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

TL;DR

This paper derives asymptotic expansions for planar orthogonal polynomials in the random normal matrix model and establishes boundary universality, showing the local process converges to a specific universal kernel near smooth droplet boundaries.

Contribution

It provides a detailed asymptotic expansion for orthogonal polynomials and introduces an algorithm to compute coefficient functions, advancing understanding of boundary universality in random matrix theory.

Findings

Asymptotic expansion of orthogonal polynomials as n,m→∞ with fixed ratio

Boundary universality result with a specific limiting correlation kernel

Development of an algorithm for coefficient computation via Riemann-Hilbert problems

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 \[ P_{m,n}(z)\sim m^{\frac{1}{4}}\sqrt{\phi_\tau'(z)}[\phi_\tau(z)]^n \mathrm{e}^{m\mathcal{Q}_\tau(z)}\left(\mathcal{B}_{\tau, 0}(z) +m^{-1}\mathcal{B}_{\tau, 1}(z)+m^{-2} \mathcal{B}_{\tau,2}(z)+\ldots\right), \] 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 $\phi_\tau$ is a conformal mapping from the complement $\mathcal{S}_\tau^c$ to the exterior disk $\Bbb{D}_\mathrm{e}$. 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 \[ \mathrm{k}(\xi,\eta)= \mathrm{e}^{\xi\bar\eta\,-\frac12(\lvert\xi\rvert^2+\lvert \eta\rvert^2)} \,\mathrm{erf}\,(\xi+\bar{\eta}). \] 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.