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

2025/11/23 by Cronvall, Joakim, Wennman, Aron
Mathematics · #30E15 #41A10 #47B32 #60B20 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2511.18628

openalex publication_date 2025/11/23 · openalex created_date 2025/11/27 · openalex updated_date 2026/07/28

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.

Citations

• On the spectral edge of non-Hermitian random matrices
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• On the Distribution of the Roots of Certain Symmetric Matrices

Cited by

• Universality for fluctuations of counting statistics of random normal matrices

Related

• The Edge Universality of Correlated Matrices
• Universality for random matrices with an edge spectrum singularity
• Non-Hermitian spectral universality at critical points
• Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge
• Cusp Universality for Correlated Random Matrices