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