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We show that the planar normalized orthogonal polynomials $P_{m,n}(z)$ ofdegree $n$ with respect to an exponentially varying planar measure$\mathrm{e}^{-2mQ}\mathrm{dA}$ enjoy an asymptotic expansion \[ P_{m,n}(z)\simm^{\frac{1}{4}}\sqrt{\phi_\tau'(z)}[\phi_\tau(z)]^n\mathrm{e}^{m\mathcal{Q}_\tau(z)}\left(\mathcal{B}_{\tau, 0}(z)+m^{-1}\mathcal{B}_{\tau, 1}(z)+m^{-2} \mathcal{B}_{\tau,2}(z)+\ldots\right),\] 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 bea smooth simple closed curve, and $\phi_\tau$ is a conformal mapping from thecomplement $\mathcal{S}_\tau^c$ to the exterior disk $\Bbb{D}_\mathrm{e}$. Thefunctions $\mathcal{Q}_\tau$ and $\mathcal{B}_{\tau, j}$ are boundedholomorphic functions which may be expressed in terms of $Q$ and$\mathcal{S}_\tau$. We apply these results to obtain boundary universality inthe random normal matrix model for smooth droplets, i.e., that the limitingrescaled 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 theasymptotic expansion of the orthogonal polynomials is the construction of anorthogonal foliation -- a smooth flow of closed curves near$\partial\mathcal{S}_\tau$, on each of which $P_{m,n}$ is appropriatelyorthogonal to lower order polynomials. To compute the coefficient functions, wedevelop an algorithm which determines the coefficients $\mathcal{B}_{\tau, j}$successively in terms of inhomogeneous Toeplitz kernel conditions. Theseinhomogeneous Toeplitz kernel conditions may be understood in terms of scalarRiemann-Hilbert problems.

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