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

Abstract

AbstractWe consider the orthogonal polynomial pn(z) with respect to the planar measure supported on the whole complex plane where dA is the Lebesgue measure of the plane, N is a positive constant, {c1, …, cν} are nonzero real numbers greater than −1 and are distinct points inside the unit disk. In the scaling limit when n/N = 1 and n → ∞ we obtain the strong asymptotics of the polynomial pn(z). We show that the support of the roots converges to what we call the “multiple Szegő curve,” a certain connected curve having ν + 1 components in its complement. We apply the nonlinear steepest descent method [9,10] on the matrix Riemann‐Hilbert problem of size (ν + 1) × (ν + 1) posed in [22]. © 2023 Wiley Periodicals, LLC.