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 \[ P_{m,n}(z)\sim m^{\frac{1}{4}}\sqrt{\phi_\tau'(z)}[\phi_\tau(z)]^n \mathrm{e}^{m\mathcal{Q}_\tau(z)}\left(\mathcal{B}_{\tau, 0}(z) +m^{-1}\mathcal{B}_{\tau, 1}(z)+m^{-2} \mathcal{B}_{\tau,2}(z)+\ldots\right), \] 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.

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