d} A(z)$$ where ${\rm d} A$ is the Lebesgue measure of the plane, $N$ is a positive constant, $\{c_1,\cdots,c_ν\}$ are nonzero real numbers greater than $-1$ and $\{a_1,\cdots,a_ν\}\subset{\mathbb D}\setminus\{0\}$ are distinct points inside the unit disk. In the scaling limit when $n/N = 1$ and $n\to \infty$ we obtain the strong asymptotics of the polynomial $p_n(z)$. We show that the support of the roots converges to what we call the "multiple Szego curve," a certain connected curve having $ν+1$ components in its complement. We apply the nonlinear steepest descent method on the matrix Riemann-Hilbert problem of size $(ν+1)\times(ν+1)$.