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Orthogonal polynomials in the spherical ensemble with two insertions

Sung-Soo Byun, Peter J. Forrester, Arno B. J. Kuijlaars, Sampad Lahiry

TL;DR

This work analyzes the large-$N$ asymptotics of planar orthogonal polynomials $P_{n,N}$ with a two-insertion weight on the complex plane, mapping the problem to a pre-critical 2D Coulomb gas. The authors develop a mother body framework via a spherical Schwarz function and a spectral curve, and show that planar orthogonality can be recast as non-Hermitian contour orthogonality, enabling a Deift-Zhou RH analysis. They derive strong asymptotics for $P_{n,N}$, determine the limiting zero counting measure supported on a mother-body contour $\Gamma_0$, and obtain explicit large-$N$ norms $h_{n,N}$; the results coherently tie to the underlying electrostatic problem through a precise relation between the 2D Robin constants. The methods yield rigorous asymptotics in the pre-critical phase, with a detailed contour/parametrix construction around branch points and a fully explicit spectral data description, providing a benchmark for universality in non-Hermitian planar ensembles.

Abstract

We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n, N\rightarrow\infty$ and $N-n$ fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted $L^2$ norm and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift-Zhou steepest descent analysis of the associated $2\times 2$ Riemann-Hilbert problem.

Orthogonal polynomials in the spherical ensemble with two insertions

TL;DR

This work analyzes the large- asymptotics of planar orthogonal polynomials with a two-insertion weight on the complex plane, mapping the problem to a pre-critical 2D Coulomb gas. The authors develop a mother body framework via a spherical Schwarz function and a spectral curve, and show that planar orthogonality can be recast as non-Hermitian contour orthogonality, enabling a Deift-Zhou RH analysis. They derive strong asymptotics for , determine the limiting zero counting measure supported on a mother-body contour , and obtain explicit large- norms ; the results coherently tie to the underlying electrostatic problem through a precise relation between the 2D Robin constants. The methods yield rigorous asymptotics in the pre-critical phase, with a detailed contour/parametrix construction around branch points and a fully explicit spectral data description, providing a benchmark for universality in non-Hermitian planar ensembles.

Abstract

We consider asymptotics of planar orthogonal polynomials (where ) with respect to the weight in the whole complex plane. With and fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted norm and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift-Zhou steepest descent analysis of the associated Riemann-Hilbert problem.

Paper Structure

This paper contains 21 sections, 17 theorems, 211 equations, 9 figures.

Table of Contents

  1. Introduction and statement of results
  2. Underlying Coulomb gas, random matrix ensemble and relation to planar orthogonal polynomials
  3. Statement of results
  4. Overview of the rest of the paper
  5. Construction of a meromorphic function on a Riemann surface
  6. Spherical Schwarz function
  7. Spectral curve
  8. The mother body
  9. Analysis of the trajectories of the quadratic differential
  10. Potential theoretic preliminaries
  11. Proof of Theorem \ref{['thmmoth']}
  12. Contour orthogonality
  13. Riemann-Hilbert analysis
  14. Some preliminaries
  15. Opening the lenses
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose $w> w_{\rm cri}$ (see def of w critical point). There exists a Borel probability measure $\mu_0$, supported on a curve $\Gamma_0$, which lies in the interior of $\Omega$ and intersects the real line between $(-1/w,0)$ with the following properties:

Figures (9)

  • Figure 1: Evolution of the droplet (shaded blue) and the mother body (dashed red) as $w$ increases. The figures correspond to $Q_0=Q_1=1$ and $w=0.5 , 1, 2$ (from left to right).
  • Figure 2: The plot of the function $F_1$ in red and its analytic continuation $F_2$ in blue on the real line. One observes $F_2$ has a pole at $0$.
  • Figure 3: The Riemann surface $\mathcal{R}$
  • Figure 4: The Schwarz function $S_1$ and $S_2$ on the real axis. One can observe three poles, one zero and a node at the real axis.
  • Figure 5: Possible topological configuration of critical trajectories. Case I in left, Case II (in box) in right which satisfies the inequality in Proposition \ref{['traj1']}.
  • ...and 4 more figures

Theorems & Definitions (50)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 40 more