Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
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We consider the orthogonal polynomial pn(z) with respect to the planar measure supported on the whole complex plane (Formula presented.) where dA is the Lebesgue measure of the plane, N is a positive constant, {c1, …, cν} are nonzero real numbers greater than −1 and (Formula presented.) 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].
Original language | English |
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Journal | Communications on Pure and Applied Mathematics |
Volume | 76 |
Issue number | 10 |
Pages (from-to) | 2888-2956 |
ISSN | 0010-3640 |
DOIs | |
Publication status | Published - 2023 |
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