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

Abstract

We prove that the planar normalized orthogonal polynomials $P_{n,m}(z)$ of degree $n$ with respect to an exponentially varying planar measure $e^{-2mQ(z)}\mathrm{dA}(z)$ enjoy an asymptotic expansion \[ P_{n,m}(z)\sim m^{\frac{1}{4}}\sqrt{\phi_\tau'(z)}[\phi_\tau(z)]^n e^{m\mathcal{Q}_\tau(z)}\left(\mathcal{B}_{0,\tau}(z) +\frac{1}{m}\mathcal{B}_{1,\tau}(z)+\frac{1}{m^2} \mathcal{B}_{2,\tau}(z)+\ldots\right), \] as $n=m\tau\to\infty$. 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 $\mathcal{S}_\tau^c\to\mathbb{D}_e$. The functions $\mathcal{Q}_\tau$ and $\mathcal{B}_{j,\tau}(z)$ are bounded holomorphic functions which may be computed in terms of $Q$ and $\mathcal{S}_\tau$. We apply these results to prove universality at the boundary for regular droplets in the random normal matrix model, i.e., that the limiting rescaled process is the random process with correlation kernel $$ \mathrm{k}(\xi,\eta)= e^{\xi\bar\eta\,-\frac12(\lvert\xi\rvert^2+\lvert \eta\rvert^2)} \operatorname{erf}(\xi+\bar{\eta}). $$ A key ingredient in the proof of the asymptotic expansion is the construction of an {orthogonal foliation} -- a smooth flow of closed curves near $\partial\mathcal{S}_\tau$, on each of which $P_{n,m}$ is orthogonal to lower order polynomials, with respect to an induced measure. To compute the coefficients, we develop an algorithm which determines $\mathcal{B}_{j,\tau}$ up to any desired order in terms of inhomogeneous Toeplitz kernel conditions.