Orthogonal polynomials in the normal matrix model with two insertions

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Abstract
We consider orthogonal polynomials with respect to the weight | z 2 + a 2 | c N e − N | z | 2 in the whole complex plane. We obtain strong asymptotics and the limiting normalised zero counting measure (mother body) of the orthogonal polynomials of degree n in the scaling limit n , N → ∞ such that n N → t . We restrict ourselves to the case a 2 ⩾ 2 c , cN integer, and t < t ∗ where t ∗ is a constant depending only on a , c . Due to this restriction, the mother body is supported on an interval. We also find the two dimensional equilibrium measure (droplet) associated with the eigenvalues in the corresponding normal matrix model. Our method relies on the recent result that the planar orthogonal polynomials are a part of a vector of type I multiple orthogonal polynomials, and this enables us to apply the steepest descent method to the associated Riemann–Hilbert problem.
Funding Information
  • International Research Training Group, KU Leuven and University of Melbourne
  • Methusalem Grant of the Flemish Government (METH/21/03)
  • Australian Research Council Discovery Project Grant (DP210102887)
  • Melbourne Research Scholarship, University of Melbourne

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