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

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 $\tau =\frac{n}{m}$ is fixed. Here ${\mathcal{S}}_{\tau }$ denotes the droplet, the boundary of which is assumed to be a smooth simple closed curve, and ${\varphi }_{\tau }$ is a conformal mapping from the complement ${\mathcal{S}}_{\tau }^c$ to the exterior disk ${\mathbb{D}}_{\mathrm{e}}$. The functions ${\mathcal{Q}}_{\tau }$ and ${\mathcal{B}}_{\tau ,j}$ are bounded holomorphic functions which may be expressed in terms of $Q$ and ${\mathcal{S}}_{\tau }$. 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 $\partial {\mathcal{S}}_{\tau }$, 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 ${\mathcal{B}}_{\tau ,j}$ successively in terms of inhomogeneous Toeplitz kernel conditions. These inhomogeneous Toeplitz kernel conditions may be understood in terms of scalar Riemann-Hilbert problems.

67 pages. 3 figures. Current version: Restructured presentation, added references, and corrected typos

Cited by in corpus (4)

• A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
• The high temperature crossover for general 2D Coulomb gases
• Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
• Products of random matrices from fixed trace and induced Ginibre ensembles