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A direct approach to soft and hard edge universality for random normal matrices

Joakim Cronvall, Aron Wennman

TL;DR

The paper proves universal local eigenvalue statistics for random normal matrices (planar Coulomb gases with $\beta=2$) across hard edges, soft edges, and soft/hard-edge regimes without symmetry assumptions. It develops a direct, polynomial-free approach using Paley-Wiener spectral embeddings and boundary-peaking weighted polynomials, bypassing orthogonal polynomial methods. Hard edges yield the universal kernel $B(\zeta,\eta)=\int_0^1 t e^{-t(\zeta+\bar{\eta})}dt$, while soft edges produce the Faddeeva-type kernel $H(\zeta,\eta)=\Phi(\zeta+\bar{\eta})e^{\zeta\bar{\eta}}$, and soft/hard edges yield $H_{\mathbb{L}}(\zeta,\eta)=e^{\zeta\bar{\eta}}\Psi(\zeta+\bar{\eta})$ on the left half-plane. The authors employ spectral embeddings, approximate peak kernels, and polynomial corrections to establish tight upper and lower bounds and show convergence to the respective universal kernels, with a unified treatment applicable to droplets with multiple components. These results extend edge universality beyond soft, single-component droplets to hard edges and multi-component interfaces, providing explicit universal kernels and a conceptual Hilbert-space perspective. The work has implications for the analysis of planar Coulomb gases and non-Hermitian random matrix models by supplying a robust, direct framework for edge universality.

Abstract

We develop a unified approach to universality of local scaling limits for eigenvalues of random normal matrices, or equivalently for planar Coulomb gases at inverse temperature $β=2$. The approach is direct in that it does not rely on expressing the kernels in terms of orthogonal polynomials. There are three main results. The first is a proof of universality at hard edges with no symmetry assumptions on either the potential or the hard edge. We also prove local universality at regular soft edges for droplets with several components, and lastly for soft/hard edges where a hard edge perfectly aligns with the droplet boundary. The main ingredients are Paley-Wiener type spectral embeddings for the Hilbert space associated with a limiting kernel, and the construction of weighted polynomials peaking near a given boundary point.

A direct approach to soft and hard edge universality for random normal matrices

TL;DR

The paper proves universal local eigenvalue statistics for random normal matrices (planar Coulomb gases with ) across hard edges, soft edges, and soft/hard-edge regimes without symmetry assumptions. It develops a direct, polynomial-free approach using Paley-Wiener spectral embeddings and boundary-peaking weighted polynomials, bypassing orthogonal polynomial methods. Hard edges yield the universal kernel , while soft edges produce the Faddeeva-type kernel , and soft/hard edges yield on the left half-plane. The authors employ spectral embeddings, approximate peak kernels, and polynomial corrections to establish tight upper and lower bounds and show convergence to the respective universal kernels, with a unified treatment applicable to droplets with multiple components. These results extend edge universality beyond soft, single-component droplets to hard edges and multi-component interfaces, providing explicit universal kernels and a conceptual Hilbert-space perspective. The work has implications for the analysis of planar Coulomb gases and non-Hermitian random matrix models by supplying a robust, direct framework for edge universality.

Abstract

We develop a unified approach to universality of local scaling limits for eigenvalues of random normal matrices, or equivalently for planar Coulomb gases at inverse temperature . The approach is direct in that it does not rely on expressing the kernels in terms of orthogonal polynomials. There are three main results. The first is a proof of universality at hard edges with no symmetry assumptions on either the potential or the hard edge. We also prove local universality at regular soft edges for droplets with several components, and lastly for soft/hard edges where a hard edge perfectly aligns with the droplet boundary. The main ingredients are Paley-Wiener type spectral embeddings for the Hilbert space associated with a limiting kernel, and the construction of weighted polynomials peaking near a given boundary point.

Paper Structure

This paper contains 15 sections, 27 theorems, 229 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Interface transitions in the random normal matrix model
  3. Main results
  4. Outline of the main idea
  5. Hard edges
  6. Limiting kernels and the upper bound
  7. Approximate kernel and the lower bound
  8. Correction to polynomial
  9. Proof of Theorem \ref{['thm:main-hard-edge']}
  10. Soft and Soft/hard edges
  11. Limiting kernel at soft edges
  12. Limiting kernel soft/hard edge
  13. Approximate kernel and the lower bound
  14. Correction to polynomial
  15. Proof of Theorem \ref{['thm:main-soft-edge']} and Theorem \ref{['thm:main-soft-hard-edge']}

Key Result

Theorem A

Assume that $\partial\Omega$ is a real-analytic Jordan curve, compactly contained in $\operatorname{Int} (S)$ and let $z_0\in\partial\Omega$. Then there exists a cocycle $c(\zeta,\eta)$ such that, locally uniformly for $\zeta,\eta\in\mathbb{C}$, where $B$ is the Hermitian-entire kernel

Figures (1)

  • Figure 1: Illustration of the Ginibre ensemble ($n=2000$) with a hard edge along an interior "hole" $\Omega$. The outer edge is a regular soft edge.

Theorems & Definitions (45)

  • Theorem A: Hard edges
  • Theorem B: Soft edges
  • Theorem C: Soft/hard edges
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • proof
  • ...and 35 more