[Paper Review] Planar orthogonal polynomials and boundary universality in the random normal matrix model

Håakan Hedenmalm, Aron Wennman|arXiv (Cornell University)|Oct 17, 2017

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes an asymptotic expansion for planar orthogonal polynomials in the random normal matrix model with exponentially varying weights, using a novel orthogonal foliation flow and Riemann-Hilbert methods. The key result is boundary universality, showing the limiting correlation kernel converges to the error function kernel, confirming a long-standing conjecture for smooth droplets.

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 \[ P_{m,n}(z)\sim m^{\frac{1}{4}}\sqrt{ϕ_τ'(z)}[ϕ_τ(z)]^n \mathrm{e}^{m\mathcal{Q}_τ(z)}\left(\mathcal{B}_{τ, 0}(z) +m^{-1}\mathcal{B}_{τ, 1}(z)+m^{-2} \mathcal{B}_{τ,2}(z)+\ldots ight), \] as $n,m o\infty$ while the ratio $τ=\frac{n}{m}$ is fixed. Here $\mathcal{S}_τ$ denotes the droplet, the boundary of which is assumed to be a smooth simple closed curve, and $ϕ_τ$ is a conformal mapping from the complement $\mathcal{S}_τ^c$ to the exterior disk $\Bbb{D}_\mathrm{e}$. The functions $\mathcal{Q}_τ$ and $\mathcal{B}_{τ, j}$ are bounded holomorphic functions which may be expressed in terms of $Q$ and $\mathcal{S}_τ$. 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 \[ \mathrm{k}(ξ,η)= \mathrm{e}^{ξ\barη\,-\frac12(\lvertξ vert^2+\lvert η vert^2)} \,\mathrm{erf}\,(ξ+\barη). \] 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 $\partial\mathcal{S}_τ$, 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 $\mathcal{B}_{τ, j}$ successively in terms of inhomogeneous Toeplitz kernel conditions. These inhomogeneous Toeplitz kernel conditions may be understood in terms of scalar Riemann-Hilbert problems.

Motivation & Objective

• To derive a complete asymptotic expansion for planar orthogonal polynomials $P_{m,n}(z)$ with respect to the measure $e^{-2mQ}\,dA$ as $n,m \to \infty$ with $\tau = n/m$ fixed.
• To establish the existence of an orthogonal foliation flow—a smooth family of closed curves near the droplet boundary—on which the polynomials remain approximately orthogonal to lower-degree polynomials.
• To resolve the coefficient functions $\mathcal{B}_{\tau,j}$ in the asymptotic expansion via an algorithmic solution to inhomogeneous Toeplitz kernel conditions.
• To prove boundary universality in the random normal matrix model, showing the limiting correlation kernel converges to the error function kernel $\mathrm{erf}(\xi + \bar{\eta})$.
• To connect the matrix $\bar{\partial}$-problem of Its and Takhtajan to the orthogonal foliation framework, enabling the construction of solutions via integration over curve families.

Proposed method

• Derive an asymptotic expansion for $P_{m,n}(z)$ in powers of $m^{-1}$, involving conformal mappings $\phi_\tau$ from the exterior of the droplet $\mathcal{S}_\tau^c$ to the exterior unit disk $\mathbb{D}_e$, and holomorphic functions $\mathcal{Q}_\tau$, $\mathcal{B}_{\tau,j}$.
• Construct an orthogonal foliation flow: a smooth family of closed curves $\Gamma_t$ near $\partial\mathcal{S}_\tau$, where $P_{m,n}$ is approximately orthogonal to $\operatorname{Pol}_k$ for $k < n$, using a master equation derived from the $\bar{\partial}$-problem.
• Use a quasipolynomial ansatz and $L^2$-orthogonality to derive an $L^2$-expansion, then polynomialize to recover the full asymptotic expansion.
• Develop an algorithm to compute the coefficients $\mathcal{B}_{\tau,j}$ successively by solving inhomogeneous Toeplitz kernel conditions, interpreted as scalar Riemann-Hilbert problems.
• Apply steepest descent analysis and weighted Laplacian growth techniques to control the behavior of the weight $e^{-2mQ}$ and the density of the measure.
• Integrate solutions of Riemann-Hilbert problems over the curve family $\{\Gamma_t\}$ to construct a solution to the matrix $\bar{\partial}$-problem, linking to the Its-Takhtajan framework.

Experimental results

Research questions

• RQ1 How do planar orthogonal polynomials $P_{m,n}(z)$ behave asymptotically as $n,m \to \infty$ with $\tau = n/m$ fixed, under an exponentially varying weight $e^{-2mQ}\,dA$?
• RQ2 Can an orthogonal foliation flow be constructed near the boundary of the droplet $\mathcal{S}_\tau$, such that $P_{m,n}$ remains approximately orthogonal to lower-degree polynomials along each loop of the flow?
• RQ3 What is the structure of the coefficient functions $\mathcal{B}_{\tau,j}$ in the asymptotic expansion of $P_{m,n}(z)$, and can they be computed algorithmically?
• RQ4 Does the limiting correlation kernel in the random normal matrix model converge to the error function kernel $\mathrm{erf}(\xi + \bar{\eta})$ at the boundary, confirming boundary universality?
• RQ5 How can the matrix $\bar{\partial}$-problem of Its and Takhtajan be adapted to construct solutions via integration over a family of curves, and how does this relate to the orthogonal foliation?

Key findings

• The planar orthogonal polynomial $P_{m,n}(z)$ admits an asymptotic expansion of the form $P_{m,n}(z) \sim m^{1/4}\sqrt{\phi_\tau'(z)}[\phi_\tau(z)]^n e^{m\mathcal{Q}_\tau(z)}\left(\mathcal{B}_{\tau,0}(z) + m^{-1}\mathcal{B}_{\tau,1}(z) + \cdots\right)$ as $n,m \to \infty$ with $\tau = n/m$ fixed.
• The orthogonal foliation flow exists as a smooth family of closed curves $\Gamma_t$ near $\partial\mathcal{S}_\tau$, constructed via the implicit function theorem and a master equation derived from the $\bar{\partial}$-problem.
• The coefficient functions $\mathcal{B}_{\tau,j}$ are determined algorithmically through successive solutions of inhomogeneous Toeplitz kernel conditions, which are equivalent to scalar Riemann-Hilbert problems.
• Boundary universality is established: the limiting rescaled correlation kernel converges to $\mathrm{k}(\xi,\eta) = e^{\xi\bar{\eta} - \frac{1}{2}(|\xi|^2 + |\eta|^2)}\mathrm{erf}(\xi + \bar{\eta})$, confirming the conjecture for smooth droplets.
• The matrix $\bar{\partial}$-problem of Its and Takhtajan is connected to the orthogonal foliation via integration over the curve family $\{\Gamma_t\}$, yielding a solution to the $\bar{\partial}$-problem with the correct asymptotics.
• The algorithmic resolution of the master equation for the orthogonal foliation flow is achieved via multivariate Faà di Bruno formulas and Taylor expansions of the weight and density terms, ensuring polynomial complexity.