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Planar orthogonal polynomials and boundary universality in the random normal matrix model

Haakan Hedenmalm, Aron Wennman

Abstract

We show that the planar normalized orthogonal polynomials $P_{m,n}(z)$ of degree $n$ with respect to an exponentially varying planar measure $\mathrm{e}^{-2mQ}\mathrm{dA}$ enjoy an asymptotic expansion \[ P_{m,n}(z)\sim m^{\frac{1}{4}}\sqrt{φ_τ'(z)}[φ_τ(z)]^n \mathrm{e}^{m\mathcal{Q}_τ(z)}\left(\mathcal{B}_{τ, 0}(z) +m^{-1}\mathcal{B}_{τ, 1}(z)+m^{-2} \mathcal{B}_{τ,2}(z)+\ldots\right), \] as $n,m\to\infty$ while the ratio $τ=\frac{n}{m}$ is fixed. Here $\mathcal{S}_τ$ denotes the droplet, the boundary of which is assumed to be a smooth simple closed curve, and $φ_τ$ is a conformal mapping from the complement $\mathcal{S}_τ^c$ to the exterior disk $\Bbb{D}_\mathrm{e}$. The functions $\mathcal{Q}_τ$ and $\mathcal{B}_{τ, j}$ are bounded holomorphic functions which may be expressed in terms of $Q$ and $\mathcal{S}_τ$. We apply these results to obtain boundary universality in the random normal matrix model for smooth droplets, i.e., that the limiting rescaled process is the random process with correlation kernel \[ \mathrm{k}(ξ,η)= \mathrm{e}^{ξ\barη\,-\frac12(\lvertξ\rvert^2+\lvert η\rvert^2)} \,\mathrm{erf}\,(ξ+\barη). \] A key ingredient in the proof of the asymptotic expansion of the orthogonal polynomials is the construction of an orthogonal foliation -- a smooth flow of closed curves near $\partial\mathcal{S}_τ$, on each of which $P_{m,n}$ is appropriately orthogonal to lower order polynomials. To compute the coefficient functions, we develop an algorithm which determines the coefficients $\mathcal{B}_{τ, j}$ successively in terms of inhomogeneous Toeplitz kernel conditions. These inhomogeneous Toeplitz kernel conditions may be understood in terms of scalar Riemann-Hilbert problems.

Planar orthogonal polynomials and boundary universality in the random normal matrix model

Abstract

We show that the planar normalized orthogonal polynomials of degree with respect to an exponentially varying planar measure enjoy an asymptotic expansion \[ P_{m,n}(z)\sim m^{\frac{1}{4}}\sqrt{φ_τ'(z)}[φ_τ(z)]^n \mathrm{e}^{m\mathcal{Q}_τ(z)}\left(\mathcal{B}_{τ, 0}(z) +m^{-1}\mathcal{B}_{τ, 1}(z)+m^{-2} \mathcal{B}_{τ,2}(z)+\ldots\right), \] as while the ratio is fixed. Here denotes the droplet, the boundary of which is assumed to be a smooth simple closed curve, and is a conformal mapping from the complement to the exterior disk . The functions and are bounded holomorphic functions which may be expressed in terms of and . We apply these results to obtain boundary universality in the random normal matrix model for smooth droplets, i.e., that the limiting rescaled process is the random process with correlation kernel A key ingredient in the proof of the asymptotic expansion of the orthogonal polynomials is the construction of an orthogonal foliation -- a smooth flow of closed curves near , on each of which is appropriately orthogonal to lower order polynomials. To compute the coefficient functions, we develop an algorithm which determines the coefficients successively in terms of inhomogeneous Toeplitz kernel conditions. These inhomogeneous Toeplitz kernel conditions may be understood in terms of scalar Riemann-Hilbert problems.

Paper Structure

This paper contains 50 sections, 33 theorems, 428 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Orthogonal polynomials
  3. Carleman-Szego asymptotics
  4. Exponentially varying weights
  5. The boundary universality conjecture
  6. Summary of the results
  7. Sketch of the main ideas
  8. An outline of the presentation
  9. Acknowledgments
  10. Notation and conventions
  11. Preliminaries
  12. An obstacle problem and logarithmic potential theory
  13. A weighted Bernstein-Walsh lemma
  14. Weighted Laplacian growth
  15. Polynomial d-bar-methods
  16. ...and 35 more sections

Key Result

Theorem 1.5.2

Assume that $Q$ is $1$-admissible. Given a positive integer $\kappa$ there exist bounded holomorphic functions ${\mathcal{B}}_{\tau,j}$ defined in a fixed neighborhood of $\mathcal{S}_\tau^c$ such that for any positive real $A$, the asymptotic formula holds, where the error term is uniform over all $z\in{\mathbb C}$ with $\mathrm{dist}_{\mathbb C}(z,\mathcal{S}_\tau^c)\le A(m^{-1}\log m)^{\frac{1

Figures (3)

  • Figure 1.1: (left) The Berezin density $\mathrm{K}_m(z_0,z_0)^{-1}|\mathrm{K}_m(z,z_0)|^2$ with $Q(z)=\frac{1}{2}|z|^2$ for the boundary point $z_0=1$ and $m=30$. (right) The orthogonal polynomial density $\lvert P_{m,n}(z)\rvert^2 \mathrm e^{-2mQ(z)}$ for $n=25, m=20$ and $Q(z)=\tfrac{1}{2}\vert z\vert^2-\operatorname{Re}(t z^2)$, where $t=0.2$.
  • Figure 1.2: The RNM process associated to a quadratic potential (The Ginibre ensemble) with blow-up at a boundary point (courtesy of Nam-Gyu Kang).
  • Figure 1.3: Laplacian growth of the compacts $\mathcal{S}_{\tau}$ for the potential $Q(z)=\frac{1}{2}|z|^2-2^{-\frac{1}{2}}\log|z+{\mathrm i}|$ (boundary curves indicated).

Theorems & Definitions (82)

  • Conjecture 1.4.1: boundary universality
  • Definition 1.5.1
  • Theorem 1.5.2
  • Remark 1.5.3
  • Theorem 1.5.4
  • Remark 1.5.5
  • Corollary 1.5.6
  • Remark 1.5.7
  • Definition 2.1.1
  • Lemma 2.2.1
  • ...and 72 more